%					
%	dja.tex	
%	Daniel Allcock   (allcock@math.utah.edu)
%	17 December 1996			
%						
%	A simple set of macros for plain tex
%	documentation is in dja.doc, a text file
%
% suppresses duplicate inputs of this file:
\ifx\djatexLoaded\relax\endinput\else\let\djatexLoaded=\relax\fi

% FONTS
%
% calligraphic font
\newfam\calfont
\font\tencal=eusm10 \font\sevencal=eusm7 \font\fivecal=eusm5
\textfont\calfont=\tencal 
\scriptfont\calfont=\sevencal 
\scriptscriptfont\calfont=\fivecal
\def\cal{\fam=\calfont}	
% fraktur
\newfam\frakturfont
\font\tenfrak=eufm10 \font\sevenfrak=eufm7 \font\fivefrak=eufm5
\textfont\frakturfont=\tenfrak 
\scriptfont\frakturfont=\sevenfrak 
\scriptscriptfont\frakturfont=\fivefrak
\def\frak{\fam=\frakturfont}
% blackboard bold (includes many math symbols)
\newfam\bbbfont
\font\tenbbb=msbm10 \font\sevenbbb=msbm7 \font\fivebbb=msbm5	
\textfont\bbbfont=\tenbbb 
\scriptfont\bbbfont=\sevenbbb 
\scriptscriptfont\bbbfont=\fivebbb
\def\bbb{\fam=\bbbfont}
% script and script-script fonts for typewriter font (all in
% 10pt for portability)
\scriptfont\ttfam=\tentt
\scriptscriptfont\ttfam=\tentt

% PROGRAMMING STUFF
% 
% `if not defined' :
\long\def\ifndef#1{\expandafter\ifx\csname#1\endcsname\relax}
% sometimes more convenient, but you can't say
% ``\ifdef...\else...\fi''
\long\def\ifdef#1{\ifndef{#1}\else}
% My warning messages
\def\mywarning#1{\ifnum\warningsoff=0
	\immediate\write16{l.\the\inputlineno: #1}\fi}
\ifndef{warningsoff}\def\warningsoff{0}\fi
\def\givenowarnings{\def\warningsoff{1}}
\def\givewarnings{\def\warningsoff{0}}

% FRONT MATTER
%
% note: \author, \title and \note get redefined when
% \bibliography is called.
\long\def\author#1{\def\Zauthor{#1}}
\long\def\title#1{\def\Ztitle{#1}}
\long\def\address#1{\def\Zaddress{#1}}
\long\def\email#1{\def\Zemail{#1}}
\long\def\homepage#1{\def\Zhomepage{#1}}
\long\def\date#1{\def\Zdate{#1}}
\long\def\subject#1{\def\Zsubject{#1}}
\long\def\note#1{\def\Znote{#1}}
\def\plaintitlepage{{\parindent=0pt
	\ifdef{Ztitle}\par{\bf\Ztitle}\medskip\fi
	\ifdef{Zauthor}\par{\Zauthor}\fi
	\ifdef{Zdate}\par{\Zdate}\fi
	\smallskip
	\ifdef{Zemail}\par{\it \Zemail}\fi
	\ifdef{Zhomepage}\par{web page: \it\Zhomepage}\fi
	\ifdef{Zaddress}\par{\Zaddress}\fi
	\ifdef{Zsubject}\smallskip\par{1991 mathematics subject
		classification: \Zsubject}\fi
	\ifndef{Znote}\else\smallskip\par{\Znote}\fi
	\vskip 0pt plus 10pt}}
\def\abstract{\bigbreak\noindent{\bf Abstract.}\par\noindent} 

% SECTIONS, THEOREMS, ETC.
%
%\let\section=\beginsection
\outer\def\section#1\par{\bigbreak\noindent{\bf #1}\nobreak\medskip\noindent}
% Proclaimed Things (theorems, lemmas, etc.)
%   (modeled on plain tex's ``\proclaim'', but theorems can be more
%   than one paragraph long.)
\outer\def\beginproclaim#1. {\medbreak
	\noindent{\bf #1.\enspace}\begingroup\sl}
\def\endproclaim{\endgroup\par\ifdim\lastskip<\medskipamount 
	\removelastskip\penalty55\medskip\fi}
% Proofs (n.b. \drawbox is defined in `OTHER')
\def\beginproof#1{\smallskip{\it#1\/}}
\def\endproof{\leavevmode\vrule height0pt width0pt
	depth0pt\nobreak\hfill\proofbox\smallskip}
\def\QED{\endproof}
\def\proofbox{\drawbox{1.2ex}{1.2ex}{.1ex}}
% Remarks (n.b. remarks are only one par long)
\def\remark#1#2\par{\ifdim\lastskip<\smallskipamount
	\removelastskip\penalty55\smallskip\fi
	{\it #1\/}#2\smallbreak}

% CROSS-REFERENCES AND BIBLIOGRAPHY
%
% If the tag has not been defined (say by \deftag), then
% \tag warns and inserts question
% marks. If the tag is defined, it
% just produces the tag (symbolic if \userawtags=0, 
% or symbolic:raw, otherwise.) Default is \userawtags=1.
\def\rawtags{\def\userawtags{1}}
\def\symbolictags{\def\userawtags{0}}
\ifndef{userawtags}\def\userawtags{1}\fi
\def\tag#1{\ifndef{#1}\mywarning{tag `#1' 
	undefined.}{\hbox{\bf?`}\tt #1\hbox{\bf ?}}\else
	\ifnum\userawtags=0 \csname #1\endcsname
	\else\csname #1\endcsname{ \tt [#1]}\fi\fi}
\def\Tag#1{\tag{#1}}
\def\eqtag#1{(\tag{#1})}
\def\eqTag#1{\ifnum\userawtags=0
	\eqtag{#1}\else
	\lower12pt\hbox{\eqtag{#1}}\fi}
\def\cite#1{[\tag{#1}]}
\def\ecite#1#2{[\tag{#1}, #2]}
\def\nocite#1{}
% syntax: \deftag{label to insert in text}{label in file}
\def\deftag#1#2{\ifndef{#2}\else\mywarning{tag `#2' defined
	more than once.}\fi 
	\expandafter\def\csname #2\endcsname{#1}}
\def\defcite#1#2{\deftag{#1}{#2}}
% bibliography entries
\def\bibitem#1{\ifnum\userawtags=0
	\item{[{\tag{#1}}]}\else
	\noindent\hangindent=\parindent\hangafter=1
			\cite{#1}\enskip\fi}

% FONT SHORTCUTS 
%
\def\rom#1{({\it\romannumeral#1\/})}
\def\Rom#1{{\rm\uppercase\expandafter{\romannumeral#1}}}
\def\a{\alpha}	\def\b{\beta}	\def\c{\gamma}
\def\d{\delta}  \def\e{\varepsilon}	
\def\cala{{\cal A}} \def\calj{{\cal J}} \def\cals{{\cal S}}
\def\calb{{\cal B}} \def\calk{{\cal K}} \def\calt{{\cal T}}
\def\calc{{\cal C}} \def\call{{\cal L}} \def\calu{{\cal U}}
\def\cald{{\cal D}} \def\calm{{\cal M}} \def\calv{{\cal V}}
\def\cale{{\cal E}} \def\caln{{\cal N}} \def\calw{{\cal W}}
\def\calf{{\cal F}} \def\calo{{\cal O}} \def\calx{{\cal X}}
\def\calg{{\cal G}} \def\calp{{\cal P}} \def\caly{{\cal Y}}
\def\calh{{\cal H}} \def\calq{{\cal Q}} \def\calz{{\cal Z}}
\def\cali{{\cal I}} \def\calr{{\cal R}}

% SHORTCUTS AND MNEMONICS FOR STANDARD MATH SYMBOLS
%
\def\Z{{\bbb Z}} % integers
\def\Q{{\bbb Q}} % rational numbers 
\def\R{{\bbb R}} % real numbers
\def\C{{\bbb C}} % complex numbers
\def\F{{\bbb F}} % A finite field
\let\sset=\subseteq		\let\dimension=\dim%
\let\equivalent=\equiv% 	
\let\congruent=\equiv%
\def\setminus{\mathop{\bbb \char"72}\nolimits}
\def\semidirect{\mathop{\bbb \char"6E}\nolimits}
\let\tensor=\otimes	
\let\del=\partial
\let\isomorphism=\cong		\let\iso=\cong
\let\to=\rightarrow
\def\aut{\mathop{\rm Aut}\nolimits}
\def\hom{\mathop{\rm Hom}\nolimits}
\def\re{\mathop{\rm Re}\nolimits}
\def\im{\mathop{\rm Im}\nolimits}
\def\tr{\mathop{\rm Tr}\nolimits}
\def\mod{\mathop{\rm mod}\nolimits}
% quick parentheses and brackets
\def\({\left(} \def\){\right)}
\def\[{\left[} \def\]{\right]}

% OTHER STUFF	
%
\def\hideboxes{\overfullrule=0pt}
\def\frac#1/#2{\leavevmode\kern.1em\raise.5ex\hbox{\the
	\scriptfont0 #1}\kern-.1em/\kern-.15em\lower.25ex
	\hbox{\the\scriptfont0 #2}} 
% 2x2 matrices for use in text:
\def\smallmatrix#1#2#3#4{%
	\left({#1\atop #3}\;{#2\atop #4}\right)}
% syntax: ``\drawbox{internal width}{internal height}{line
% 	thickness}''; arguments should be dimens.
\def\drawbox#1#2#3{\leavevmode\vbox{\hrule height #3%
	\hbox{\vrule width #3 height #2\kern #1%
	\vrule width #3}\hrule height #3}}

% does an \halign but centers it with a little vertical space
% around it. Uses diplay math mode.
\def\calign#1{$$\vbox{\halign{#1}}$$}

% an \llap that you can use inside formulas without
% tex changing to textstyle inside the \hbox of \llap
\def\mathllap#1{\mathchoice
{\llap{$\displaystyle #1$}}%
{\llap{$\textstyle #1$}}%
{\llap{$\scriptstyle #1$}}%
{\llap{$\scriptscriptstyle #1$}}}
% the set of all #1 such that #2; makes the braces and the
% middle bar big enough to accomodate everything.
\def\set#1#2{\left\{\,#1\mathllap{\phantom{#2}}\mathrel{}\right|\left.#2\mathllap{\phantom{#1}}\,\right\}}

\parskip=0pt

% -----------------end-of-dja.tex-------------------------------
% This is PiCTeX, Version 1.1   9/21/87
 
% CAVEAT: The PiCTeX manual often has a more lucid explanation
%   of any given topic than you will find in the internal documentation
%   of the macros.
 
% PiCTeX's commands can be classified into two groups: (1) public (or
%   external), and (2) private (or internal). The public macros are
%   discussed at length in the manual. The only discussion of the private
%   macros is the internal documentation. The private macros all have
%   names beginning with an exclamation point (!) of category code 11. 
%   Since in normal usage "!" has category code 12, these macros can't
%   be accessed or modified by the general user.
 
% The macros are organized into thematically related groups. For example,
%   the macros dealing with dots & dashes are all in the DASHPATTERN group.
%   The table below shows which macros are in which groups. The table
%   covers all public macros, and many (but not all) of PiCTeX's upper level
%   private macros. Following the table, the various groups are listed
%   in the order in which they appear in the table.
 
% *********************** TABLE OF GROUPS OF MACROS **********************
 
% HACKS:  Utility macros
%    \PiC
%    \PiCTeX
%    \placevalueinpts 
%    \!!loop 
%    \!cfor
%    \!copylist
%    \!ecfor
%    \!etfor
%    \!getnext
%    \!getnextvalueof
%    \!ifempty
%    \!ifnextchar
%    \!leftappend
%    \!listaddon 
%    \!loop
%    \!lop
%    \!mlap
%    \!not
%    \!removept
%    \!rightappend
%    \!tfor 
%    \!vmlap
%    \!wlet
 
% ALLOCATION:  Allocates registers
 
% AREAS: Deals with plot areas 
%    \axis
%    \grid 
%    \invisibleaxes
%    \normalgraphs 
%    \plotheading 
%    \setplotarea
%    \visibleaxes
 
% ARROWS:  Draws arrows
%    \arrow
%    \betweenarrows
 
% BARS:  Draws bars
%    \putbar 
%    \setbars
 
% BOXES:  Draws rectangles
%    \frame
%    \putrectangle 
%    \rectangle
%    \shaderectangleson
%    \shaderectanglesoff
 
% CURVES:  Upper level plot commands
%    \hshade 
%    \plot 
%    \sethistograms
%    \setlinear
%    \setquadratic
%    \vshade
 
% DASHPATTERNS:  Sets up dash patterns
%    \findlength 
%    \setdashes 
%    \setdashesnear
%    \setdashpattern
%    \setdots 
%    \setdotsnear 
%    \setsolid
%    \!dashingoff
%    \!dashingon
 
% DIVISION:  Does long division of dimension registers
%    \Divide 
%    \!divide
 
% ELLIPSES:  Draws ellipses and circles
%    \circulararc 
%    \ellipticalarc 
 
% RULES:  Draws rules, i.e., horizontal & vertical lines
%    \putrule 
%    \!putdashedhline
%    \!putdashedvline
%    \!puthline 
%    \!putsolidhline  
%    \!putsolidvline  
%    \!putvline
 
% LINEAR ARC:  Draws straight lines -- solid and dashed
%    \inboundscheckoff
%    \inboundscheckon
%    \!advancedashing 
%    \!drawlinearsegment
%    \!initinboundscheck
%    \!linearsolid
%    \!lineardashed
%    \!ljoin
%    \!plotifinbounds     
%    \!start 
 
% LOGTEN:  Log_10 function
%    \!logten
 
% PICTURES:  Basic setups for PiCtures; \put commands
%    \accountingoff
%    \accountingon
%    \beginpicture
%    \endpicture    
%    \endpicturesave 
%    \lines
%    \multiput
%    \put 
%    \setcoordinatemode
%    \setcoordinatesystem
%    \setdimensionmode
%    \stack 
%    \Lines
%    \Xdistance
%    \Ydistance
%    \!dimenput
%    \!ifcoordmode
%    \!ifdimenmode
%    \!setcoordmode
%    \!setdimenmode
%    \!setputobject
   
% PLOTTING:  Things to do with plotting
%    \dontsavelinesandcurves
%    \replot
%    \savelinesandcurves 
%    \setplotsymbol
%    \writesavefile 
%    \!plot
 
% PYTHAGORAS:  Euclidean distance function
%    \placehypotenuse 
%    \!Pythag
 
% QUADRATIC ARC:  Draws a quadratic arc
%    \!qjoin 
 
% ROTATIONS:  Handles rotations
%    \startrotation 
%    \stoprotation
%    \!rotateaboutpivot
%    \!rotateonly
 
% SHADING:  Handles shading
%    \setshadegrid 
%    \setshadesymbol
%    \!lshade
%    \!qshade
%    \!starthshade
%    \!startvshade  
 
% TICKS:  Draws ticks on graphs
%    \gridlines
%    \loggedticks
%    \nogridlines
%    \ticksin
%    \ticksout
%    \unloggesticks
 
% ***************** END OF TABLE OF GROUPS OF MACROS ********************


\catcode`!=11 %  ***** THIS MUST NEVER BE OMITTED
% *******************************
% *** HACKS  (Utility macros) ***
% *******************************
 
% ** User commands
% **   \PiC{P\kern-.12em\lower.5ex\hbox{I}\kern-.075emC}
% **   \PiCTeX{\PiC\kern-.11em\TeX}
% **   \placevalueinpts of <DIMENSION REGISTER> in {CONTROL SEQUENCE}
  
% ** Internal commands
% **   \!ifnextchar{CHARACTER}{TRUE ACTION}{FALSE ACTION}
% **   \!tfor NAME := LIST \do {BODY}
% **   \!etfor NAME:= LIST \do {BODY}
% **   \!cfor NAME := LIST \do {BODY}
% **   \!ecfor NAME:= LIST \do {BODY}
% **   \!ifempty{MACRO}{TRUE ACTION}{FALSE ACTION}
% **   \!getnext\\ITEMfrom\LIST
% **   \!getnextvalueof\DIMEN\from\LIST
% **   \!copylist\LISTMACRO_A\to\LISTMACRO_B
% **   \!wlet\CONTROL_SEQUENCE_A=\CONTROL_SEQUENCE_B
% **   \!listaddon ITEM LIST
% **   \!rightappendITEM\withCS\to\LISTMACRO
% **   \!leftappendITEM\withCS\to\LISTMACRO
% **   \!lop\LISTMACRO\to\ITEM
% **   \!loop ... repeat
% **   \!!loop ... repeat
% **   \!removept{DIMENSION REGISTER}{CONTROL SEQUENCE}
% **   \!mlap{...}  
% **   \!vmlap{...}
% **   \!not{TEK if-CONDITION}

% ** First, here are the the PiCTeX logo, and the syllable PiC:
\def\PiC{P\kern-.12em\lower.5ex\hbox{I}\kern-.075emC}
\def\PiCTeX{\PiC\kern-.11em\TeX}

% ** The following macro expands to parameter #2 or parameter #3 according to
% ** whether the next non-blank character following the macro is or is not #1. 
% ** Blanks following the macro are gobbled.
\def\!ifnextchar#1#2#3{%
  \let\!testchar=#1%
  \def\!first{#2}%
  \def\!second{#3}%
  \futurelet\!nextchar\!testnext}
\def\!testnext{%
  \ifx \!nextchar \!spacetoken 
    \let\!next=\!skipspacetestagain
  \else
    \ifx \!nextchar \!testchar
      \let\!next=\!first
    \else 
      \let\!next=\!second 
    \fi 
  \fi
  \!next}
\def\\{\!skipspacetestagain} 
  \expandafter\def\\ {\futurelet\!nextchar\!testnext} 
\def\\{\let\!spacetoken= } \\  %  ** set \spacetoken to a space token
 
 
% ** Borrow the "tfor" macro from Latex:
% **   \!tfor NAME := LIST \do {BODY}
% **   if, before expansion, LIST = T1 ... Tn,  where each  Ti  is a token
% **   or  {...},  then executes  BODY  n  times, with  NAME = Ti  on the
% **   i-th iteration.  Works for  n=0.
\def\!tfor#1:=#2\do#3{%
  \edef\!fortemp{#2}%
  \ifx\!fortemp\!empty 
    \else
    \!tforloop#2\!nil\!nil\!!#1{#3}%
  \fi}
\def\!tforloop#1#2\!!#3#4{%
  \def#3{#1}%
  \ifx #3\!nnil
    \let\!nextwhile=\!fornoop
  \else
    #4\relax
    \let\!nextwhile=\!tforloop
  \fi 
  \!nextwhile#2\!!#3{#4}}
 
 
% **   \!etfor NAME:= LIST\do {BODY}
% **   This is like \!cfor, but LIST is any balanced token list whose complete
% **     expansion has the form  T1 ... Tn
\def\!etfor#1:=#2\do#3{%
  \def\!!tfor{\!tfor#1:=}%
  \edef\!!!tfor{#2}%
  \expandafter\!!tfor\!!!tfor\do{#3}}
 
 
% **   modify the Latex \tfor (token-for) loop to a \cfor (comma-for) loop.
% **   \!cfor NAME := LIST \do {BODY}
% **     if, before expansion, LIST = a1,a2,...an, then executes  BODY n times,
% **     with  NAME = ai  on the i-th iteration.  Works for  n=0.
\def\!cfor#1:=#2\do#3{%
  \edef\!fortemp{#2}%
  \ifx\!fortemp\!empty 
  \else
    \!cforloop#2,\!nil,\!nil\!!#1{#3}%
  \fi}
\def\!cforloop#1,#2\!!#3#4{%
  \def#3{#1}%
  \ifx #3\!nnil
    \let\!nextwhile=\!fornoop 
  \else
    #4\relax
    \let\!nextwhile=\!cforloop
  \fi
  \!nextwhile#2\!!#3{#4}}
 
 
% **   \!ecfor NAME:= LIST\do {BODY}
% **   This is like \!cfor, but LIST is any balanced token list whose complete
% **     expansion has the form  a1,a2,...,an.
\def\!ecfor#1:=#2\do#3{%
  \def\!!cfor{\!cfor#1:=}%
  \edef\!!!cfor{#2}%
  \expandafter\!!cfor\!!!cfor\do{#3}}
 
 
\def\!empty{}
\def\!nnil{\!nil}
\def\!fornoop#1\!!#2#3{}
 
 
% **  \!ifempty{ARG}{TRUE ACTION}{FALSE ACTION}
\def\!ifempty#1#2#3{%
  \edef\!emptyarg{#1}%
  \ifx\!emptyarg\!empty
    #2%
  \else
    #3%
  \fi}
 
% **  \!getnext\\ITEMfrom\LIST
% **    \LIST has the form \\{item1}\\{item2}\\{item3}...\\{itemk}
% **    This routine sets \ITEM to item1, and cycles \LIST to
% **    \\{item2}\\{item3}...\\{itemk}\\{item1}
\def\!getnext#1\from#2{%
  \expandafter\!gnext#2\!#1#2}%
\def\!gnext\\#1#2\!#3#4{%
  \def#3{#1}%
  \def#4{#2\\{#1}}%
  \ignorespaces}
 
 
% ** \!getnextvalueof\DIMEN\from\LIST
% **   Similar to !getnext.  
% **   \LIST has the form \\{dimen1}\\{dimen2}\\{dimen3} ... 
% **   \DIMEN is a dimension register
% **   Works also for counts
%
\def\!getnextvalueof#1\from#2{%
  \expandafter\!gnextv#2\!#1#2}%
\def\!gnextv\\#1#2\!#3#4{%
  #3=#1%
  \def#4{#2\\{#1}}%
  \ignorespaces}
 
 
% ** \!copylist\LISTMACROA\to\LISTMACROB
% **   makes the replacement text of LISTMACRO B identical to that of
% **   list macro A.
\def\!copylist#1\to#2{%
  \expandafter\!!copylist#1\!#2}
\def\!!copylist#1\!#2{%
  \def#2{#1}\ignorespaces}
 
 
% **  \!wlet\CSA=\CSB
% **  lets control sequence \CSB = control sequence \CSA, and writes a
% **    message to that effect in the log file using plain TEK's \wlog
\def\!wlet#1=#2{%
  \let#1=#2 
  \wlog{\string#1=\string#2}}
 
% ** \!listaddon ITEM LIST
% ** LIST <-- LIST \\ ITEM
\def\!listaddon#1#2{%
  \expandafter\!!listaddon#2\!{#1}#2}
\def\!!listaddon#1\!#2#3{%
  \def#3{#1\\#2}}
 
% **  \!rightappendITEM\to\LISTMACRO
% **    \LISTMACRO --> \LISTMACRO\\{ITEM}
%\def\!rightappend#1\to#2{\expandafter\!!rightappend#2\!{#1}#2}
%\def\!!rightappend#1\!#2#3{\def#3{#1\\{#2}}}
 
 
% **  \!rightappendITEM\withCS\to\LISTMACRO
% **    \LISTMACRO --> \LISTMACRO||CS||{ITEM}
\def\!rightappend#1\withCS#2\to#3{\expandafter\!!rightappend#3\!#2{#1}#3}
\def\!!rightappend#1\!#2#3#4{\def#4{#1#2{#3}}}
 
 
% **  \!leftappendITEM\withCS\to\LISTMACRO
% **    \LISTMACRO --> CS||{ITEM}||\LISTMACRO
\def\!leftappend#1\withCS#2\to#3{\expandafter\!!leftappend#3\!#2{#1}#3}
\def\!!leftappend#1\!#2#3#4{\def#4{#2{#3}#1}}
 
 
% **  \!lop\LISTMACRO\to\ITEM
% **    \\{item1}\\{item2}\\{item3} ... --> \\{item2}\\{item3} ...
% **    item1 --> \ITEM
\def\!lop#1\to#2{\expandafter\!!lop#1\!#1#2}
\def\!!lop\\#1#2\!#3#4{\def#4{#1}\def#3{#2}}
 
 
% **  \!placeNUMBER\of\LISTMACRO\in\ITEM
% **    the NUMBERth item of \LISTMACRO --> replacement text of \ITEM
%\def\!place#1\of#2\in#3{\def#3{\outofrange}%
%{\count0=#1\def\\##1{\advance\count0-1 \ifnum\count0=0 \gdef#3{##1}\fi}#2}}
 
 
% **  Following code converts a commalist to a list macro, with all items 
% **    fully expanded.
%\!ecfor\item:=\commalist\do{\expandafter\!rightappend\item\to\list}
 
 
% ** \!loop ... repeat
% ** This is exactly like TEX's \loop ... repeat.  It can be used in nesting
% ** two loops, without puting the inner one inside a group.
\def\!loop#1\repeat{\def\!body{#1}\!iterate}
\def\!iterate{\!body\let\!next=\!iterate\else\let\!next=\relax\fi\!next}
 
% ** \!!loop ... repeat
% ** This is exactly like TEX's \loop ... repeat.  It can be used in nesting
% ** two loops, without puting the inner one inside a group.
\def\!!loop#1\repeat{\def\!!body{#1}\!!iterate}
\def\!!iterate{\!!body\let\!!next=\!!iterate\else\let\!!next=\relax\fi\!!next}
%  (\multiput uses \!!loop)
 
% ** \!removept{DIMENREG}{\CS}
% ** Defines the control sequence CS to be the value (in points) in the
% ** dimension register DIMENREG (but without the "pt" TEK usually adds)
% ** E.g., after  \dimen0=12.3pt \!removept\dimen0\A, \A expands to 12.3
\def\!removept#1#2{\edef#2{\expandafter\!!removePT\the#1}}
{\catcode`p=12 \catcode`t=12 \gdef\!!removePT#1pt{#1}}

% ** \pladevalueinpts of <DIMENSION REGISTER> in {CONTROL SEQUENCE}
\def\placevalueinpts of <#1> in #2 {%
  \!removept{#1}{#2}}
 
% ** \!mlap{...}  \!vmlap{...}
% ** Center  ...  in a box of width 0.
\def\!mlap#1{\hbox to 0pt{\hss#1\hss}}
\def\!vmlap#1{\vbox to 0pt{\vss#1\vss}}
 
% ** \!not{TEK if-CONDITION}
% ** By a TEK if-CONDITION is meant something like 
% **     \ifnum\N<0,   or   \ifdim\A>\B
% ** \!not produces an if-condition which is false if the original condition
% ** is true, and true if the original condition is false.
\def\!not#1{%
  #1\relax
    \!switchfalse
  \else
    \!switchtrue
  \fi
  \if!switch
  \ignorespaces}
 

% *******************
% *** ALLOCATIONS ***
% *******************

% This section allocates all the registers PiCTeX uses. Following
% each allocation is a string of the form  ....N.D...L......... ;
% the various letters show which sections of PiCTeX make explicit
% reference to that register, according to the following code:
 
%   H Hacks
%   A Areas
%   W arroWs
%   B Bars
%   X boXes
%   C Curves
%   D Dashpattterns
%   V diVision
%   E Ellipses
%   U rUles
%   L Linear arc
%   G loGten
%   P Pictures
%   O plOtting
%   Y pYthagoras
%   Q Quadratic arc
%   R Rotations
%   S Shading
%   T Ticks

% Turn off messages from TeX's allocation macros
\let\!!!wlog=\wlog              % "\wlog" is defined in plain TeX
\def\wlog#1{}    

\newdimen\headingtoplotskip     %.A.................
\newdimen\linethickness         %.A..X....U........T
\newdimen\longticklength        %.A................T
\newdimen\plotsymbolspacing     %......D...L....Q...
\newdimen\shortticklength       %.A................T
\newdimen\stackleading          %.A..........P......
\newdimen\tickstovaluesleading  %.A................T
\newdimen\totalarclength        %......D...L....Q...
\newdimen\valuestolabelleading  %.A.................

\newbox\!boxA                   %.AW...............T
\newbox\!boxB                   %..W................
\newbox\!picbox                 %............P......
\newbox\!plotsymbol             %..........L..O.....
\newbox\!putobject              %............PO...S.
\newbox\!shadesymbol            %.................S.

\newcount\!countA               %.A....D..UL....Q.ST
\newcount\!countB               %......D..U.....Q.ST
\newcount\!countC               %...............Q..T
\newcount\!countD               %...................
\newcount\!countE               %.............O....T
\newcount\!countF               %.............O....T
\newcount\!countG               %..................T
\newcount\!fiftypt              %.........U.........
\newcount\!intervalno           %..........L....Q...
\newcount\!npoints              %..........L........
\newcount\!nsegments            %.........U.........
\newcount\!ntemp                %............P......
\newcount\!parity               %.................S.
\newcount\!scalefactor          %..................T
\newcount\!tfs                  %.......V...........
\newcount\!tickcase             %..................T

\newdimen\!Xleft                %............P......
\newdimen\!Xright               %............P......
\newdimen\!Xsave                %.A................T
\newdimen\!Ybot                 %............P......
\newdimen\!Ysave                %.A................T
\newdimen\!Ytop                 %............P......
\newdimen\!angle                %........E..........
\newdimen\!arclength            %..W......UL....Q...
\newdimen\!areabloc             %.A........L........
\newdimen\!arealloc             %.A........L........
\newdimen\!arearloc             %.A........L........
\newdimen\!areatloc             %.A........L........
\newdimen\!bshrinkage           %.................S.
\newdimen\!checkbot             %..........L........
\newdimen\!checkleft            %..........L........
\newdimen\!checkright           %..........L........
\newdimen\!checktop             %..........L........
\newdimen\!dimenA               %.AW.X.DVEUL..OYQRST
\newdimen\!dimenB               %....X.DVEU...O.QRS.
\newdimen\!dimenC               %..W.X.DVEU......RS.
\newdimen\!dimenD               %..W.X.DVEU....Y.RS.
\newdimen\!dimenE               %..W........G..YQ.S.
\newdimen\!dimenF               %...........G..YQ.S.
\newdimen\!dimenG               %...........G..YQ.S.
\newdimen\!dimenH               %...........G..Y..S.
\newdimen\!dimenI               %...BX.........Y....
\newdimen\!distacross           %..........L....Q...
\newdimen\!downlength           %..........L........
\newdimen\!dp                   %.A..X.......P....S.
\newdimen\!dshade               %.................S.
\newdimen\!dxpos                %..W......U..P....S.
\newdimen\!dxprime              %...............Q...
\newdimen\!dypos                %..WB.....U..P......
\newdimen\!dyprime              %...............Q...
\newdimen\!ht                   %.A..X.......P....S.
\newdimen\!leaderlength         %......D..U.........
\newdimen\!lshrinkage           %.................S.
\newdimen\!midarclength         %...............Q...
\newdimen\!offset               %.A................T
\newdimen\!plotheadingoffset    %.A.................
\newdimen\!plotsymbolxshift     %..........L..O.....
\newdimen\!plotsymbolyshift     %..........L..O.....
\newdimen\!plotxorigin          %..........L..O.....
\newdimen\!plotyorigin          %..........L..O.....
\newdimen\!rootten              %...........G.......
\newdimen\!rshrinkage           %.................S.
\newdimen\!shadesymbolxshift    %.................S.
\newdimen\!shadesymbolyshift    %.................S.
\newdimen\!tenAa                %...........G.......
\newdimen\!tenAc                %...........G.......
\newdimen\!tenAe                %...........G.......
\newdimen\!tshrinkage           %.................S.
\newdimen\!uplength             %..........L........
\newdimen\!wd                   %....X.......P....S.
\newdimen\!wmax                 %...............Q...
\newdimen\!wmin                 %...............Q...
\newdimen\!xB                   %...............Q...
\newdimen\!xC                   %...............Q...
\newdimen\!xE                   %..W.....E.L....Q.S.
\newdimen\!xM                   %..W.....E......Q.S.
\newdimen\!xS                   %..W.....E.L....Q.S.
\newdimen\!xaxislength          %.A................T
\newdimen\!xdiff                %..........L........
\newdimen\!xleft                %............P......
\newdimen\!xloc                 %..WB.....U.......S.
\newdimen\!xorigin              %.A........L.P....S.
\newdimen\!xpivot               %................R..
\newdimen\!xpos                 %..........L.P..Q.ST
\newdimen\!xprime               %...............Q...
\newdimen\!xright               %............P......
\newdimen\!xshade               %.................S.
\newdimen\!xshift               %..W.........PO...S.
\newdimen\!xtemp                %............P......
\newdimen\!xunit                %.AWBX...EUL.P..QRS.
\newdimen\!xxE                  %........E..........
\newdimen\!xxM                  %........E..........
\newdimen\!xxS                  %........E..........
\newdimen\!xxloc                %..WB....EU.........
\newdimen\!yB                   %...............Q...
\newdimen\!yC                   %...............Q...
\newdimen\!yE                   %..W.....E.L....Q...
\newdimen\!yM                   %..W.....E......Q...
\newdimen\!yS                   %..W.....E.L....Q...
\newdimen\!yaxislength          %.A................T
\newdimen\!ybot                 %............P......
\newdimen\!ydiff                %..........L........
\newdimen\!yloc                 %..WB.....U.......S.
\newdimen\!yorigin              %.A........L.P....S.
\newdimen\!ypivot               %................R..
\newdimen\!ypos                 %..........L.P..Q.ST
\newdimen\!yprime               %...............Q...
\newdimen\!yshade               %.................S.
\newdimen\!yshift               %..W.........PO...S.
\newdimen\!ytemp                %............P......
\newdimen\!ytop                 %............P......
\newdimen\!yunit                %.AWBX...EUL.P..QRS.
\newdimen\!yyE                  %........E..........
\newdimen\!yyM                  %........E..........
\newdimen\!yyS                  %........E..........
\newdimen\!yyloc                %..WB....EU.........
\newdimen\!zpt                  %.AWBX.DVEULGP.YQ.ST

\newif\if!axisvisible           %.A.................
\newif\if!gridlinestoo          %..................T
\newif\if!keepPO                %...................
\newif\if!placeaxislabel        %.A.................
\newif\if!switch                %H..................
\newif\if!xswitch               %.A................T

\newtoks\!axisLaBeL             %.A.................
\newtoks\!keywordtoks           %.A.................

\newwrite\!replotfile           %.............O.....

\newhelp\!keywordhelp{The keyword mentioned in the error message in unknown. 
Replace NEW KEYWORD in the indicated response by the keyword that 
should have been specified.}    %.A.................

% The following commands assign alternate names to some of the 
% above registers.  "\!wlet"  is defined in  Hacks.
\!wlet\!!origin=\!xM                   %.A................T
\!wlet\!!unit=\!uplength               %.A................T
\!wlet\!Lresiduallength=\!dimenG       %.........U.........
\!wlet\!Rresiduallength=\!dimenF       %.........U.........
\!wlet\!axisLength=\!distacross        %.A................T
\!wlet\!axisend=\!ydiff                %.A................T
\!wlet\!axisstart=\!xdiff              %.A................T
\!wlet\!axisxlevel=\!arclength         %.A................T
\!wlet\!axisylevel=\!downlength        %.A................T
\!wlet\!beta=\!dimenE                  %...............Q...
\!wlet\!gamma=\!dimenF                 %...............Q...
\!wlet\!shadexorigin=\!plotxorigin     %.................S.
\!wlet\!shadeyorigin=\!plotyorigin     %.................S.
\!wlet\!ticklength=\!xS                %..................T
\!wlet\!ticklocation=\!xE              %..................T
\!wlet\!ticklocationincr=\!yE          %..................T
\!wlet\!tickwidth=\!yS                 %..................T
\!wlet\!totalleaderlength=\!dimenE     %.........U.........
\!wlet\!xone=\!xprime                  %....X..............
\!wlet\!xtwo=\!dxprime                 %....X..............
\!wlet\!ySsave=\!yM                    %...................
\!wlet\!ybB=\!yB                       %.................S.
\!wlet\!ybC=\!yC                       %.................S.
\!wlet\!ybE=\!yE                       %.................S.
\!wlet\!ybM=\!yM                       %.................S.
\!wlet\!ybS=\!yS                       %.................S.
\!wlet\!ybpos=\!yyloc                  %.................S.
\!wlet\!yone=\!yprime                  %....X..............
\!wlet\!ytB=\!xB                       %.................S.
\!wlet\!ytC=\!xC                       %.................S.
\!wlet\!ytE=\!downlength               %.................S.
\!wlet\!ytM=\!arclength                %.................S.
\!wlet\!ytS=\!distacross               %.................S.
\!wlet\!ytpos=\!xxloc                  %.................S.
\!wlet\!ytwo=\!dyprime                 %....X..............


% Initial values for registers
\!zpt=0pt                              % static
\!xunit=1pt
\!yunit=1pt
\!arearloc=\!xunit
\!areatloc=\!yunit
\!dshade=5pt
\!leaderlength=24in
\!tfs=256                              % static
\!wmax=5.3pt                           % static
\!wmin=2.7pt                           % static
\!xaxislength=\!xunit
\!xpivot=\!zpt
\!yaxislength=\!yunit 
\!ypivot=\!zpt
\plotsymbolspacing=.4pt
  \!dimenA=50pt \!fiftypt=\!dimenA     % static

\!rootten=3.162278pt                   % static
\!tenAa=8.690286pt                     % static  (A5)
\!tenAc=2.773839pt                     % static  (A3)
\!tenAe=2.543275pt                     % static  (A1)

% Initial values for control sequences
\def\!cosrotationangle{1}      %................R..
\def\!sinrotationangle{0}      %................R..
\def\!xpivotcoord{0}           %................R..
\def\!xref{0}                  %............P......
\def\!xshadesave{0}            %.................S.
\def\!ypivotcoord{0}           %................R..
\def\!yref{0}                  %............P......
\def\!yshadesave{0}            %.................S.
\def\!zero{0}                  %..................T

% Reset TeX to report allocations
\let\wlog=\!!!wlog
%  *************************************
%  ***  AREAS: Deals with plot areas ***
%  *************************************
%
%  ** User commands
%  **   \setplotarea x from LEFT XCOORD to RIGTH XCOORD, y from BOTTOM YCOORD
%  **      to TOP YCOORD
%  **   \axis BOTTOM-LEFT-TOP-RIGHT  [SHIFTEDTO xy=COORD] [VISIBLE-INVISIBLE]
%  **      [LABEL {label}] [TICKS] /
%  **   \visibleaxes
%  **   \invisibleaxes
%  **   \plotheading {HEADING}
%  **   \grid {# of columns} {# of rows}
%  **   \normalgraphs 
  
%  **  \normalgraphs
%  **    Sets defaults for graph setup. See Subsection 3.4 of manual.
\def\normalgraphs{%
  \longticklength=.4\baselineskip
  \shortticklength=.25\baselineskip
  \tickstovaluesleading=.25\baselineskip
  \valuestolabelleading=.8\baselineskip
  \linethickness=.4pt
  \stackleading=.17\baselineskip
  \headingtoplotskip=1.5\baselineskip
  \visibleaxes
  \ticksout
  \nogridlines
  \unloggedticks}
%
% **  \setplotarea x from LEFT XCOORD to RIGTH XCOORD, y from BOTTOM YCOORD
% **    to TOP YCOORD
% **  Reserves space in PICBOX for a rectangular box with the indicated
% **   coordinates.  Must be specified before calls to  \axis, 
% **   \grid, \plotheading.
% **  See Subsection 3.1 of the manual.
\def\setplotarea x from #1 to #2, y from #3 to #4 {%
  \!arealloc=\!M{#1}\!xunit \advance \!arealloc -\!xorigin
  \!areabloc=\!M{#3}\!yunit \advance \!areabloc -\!yorigin
  \!arearloc=\!M{#2}\!xunit \advance \!arearloc -\!xorigin
  \!areatloc=\!M{#4}\!yunit \advance \!areatloc -\!yorigin
  \!initinboundscheck
  \!xaxislength=\!arearloc  \advance\!xaxislength -\!arealloc
  \!yaxislength=\!areatloc  \advance\!yaxislength -\!areabloc
  \!plotheadingoffset=\!zpt
  \!dimenput {{\setbox0=\hbox{}\wd0=\!xaxislength\ht0=\!yaxislength\box0}}
     [bl] (\!arealloc,\!areabloc)}
%
% ** \visibleaxes, \invisibleaxes 
% ** Switches for setting visibility of subsequent axes.
% ** See Subsection 3.2 of the manual.
\def\visibleaxes{%
  \def\!axisvisibility{\!axisvisibletrue}}
\def\invisibleaxes{%
  \def\!axisvisibility{\!axisvisiblefalse}}
%
% ** The next few macros enable the user to fix up an erroneous keyword
% **   in the \axis command.
%  \newhelp is in ALLOCATIONS
%  \newhelp\!keywordhelp{The keyword mentioned in the error message in unknown. 
%  Replace NEW KEYWORD in the indicated response by the keyword that 
%  should have been specified.}

\def\!fixkeyword#1{%
  \errhelp=\!keywordhelp
  \errmessage{Unrecognized keyword `#1': \the\!keywordtoks{NEW KEYWORD}'}}

%  \newtoks\!keywordtoks    In ALLOCATIONS.
\!keywordtoks={enter `i\fixkeyword}

\def\fixkeyword#1{%
  \!nextkeyword#1 }

% ** \axis BOTTOM-LEFT-TOP-RIGHT  [SHIFTEDTO xy=COORD] [VISIBLE-INVISIBLE]
% **   [LABEL {label}] [TICKS] /
% ** Exactly one of the keywords BOTTOM, LEFT, TOP, RIGHT must be
% ** specified. Axis is drawn along the indicated edge of the current
% ** plot area, shifted if the SHIFTEDTO option is used, visible or
% ** invisible according the selected option, with an optional LABEL,
% ** and optional TICKS (see ticks.tex for the options avialabel with
% ** TICKS). The TICKS option must be the last one specified. The \axis
% ** MUST be terminated with a / followed by a space.
% ** See Subsection 3.2 of the manual for more information.

% ** The various options of the \axis command are processed by the
% ** \!nextkeyword macro defined below. For example, 
% ** `\!nextkeyword shiftedto ' expands to `\!axisshiftedto'.
\def\axis {%
  \def\!nextkeyword##1 {%
    \expandafter\ifx\csname !axis##1\endcsname \relax
      \def\!next{\!fixkeyword{##1}}%
    \else
      \def\!next{\csname !axis##1\endcsname}%
    \fi
    \!next}%
  \!offset=\!zpt
  \!axisvisibility
  \!placeaxislabelfalse
  \!nextkeyword}

% ** This and the various macros that follow handle the keyword
% ** specifications on the \axis command
% ** See Subsection 3.2 of the manual.
\def\!axisbottom{%
  \!axisylevel=\!areabloc
  \def\!tickxsign{0}%
  \def\!tickysign{-}%
  \def\!axissetup{\!axisxsetup}%
  \def\!axislabeltbrl{t}%
  \!nextkeyword}

\def\!axistop{%
  \!axisylevel=\!areatloc
  \def\!tickxsign{0}%
  \def\!tickysign{+}%
  \def\!axissetup{\!axisxsetup}%
  \def\!axislabeltbrl{b}%
  \!nextkeyword}

\def\!axisleft{%
  \!axisxlevel=\!arealloc
  \def\!tickxsign{-}%
  \def\!tickysign{0}%
  \def\!axissetup{\!axisysetup}%
  \def\!axislabeltbrl{r}%
  \!nextkeyword}

\def\!axisright{%
  \!axisxlevel=\!arearloc
  \def\!tickxsign{+}%
  \def\!tickysign{0}%
  \def\!axissetup{\!axisysetup}%
  \def\!axislabeltbrl{l}%
  \!nextkeyword}

\def\!axisshiftedto#1=#2 {%
  \if 0\!tickxsign
    \!axisylevel=\!M{#2}\!yunit
    \advance\!axisylevel -\!yorigin
  \else
    \!axisxlevel=\!M{#2}\!xunit
    \advance\!axisxlevel -\!xorigin
  \fi
  \!nextkeyword}

\def\!axisvisible{%
  \!axisvisibletrue  
  \!nextkeyword}

\def\!axisinvisible{%
  \!axisvisiblefalse
  \!nextkeyword}

\def\!axislabel#1 {%
  \!axisLaBeL={#1}%
  \!placeaxislabeltrue
  \!nextkeyword}

\expandafter\def\csname !axis/\endcsname{%
  \!axissetup % This could done already by "ticks"; if so, now \relax
  \if!placeaxislabel
    \!placeaxislabel
  \fi
  \if +\!tickysign %                 ** (A "top" axis)
    \!dimenA=\!axisylevel
    \advance\!dimenA \!offset %      ** dimA = top of the axis structure
    \advance\!dimenA -\!areatloc %   ** dimA = excess over the plot area
    \ifdim \!dimenA>\!plotheadingoffset
      \!plotheadingoffset=\!dimenA % ** Greatest excess over the plot area
    \fi
  \fi}

% ** \grid {c} {r} 
% ** Partitions the plot area into c columns and r rows; see Subsection 3.3
% ** of the manual.
% ** (Other grid patterns can be drawn with the TICKS option of the \axis 
% ** command.
\def\grid #1 #2 {%
  \!countA=#1\advance\!countA 1
  \axis bottom invisible ticks length <\!zpt> andacross quantity {\!countA} /
  \!countA=#2\advance\!countA 1
  \axis left   invisible ticks length <\!zpt> andacross quantity {\!countA} / }

% ** \plotheading{HEADING}
% ** Places HEADING centered above the top of the plotarea (and above
% ** any top axis ticks marks, tick labels, and axis label); see
% ** Subsection 3.3 of the manual.
\def\plotheading#1 {%
  \advance\!plotheadingoffset \headingtoplotskip
  \!dimenput {#1} [B] <.5\!xaxislength,\!plotheadingoffset>
    (\!arealloc,\!areatloc)}

% ** From here on, the routines are internal.
\def\!axisxsetup{%
  \!axisxlevel=\!arealloc
  \!axisstart=\!arealloc
  \!axisend=\!arearloc
  \!axisLength=\!xaxislength
  \!!origin=\!xorigin
  \!!unit=\!xunit
  \!xswitchtrue
  \if!axisvisible 
    \!makeaxis
  \fi}

\def\!axisysetup{%
  \!axisylevel=\!areabloc
  \!axisstart=\!areabloc
  \!axisend=\!areatloc
  \!axisLength=\!yaxislength
  \!!origin=\!yorigin
  \!!unit=\!yunit
  \!xswitchfalse
  \if!axisvisible
    \!makeaxis
  \fi}

\def\!makeaxis{%
  \setbox\!boxA=\hbox{% (Make a pseudo-y[x] tick for an x[y]-axis)
    \beginpicture
      \!setdimenmode
      \setcoordinatesystem point at {\!zpt} {\!zpt}   
      \putrule from {\!zpt} {\!zpt} to
        {\!tickysign\!tickysign\!axisLength} 
        {\!tickxsign\!tickxsign\!axisLength}
    \endpicturesave <\!Xsave,\!Ysave>}%
    \wd\!boxA=\!zpt
    \!placetick\!axisstart}

\def\!placeaxislabel{%
  \advance\!offset \valuestolabelleading
  \if!xswitch
    \!dimenput {\the\!axisLaBeL} [\!axislabeltbrl]
      <.5\!axisLength,\!tickysign\!offset> (\!axisxlevel,\!axisylevel)
    \advance\!offset \!dp  % ** advance offset by the "tallness"
    \advance\!offset \!ht  % ** of the label
  \else
    \!dimenput {\the\!axisLaBeL} [\!axislabeltbrl]
      <\!tickxsign\!offset,.5\!axisLength> (\!axisxlevel,\!axisylevel)
  \fi
  \!axisLaBeL={}}


% *******************************
% *** ARROWS  (Draws arrows)  ***
% *******************************
%
% ** User commands
% **  \arrow <ARROW HEAD LENGTH> [MID FRACTION, BASE FRACTION]
% **    [<XSHIFT,YSHIFT>] from XFROM YFROM to XTO YTO
% **  \betweenarrows {TEXT} [orientation & shift] from XFROM YFROM to XTO YTO

% ** \arrow <ARROW HEAD LENGTH> [MID FRACTION, BASE FRACTION]
% **    [<XSHIFT,YSHIFT>] from XFROM YFROM to XTO YTO
% ** Draws an arrow from (XFROM,YFROM) to (XTO,YTO).  The arrow head
% ** is constructed two quadratic arcs, which extend back a distance
% ** ARROW HEAD LENGTH (a dimension) on both sides of the arrow shaft.
% ** All the way back the arcs are a distance BASE FRACTION*ARROW HEAD
% ** LENGTH apart, while half-way back they are a distance MID FRACTION*
% ** ARROW HEAD LENGTH apart. <XSHIFT,YSHIFT> is optional, and has
% ** its usual interpreation. See Subsection 5.4 of the manual.

\def\arrow <#1> [#2,#3]{%
  \!ifnextchar<{\!arrow{#1}{#2}{#3}}{\!arrow{#1}{#2}{#3}<\!zpt,\!zpt> }}

\def\!arrow#1#2#3<#4,#5> from #6 #7 to #8 #9 {%
%
% ** convert to dimensions
  \!xloc=\!M{#8}\!xunit   
  \!yloc=\!M{#9}\!yunit
  \!dxpos=\!xloc  \!dimenA=\!M{#6}\!xunit  \advance \!dxpos -\!dimenA
  \!dypos=\!yloc  \!dimenA=\!M{#7}\!yunit  \advance \!dypos -\!dimenA
  \let\!MAH=\!M%                         ** save current c/d mode
  \!setdimenmode%                        ** go into dimension mode
%
  \!xshift=#4\relax  \!yshift=#5\relax%  ** pick up shift
  \!reverserotateonly\!xshift\!yshift%   ** back rotate shift
  \advance\!xshift\!xloc  \advance\!yshift\!yloc
%
% **  draw shaft of arrow
  \!xS=-\!dxpos  \advance\!xS\!xshift
  \!yS=-\!dypos  \advance\!yS\!yshift
  \!start (\!xS,\!yS)
  \!ljoin (\!xshift,\!yshift)
%
% ** find 32*cosine and 32*sine of angle of rotation
  \!Pythag\!dxpos\!dypos\!arclength
  \!divide\!dxpos\!arclength\!dxpos  
  \!dxpos=32\!dxpos  \!removept\!dxpos\!!cos
  \!divide\!dypos\!arclength\!dypos  
  \!dypos=32\!dypos  \!removept\!dypos\!!sin
% 
% ** construct arrowhead
  \!halfhead{#1}{#2}{#3}%                ** draw half of arrow head
  \!halfhead{#1}{-#2}{-#3}%              ** draw other half
%
  \let\!M=\!MAH%                         ** restore old c/d mode
  \ignorespaces}
%
% ** draw half of arrow head
  \def\!halfhead#1#2#3{%
    \!dimenC=-#1%                
    \divide \!dimenC 2 %                 ** half way back
    \!dimenD=#2\!dimenC%                 ** half the mid width
    \!rotate(\!dimenC,\!dimenD)by(\!!cos,\!!sin)to(\!xM,\!yM)
    \!dimenC=-#1%                        ** all the way back
    \!dimenD=#3\!dimenC
    \!dimenD=.5\!dimenD%                 ** half the full width
    \!rotate(\!dimenC,\!dimenD)by(\!!cos,\!!sin)to(\!xE,\!yE)
    \!start (\!xshift,\!yshift)
    \advance\!xM\!xshift  \advance\!yM\!yshift
    \advance\!xE\!xshift  \advance\!yE\!yshift
    \!qjoin (\!xM,\!yM) (\!xE,\!yE) 
    \ignorespaces}


% ** \betweenarrows {TEXT} [orientation & shift] from XFROM YFROM to XTO YTO
% **   Makes things like <--- text --->, using arrow heads from TeX's fonts.
% **   See Subsection 5.4 of the manual.
\def\betweenarrows #1#2 from #3 #4 to #5 #6 {%
  \!xloc=\!M{#3}\!xunit  \!xxloc=\!M{#5}\!xunit%   
  \!yloc=\!M{#4}\!yunit  \!yyloc=\!M{#6}\!yunit%           
  \!dxpos=\!xxloc  \advance\!dxpos by -\!xloc
  \!dypos=\!yyloc  \advance\!dypos by -\!yloc
  \advance\!xloc .5\!dxpos
  \advance\!yloc .5\!dypos
%
  \let\!MBA=\!M%           ** save current coord\dimen mode
  \!setdimenmode%          ** express locations in dimens
  \ifdim\!dypos=\!zpt
    \ifdim\!dxpos<\!zpt \!dxpos=-\!dxpos \fi
    \put {\!lrarrows{\!dxpos}{#1}}#2{} at {\!xloc} {\!yloc}
  \else
    \ifdim\!dxpos=\!zpt
      \ifdim\!dypos<\!zpt \!dypos=-\!dypos \fi
      \put {\!udarrows{\!dypos}{#1}}#2{} at {\!xloc} {\!yloc}
    \fi
  \fi
  \let\!M=\!MBA%           ** restore previous c/d mode
  \ignorespaces}

% ** Subroutine for left-right between arrows 
\def\!lrarrows#1#2{% #1=width, #2=text
  {\setbox\!boxA=\hbox{$\mkern-2mu\mathord-\mkern-2mu$}%
   \setbox\!boxB=\hbox{$\leftarrow$}\!dimenE=\ht\!boxB
   \setbox\!boxB=\hbox{}\ht\!boxB=2\!dimenE
   \hbox to #1{$\mathord\leftarrow\mkern-6mu
     \cleaders\copy\!boxA\hfil
     \mkern-6mu\mathord-$%
     \kern.4em $\vcenter{\box\!boxB}$$\vcenter{\hbox{#2}}$\kern.4em
     $\mathord-\mkern-6mu
     \cleaders\copy\!boxA\hfil
     \mkern-6mu\mathord\rightarrow$}}}

% ** Subroutine for up-down between arrows 
\def\!udarrows#1#2{% #1=width, #2=text
  {\setbox\!boxB=\hbox{#2}%
   \setbox\!boxA=\hbox to \wd\!boxB{\hss$\vert$\hss}%
   \!dimenE=\ht\!boxA \advance\!dimenE \dp\!boxA \divide\!dimenE 2
   \vbox to #1{\offinterlineskip
      \vskip .05556\!dimenE
      \hbox to \wd\!boxB{\hss$\mkern.4mu\uparrow$\hss}\vskip-\!dimenE
      \cleaders\copy\!boxA\vfil
      \vskip-\!dimenE\copy\!boxA
      \vskip\!dimenE\copy\!boxB\vskip.4em
      \copy\!boxA\vskip-\!dimenE
      \cleaders\copy\!boxA\vfil
      \vskip-\!dimenE \hbox to \wd\!boxB{\hss$\mkern.4mu\downarrow$\hss}
      \vskip .05556\!dimenE}}}


% ***************************
% *** BARS  (Draws bars)  ***
% ***************************
%
% ** User commands:
% ** \putbar [<XSHIFT,YSHIFT>] breadth <BREADTH> from XSTART YSTART
% **   to XEND YEND
% ** \setbars [<XSHIFT,YSHIFT>] breadth <BREADTH> baseline at XY = COORD
% **   [baselabels ([B_ORIENTATION_x,B_ORIENTATION_y] <B_XSHIFT,B_YSHIFT>)]
% **   [endlabels  ([E_ORIENTATION_x,E_ORIENTATION_y] <E_XSHIFT,E_YSHIFT>)]


% ** \putbar [<XSHIFT,YSHIFT>] breadth <BREADTH> from XSTART YSTART
% **   to XEND YEND
% ** Either XSTART=XEND or YSTART=YEND. Draws a rectangle between
% **   (XSTART,YSTART) & (XEND,YEND). The "depth" of the rectangle
% **   is determined by those two plot positions; its other
% **   dimension "breadth" is specified by the dimension BREADTH.
% ** See Subsection 4.2 of the manual.
\def\putbar#1breadth <#2> from #3 #4 to #5 #6 {%
  \!xloc=\!M{#3}\!xunit  \!xxloc=\!M{#5}\!xunit%   
  \!yloc=\!M{#4}\!yunit  \!yyloc=\!M{#6}\!yunit%           
  \!dypos=\!yyloc  \advance\!dypos by -\!yloc
  \!dimenI=#2  
% 
  \ifdim \!dimenI=\!zpt %            ** If 0 breadth
    \putrule#1from {#3} {#4} to {#5} {#6} % ** Then draw line
  \else %                            ** Else, put in a rectangle
    \let\!MBar=\!M%                  ** save current c/d mode
    \!setdimenmode %                 ** go into dimension mode
    \divide\!dimenI 2
    \ifdim \!dypos=\!zpt             
      \advance \!yloc -\!dimenI %    ** Equal y coordinates
      \advance \!yyloc \!dimenI
    \else
      \advance \!xloc -\!dimenI %    ** Equal x coordinates
      \advance \!xxloc \!dimenI
    \fi
    \putrectangle#1corners at {\!xloc} {\!yloc} and {\!xxloc} {\!yyloc}
    \let\!M=\!MBar %                 ** restore c/d mode
  \fi
  \ignorespaces}


% ** \setbars [<XSHIFT,YSHIFT>] breadth <BREADTH> baseline at XY = COORD
% **   [baselabels ([B_ORIENTATION_x,B_ORIENTATION_y] <B_XSHIFT,B_YSHIFT>)]
% **   [endlabels  ([E_ORIENTATION_x,E_ORIENTATION_y] <E_XSHIFT,E_YSHIFT>)]
% ** This command puts PiCTeX into the bar graph drawing mode described
% **   in Subsection 4.4 of the manual.
\def\setbars#1breadth <#2> baseline at #3 = #4 {%
  \edef\!barshift{#1}%
  \edef\!barbreadth{#2}%
  \edef\!barorientation{#3}%
  \edef\!barbaseline{#4}%
  \def\!bardobaselabel{\!bardoendlabel}%
  \def\!bardoendlabel{\!barfinish}%
  \let\!drawcurve=\!barcurve
  \!setbars}
\def\!setbars{%
  \futurelet\!nextchar\!!setbars}
\def\!!setbars{%
  \if b\!nextchar
    \def\!!!setbars{\!setbarsbget}%
  \else 
    \if e\!nextchar
      \def\!!!setbars{\!setbarseget}%
    \else
      \def\!!!setbars{\relax}%
    \fi
  \fi
  \!!!setbars}
\def\!setbarsbget baselabels (#1) {%
  \def\!barbaselabelorientation{#1}%
  \def\!bardobaselabel{\!!bardobaselabel}%
  \!setbars}
\def\!setbarseget endlabels (#1) {%
  \edef\!barendlabelorientation{#1}%
  \def\!bardoendlabel{\!!bardoendlabel}%
  \!setbars}

% ** \!barcurve
% ** Draws a bargraph with preset values of barshift, barbreadth,
% ** barorientation (x or y) and barbaseline (coordinate)
\def\!barcurve #1 #2 {%
  \if y\!barorientation
    \def\!basexarg{#1}%
    \def\!baseyarg{\!barbaseline}%
  \else
    \def\!basexarg{\!barbaseline}%
    \def\!baseyarg{#2}%
  \fi
  \expandafter\putbar\!barshift breadth <\!barbreadth> from {\!basexarg}
    {\!baseyarg} to {#1} {#2}
  \def\!endxarg{#1}%
  \def\!endyarg{#2}%
  \!bardobaselabel}

\def\!!bardobaselabel "#1" {%
  \put {#1}\!barbaselabelorientation{} at {\!basexarg} {\!baseyarg}
  \!bardoendlabel}
 
\def\!!bardoendlabel "#1" {%
  \put {#1}\!barendlabelorientation{} at {\!endxarg} {\!endyarg}
  \!barfinish}

\def\!barfinish{%
  \!ifnextchar/{\!finish}{\!barcurve}}
 

% ********************************
% *** BOXES (Draws rectangles) ***
% ********************************
%
% ** User commands:
% **   \putrectangle [<XSHIFT,YSHIFT>] corners at  XCOORD1 YCOORD1
% **     and  XCOORD2 YCOORD2 
% **   \shaderectangleson
% **   \shaderectanglesoff
% **   \frame [<SEPARATION>] {TEXT}
% **   \rectangle <WIDTH> <HEIGHT>
%
%
% **  \putrectangle [<XSHIFT,YSHIFT>] corners at XCOORD1 YCOORD1 
% **    and  XCOORD2 YCOORD2 
% **  Draws a rectangle with corners at (X1,Y1), (X2,Y1), (X1,Y2), (X2,Y2)
% **  Lines have thickness \linethickness, and overlap at the corners.
% **  The optional field  <XSHIFT,YSHIFT>  functions as with a \put command.
% **  See Subsection 4.2 of the manual.
\def\putrectangle{%
  \!ifnextchar<{\!putrectangle}{\!putrectangle<\!zpt,\!zpt> }}
\def\!putrectangle<#1,#2> corners at #3 #4 and #5 #6 {%
%
% ** get locations
  \!xone=\!M{#3}\!xunit  \!xtwo=\!M{#5}\!xunit%   
  \!yone=\!M{#4}\!yunit  \!ytwo=\!M{#6}\!yunit%           
  \ifdim \!xtwo<\!xone
    \!dimenI=\!xone  \!xone=\!xtwo  \!xtwo=\!dimenI
  \fi
  \ifdim \!ytwo<\!yone
    \!dimenI=\!yone  \!yone=\!ytwo  \!ytwo=\!dimenI
  \fi
  \!dimenI=#1\relax  \advance\!xone\!dimenI  \advance\!xtwo\!dimenI
  \!dimenI=#2\relax  \advance\!yone\!dimenI  \advance\!ytwo\!dimenI
  \let\!MRect=\!M%                  ** save current coord/dimen mode
  \!setdimenmode
%
% ** shade rectangle if appropriate
  \!shaderectangle
%
% ** draw horizontal edges
  \!dimenI=.5\linethickness
  \advance \!xone  -\!dimenI%       ** adjust x-location to overlap corners
  \advance \!xtwo   \!dimenI%       ** ditto
  \putrule from {\!xone} {\!yone} to {\!xtwo} {\!yone} 
  \putrule from {\!xone} {\!ytwo} to {\!xtwo} {\!ytwo} 
%
% ** draw vertical edges
  \advance \!xone   \!dimenI%       ** restore original x-values
  \advance \!xtwo  -\!dimenI% 
  \advance \!yone  -\!dimenI%       ** adjust y-location to overlap corners
  \advance \!ytwo   \!dimenI%       ** ditto
  \putrule from {\!xone} {\!yone} to {\!xone} {\!ytwo} 
  \putrule from {\!xtwo} {\!yone} to {\!xtwo} {\!ytwo} 
%
  \let\!M=\!MRect%                  ** restore coord/dimen mode
  \ignorespaces}
 
% ** \shaderectangleson 
% **   Subsequent rectangles will be shaded according to 
% **   the current shading pattern.  Affects \putrectangle, \putbar,
% **   \frame, \sethistograms, and \setbars. See Subsection 7.5 of the manual.
\def\shaderectangleson{%     
  \def\!shaderectangle{\!!shaderectangle}%
  \ignorespaces}
% ** \shaderectanglesoff 
% **    Suppresses  \shaderectangleson.  The default.
\def\shaderectanglesoff{%
  \def\!shaderectangle{}%
  \ignorespaces}

\shaderectanglesoff
 
% ** The following internal routine shades the current rectangle, when
% **   \!shaderectangle = \!!shaderectangle . 
\def\!!shaderectangle{%
  \!dimenA=\!xtwo  \advance \!dimenA -\!xone
  \!dimenB=\!ytwo  \advance \!dimenB -\!yone
  \ifdim \!dimenA<\!dimenB
    \!startvshade (\!xone,\!yone,\!ytwo)
    \!lshade      (\!xtwo,\!yone,\!ytwo)
  \else
    \!starthshade (\!yone,\!xone,\!xtwo)
    \!lshade      (\!ytwo,\!xone,\!xtwo)
  \fi
  \ignorespaces}
  
% ** \frame [<SEPARATION>] {TEXT}
% ** Draws a frame of thickness linethickness about the box enclosing
% **   TEXT; the frame is separated from the box by a distance of
% **   SEPARATION.  The result is an hbox with the same baseline as TEXT.
% **   If <SEPARATION> is omitted, you get the effect of <0pt>.
% ** See Subsection 4.2 of the manual.
\def\frame{%
  \!ifnextchar<{\!frame}{\!frame<\!zpt> }}
\long\def\!frame<#1> #2{%
  \beginpicture
    \setcoordinatesystem units <1pt,1pt> point at 0 0 
    \put {#2} [Bl] at 0 0 
    \!dimenA=#1\relax
    \!dimenB=\!wd \advance \!dimenB \!dimenA
    \!dimenC=\!ht \advance \!dimenC \!dimenA
    \!dimenD=\!dp \advance \!dimenD \!dimenA
    \let\!MFr=\!M
    \!setdimenmode
    \putrectangle corners at {-\!dimenA} {-\!dimenD} and {\!dimenB} {\!dimenC}
    \!setcoordmode
    \let\!M=\!MFr
  \endpicture
  \ignorespaces}
 
% ** \rectangle <WIDTH> <HEIGHT>
% ** Constructs a rectangle of width WIDTH and heigth HEIGHT. 
% ** See Subsection 4.2 of the manual.
\def\rectangle <#1> <#2> {%
  \setbox0=\hbox{}\wd0=#1\ht0=#2\frame {\box0}}


% *********************************************
% ***  CURVES  (Upper level \plot commands) ***
% *********************************************
%
% ** User commands
% **   \plot  DATA  /
% **   \plot  "FILE NAME"
% **   \setquadratic
% **   \setlinear
% **   \sethistograms
% **   \vshade  ...
% **   \hshade  ...

% \plot: multi-purpose command. Draws histograms, bar graphs, piecewise-linear
% or piecewise quadratic curves, depending on the setting of \!drawcurve.
% See Subsections 4.3-4.5, 5.1, 5.2 of the manual.
\def\plot{%
  \!ifnextchar"{\!plotfromfile}{\!drawcurve}}
\def\!plotfromfile"#1"{%
  \expandafter\!drawcurve \input #1 /}

% Command to set piecewise quadratic mode
% See Subsections 5.1, 7.3, and 7.4 of the manual.
\def\setquadratic{%
  \let\!drawcurve=\!qcurve
  \let\!!Shade=\!!qShade
  \let\!!!Shade=\!!!qShade}

% Command to set piecewise linear mode
% See Subsections 5.1, 7.3, and 7.4 of the manual.
\def\setlinear{%
  \let\!drawcurve=\!lcurve
  \let\!!Shade=\!!lShade
  \let\!!!Shade=\!!!lShade}

% Command to set histogram mode
% See Subsection 4.3 of the manual.
\def\sethistograms{%
  \let\!drawcurve=\!hcurve}

% Commands to cycle through list of coordinates in piecewise quadratic 
% interpolation mode
\def\!qcurve #1 #2 {%
  \!start (#1,#2)
  \!Qjoin}
\def\!Qjoin#1 #2 #3 #4 {%
  \!qjoin (#1,#2) (#3,#4)             % \!qjoin  is defined in QUADRATIC
  \!ifnextchar/{\!finish}{\!Qjoin}}

% Commands to cycle through list of coordinates in piecewise linear 
% interpolation mode
\def\!lcurve #1 #2 {%
  \!start (#1,#2)
  \!Ljoin}
\def\!Ljoin#1 #2 {%
  \!ljoin (#1,#2)                    % \!ljoin  is defined in LINEAR
  \!ifnextchar/{\!finish}{\!Ljoin}}

\def\!finish/{\ignorespaces}

% Command to cycle through list of coordinates in histogram mode
\def\!hcurve #1 #2 {%
  \edef\!hxS{#1}%
  \edef\!hyS{#2}%
  \!hjoin}
\def\!hjoin#1 #2 {%
  \putrectangle corners at {\!hxS} {\!hyS} and {#1} {#2}
  \edef\!hxS{#1}%
  \!ifnextchar/{\!finish}{\!hjoin}}


% \vshade: See Subsection 7.3 of the manual.
\def\vshade #1 #2 #3 {%
  \!startvshade (#1,#2,#3)
  \!Shadewhat}

% \hshade: See Subsection 7.4 of the manual.
\def\hshade #1 #2 #3 {%
  \!starthshade (#1,#2,#3)
  \!Shadewhat}

% Commands to cycle through coordinates and optional "edge effect"
% fields while shading.
\def\!Shadewhat{%
  \futurelet\!nextchar\!Shade}
\def\!Shade{%
  \if <\!nextchar
    \def\!nextShade{\!!Shade}%
  \else
    \if /\!nextchar
      \def\!nextShade{\!finish}%
    \else
      \def\!nextShade{\!!!Shade}%
    \fi
  \fi
  \!nextShade}
\def\!!lShade<#1> #2 #3 #4 {%
  \!lshade <#1> (#2,#3,#4)                 % \!lshade is defined in SHADING
  \!Shadewhat}
\def\!!!lShade#1 #2 #3 {%
  \!lshade (#1,#2,#3)
  \!Shadewhat} 
\def\!!qShade<#1> #2 #3 #4 #5 #6 #7 {%
  \!qshade <#1> (#2,#3,#4) (#5,#6,#7)      % \!qshade is defined in SHADING
  \!Shadewhat}
\def\!!!qShade#1 #2 #3 #4 #5 #6 {%
  \!qshade (#1,#2,#3) (#4,#5,#6)
  \!Shadewhat} 

% ** Set default interpolation mode
\setlinear


%  ********************************************
%  *** DASHPATTERNS (Sets up dash patterns) ***
%  ********************************************

%  **  User commands:
%  **    \setdashpattern <DIMEN1,DIMEN2,DIMEN3,...>
%  **    \setdots <INTRADOT_DISTANCE>
%  **    \setdotsnear <INTRADOT_DISTANCE> for <ARC LENGTH>
%  **    \setdashes <DASH/SKIP_DISTANCE>
%  **    \setdashesnear <DASH/SKIP_DISTANCE> for <ARC LENGTH>
%  **    \setsolid
%  **    \findlength {CURVE CMDS}
 
%  **  Internal commands:
%  **    \!dashingon
%  **    \!dashingoff
 
%  **  Dash patterns are specified by a balanced token list whose complete
%  **    expansion has the form: DIMEN1,DIMEN2,DIMEN3,DIMEN4,... ; this produces
%  **    an arc of length DIMEN1, a skip of length DIMEN2, an arc of length
%  **    DIMEN3, a skip of length DIMEN4, ... .  Any number of DIMEN values may
%  **    be given. The pattern is repeated as many times (perhaps fractional)
%  **    as necessary to draw the curve. 
%  **  A dash pattern remains in effect until it is overridden by a call to
%  **    \setdashpattern, or to \setdots, \setdotsnear ... , \setdashes, 
%  **    \setdashesnear ... , or \setsolid.
%  **  Solid lines are the default.
 
 
%  **  \def\setdashpattern <DIMEN1,DIMEN2,DIMEN3,...>
%  **  The following routine converts a balanced list of tokens whose
%  **  complete expansion has the form  DIMEN1,DIMEN2, ... , DIMENk  into 
%  **  three list macros that are used in drawing dashed rules and curves:
%  **    !Flist:   \!Rule{DIMEN1}\!Skip{DIMEN2}\!Rule{DIMEN3}\!Skip{DIMEN4} ...
%  **    !Blist:   ...\!Skip{DIMEN4}\!Rule{DIMEN3}\!Skip{DIMEN2}\!Rule{DIMEN1}
%  **    !UDlist:  \\{DIMEN1}\\{DIMEN2}\\{DIMEN3}\\{DIMEN4} ...;
%  **  calculates \!leaderlength := DIMEN1 + ... + DIMENk; and
%  **  sets the curve drawing routines to dash mode.
%  **  Those lists are used by the curve drawing routines.
%  **  Dimenj ... may be given as an explicit dimension (e.g., 5pt), or
%  **  as an expression involving a dimension register (e.g., -2.5\dimen0).
%  **  See Subsection 6.2 of the manual
\def\setdashpattern <#1>{%
  \def\!Flist{}\def\!Blist{}\def\!UDlist{}%
  \!countA=0
  \!ecfor\!item:=#1\do{%
    \!dimenA=\!item\relax
    \expandafter\!rightappend\the\!dimenA\withCS{\\}\to\!UDlist%
    \advance\!countA  1
    \ifodd\!countA
      \expandafter\!rightappend\the\!dimenA\withCS{\!Rule}\to\!Flist%
      \expandafter\!leftappend\the\!dimenA\withCS{\!Rule}\to\!Blist%
    \else 
      \expandafter\!rightappend\the\!dimenA\withCS{\!Skip}\to\!Flist%
      \expandafter\!leftappend\the\!dimenA\withCS{\!Skip}\to\!Blist%
    \fi}%
  \!leaderlength=\!zpt
  \def\!Rule##1{\advance\!leaderlength  ##1}%
  \def\!Skip##1{\advance\!leaderlength  ##1}%
  \!Flist%
  \ifdim\!leaderlength>\!zpt 
  \else
    \def\!Flist{\!Skip{24in}}\def\!Blist{\!Skip{24in}}\ignorespaces
    \def\!UDlist{\\{\!zpt}\\{24in}}\ignorespaces
    \!leaderlength=24in
  \fi
  \!dashingon}   
 
 
%  **  \!dashingon  -- puts the curve drawing routines into dash mode
%  **  \!dashingoff -- puts the curve drawing routines into solid mode
%  **  These are internal commands, invoked by \setdashpattern and \setsolid
\def\!dashingon{%
  \def\!advancedashing{\!!advancedashing}%
  \def\!drawlinearsegment{\!lineardashed}%
  \def\!puthline{\!putdashedhline}%
  \def\!putvline{\!putdashedvline}%
%  \def\!putsline{\!putdashedsline}%
  \ignorespaces}% 
\def\!dashingoff{%
  \def\!advancedashing{\relax}%
  \def\!drawlinearsegment{\!linearsolid}%
  \def\!puthline{\!putsolidhline}%
  \def\!putvline{\!putsolidvline}%
%  \def\!putsline{\!putsolidsline}%
  \ignorespaces}
 
 
%  **  \setdots <LENGTH>  --  sets up a dot/skip pattern where dot (actually
%  **    the current plotsymbol) is plunked down once for every LENGTH 
%  **    traveled along the curve.  LENGTH defaults to 5pt.
%  **    See Subsection 6.1 of the manual.
\def\setdots{%
  \!ifnextchar<{\!setdots}{\!setdots<5pt>}}
\def\!setdots<#1>{%
  \!dimenB=#1\advance\!dimenB -\plotsymbolspacing
  \ifdim\!dimenB<\!zpt
    \!dimenB=\!zpt
  \fi
\setdashpattern <\plotsymbolspacing,\!dimenB>}
 
% ** \setdotsnear <LENGTH> for <ARC LENGTH>
% ** sets up a dot pattern where the dots are approximately LENGTH apart,
% ** the total length of the pattern is ARC LENGTH, and the pattern
% ** begins and ends with a dot. See Subsection 6.3 of the manual.
\def\setdotsnear <#1> for <#2>{%
  \!dimenB=#2\relax  \advance\!dimenB -.05pt  
  \!dimenC=#1\relax  \!countA=\!dimenC 
  \!dimenD=\!dimenB  \advance\!dimenD .5\!dimenC  \!countB=\!dimenD
  \divide \!countB  \!countA
  \ifnum 1>\!countB 
    \!countB=1
  \fi
  \divide\!dimenB  \!countB
  \setdots <\!dimenB>}
 
%  **  \setdashes <LENGTH>  --  sets up a dash/skip pattern where the dash
%  **    and the skip are each of length LENGTH (the dash is formed by
%  **    plunking down the current plotsymbol over an arc of length LENGTH
%  **    and so may actually be longer than LENGTH.  LENGTH defaults to 5pt.
%  **    See Subsection 6.1 of the manual.
\def\setdashes{%
  \!ifnextchar<{\!setdashes}{\!setdashes<5pt>}}
\def\!setdashes<#1>{\setdashpattern <#1,#1>}
 
% ** \setdashesnear ...
% ** Like \setdotsnear; the pattern begins and ends with a dash.
% ** See Subsection 6.3 of the manual.
\def\setdashesnear <#1> for <#2>{%
  \!dimenB=#2\relax  
  \!dimenC=#1\relax  \!countA=\!dimenC 
  \!dimenD=\!dimenB  \advance\!dimenD .5\!dimenC  \!countB=\!dimenD
  \divide \!countB  \!countA
  \ifodd \!countB 
  \else 
    \advance \!countB  1
  \fi
  \divide\!dimenB  \!countB
  \setdashes <\!dimenB>}
 
%  **  \setsolid  --  puts the curve drawing routines in "solid line" mode,
%  **    the default mode.  See Subsection 6.1 of the manual.
\def\setsolid{%
  \def\!Flist{\!Rule{24in}}\def\!Blist{\!Rule{24in}}%  
  \def\!UDlist{\\{24in}\\{\!zpt}}%
  \!dashingoff}  
\setsolid

%  **  \findlength {CURVE CMDS}
%  **  PiCTeX executes the \start, \ljoin, and \qjoin cmds comprising
%  **  CURVE CMDS without plotting anything, but stashes the length
%  **  of the phantom curve away in \totalarclength.
%  **  See Subsection 6.3 of the manual.
\def\findlength#1{%
  \begingroup
    \setdashpattern <0pt, \maxdimen>
    \setplotsymbol ({})  
    \dontsavelinesandcurves
    #1%
  \endgroup
  \ignorespaces}


% *************************************************************
% *** DIVISION  (Does long division of dimension registers) ***
% *************************************************************
 
% ** User command:
% **   \Divide {DIVIDEND} by {DIVISOR} forming {RESULT}
  
% ** Internal command
% **   \!divide{DIVIDEND}{DIVISOR}{RESULT}
 
% **  \!divide DIVIDEND [by] DIVISOR [to get] ANSWER
% **  Divides the dimension DIVIDEND by the dimension DIVISOR, placing the 
% **  quotient in the dimension register ANSWER.  Values are understood to 
% **  be in points.  E.g.  12.5pt/1.4pt=8.92857pt.
% **  Quotient is accurate to 1/65536pt=2**[-16]pt
% **  |DIVISOR| should be < 2048pt (about 28 inches).
\def\!divide#1#2#3{%
  \!dimenB=#1%                      **  dimB  holds current remainder (r)
  \!dimenC=#2%                      **  dimC  holds divisor (d)
  \!dimenD=\!dimenB%                **  dimD  holds quotient q=r/d for this 
  \divide \!dimenD \!dimenC%        **    step, in units of scaled pts
  \!dimenA=\!dimenD%                **  dimA  eventually holds answer (a)
  \multiply\!dimenD \!dimenC%       **  r <-- r - dq
  \advance\!dimenB -\!dimenD%       **  First step complete. Have integer part
%                                   **  of a, and corresponding remainder.
  \!dimenD=\!dimenC%                **  Temporarily use dimD to hold |d|
    \ifdim\!dimenD<\!zpt \!dimenD=-\!dimenD 
  \fi
  \ifdim\!dimenD<64pt%              **  Branch on the magnitude of |d|
    \!divstep[\!tfs]\!divstep[\!tfs]%
  \else 
    \!!divide
  \fi
  #3=\!dimenA\ignorespaces}

% **  The following code handles divisors  d  with 
% **    (1)  .88in =  64pt <= d <  256pt =  3.54in
% **    (2) 3.54in = 256pt <= d < 2048pt = 28.34in
% **  Anything bigger than that may result in an overflow condition.
% **  For our purposes, we should never even see case (2).
\def\!!divide{%
  \ifdim\!dimenD<256pt
    \!divstep[64]\!divstep[32]\!divstep[32]%
  \else 
    \!divstep[8]\!divstep[8]\!divstep[8]\!divstep[8]\!divstep[8]%
    \!dimenA=2\!dimenA
  \fi}
 
 
% **  The following macro does the real long division work.
\def\!divstep[#1]{%                 **  #1 = "B"
  \!dimenB=#1\!dimenB%              **  r <-- B*r
  \!dimenD=\!dimenB%                **  dimD  holds quotient q=r/d for this 
    \divide \!dimenD by \!dimenC%   **    step, in units of scaled pts
  \!dimenA=#1\!dimenA%              **  a <-- B*a + q
    \advance\!dimenA by \!dimenD%
  \multiply\!dimenD by \!dimenC%    **  r <-- r - dq
    \advance\!dimenB by -\!dimenD}
 
% **  \Divide:  See Subsection 9.3 of the manual.
\def\Divide <#1> by <#2> forming <#3> {%
  \!divide{#1}{#2}{#3}}


% *********************************************
% *** ELLIPSES (Draws ellipses and circles) ***
% *********************************************
 
% ** User commands
% **   \ellipticalarc  axes ratio A:B  DEGREES degrees from XSTART YSTART 
% **      center at XCENTER YCENTER 
% **   \circulararc DEGREES degrees from XSTART YSTART 
% **      center at XCENTER YCENTER 
 
% ** Internal command
% **   \!sinandcos{32*ANGLE in radians}{32*SIN}{32*COS}
 
 
% **   \ellipticalarc  axes ratio A:B  DEGREES degrees from XSTART YSTART 
% **      center at XCENTER YCENTER 
% **    Draws a elliptical arc starting at the coordinate point (XSTART,YSTART).
% **    The center of the ellipse of which the arc is a segment is at 
% **      (XCENTER,YCENTER).
% **    The arc extends through an angle of DEGREES degrees (may be + or -).
% **    A:B is the ratio of the length of the xaxis to the length of
% **      the yaxis of the ellipse
% **    Sqrt{[(XSTART-XCENTER)/A]**2 + [(YSTART-YCENTER)/B]**2}
% **      must be < 512pt (about 7in).
% **    Doesn't modify the dimensions (ht, dp, wd) of the PiCture under
% **      construction.
 
% ** \circulararc  --  See Subsection 5.3 of the manual.
\def\circulararc{%
  \ellipticalarc axes ratio 1:1 }

% ** \ellipticalarc  --  See Subsection 5.3 of the manual.
\def\ellipticalarc axes ratio #1:#2 #3 degrees from #4 #5 center at #6 #7 {%
  \!angle=#3pt\relax%                    ** get angle
  \ifdim\!angle>\!zpt 
    \def\!sign{}%                        ** counterclockwise
  \else 
    \def\!sign{-}\!angle=-\!angle%       ** clockwise
  \fi
  \!xxloc=\!M{#6}\!xunit%                ** convert CENTER to dimension
  \!yyloc=\!M{#7}\!yunit     
  \!xxS=\!M{#4}\!xunit%                  ** get STARTing point on rim of ellipse
  \!yyS=\!M{#5}\!yunit
  \advance\!xxS -\!xxloc%                ** make center of ellipse (0,0)
  \advance\!yyS -\!yyloc
  \!divide\!xxS{#1pt}\!xxS %             ** scale point on ellipse to point on 
  \!divide\!yyS{#2pt}\!yyS %                 corresponding circle
%
  \let\!MC=\!M%                          ** save current c/d mode
  \!setdimenmode%                        ** go into dimension mode
%
  \!xS=#1\!xxS  \advance\!xS\!xxloc
  \!yS=#2\!yyS  \advance\!yS\!yyloc
  \!start (\!xS,\!yS)%
  \!loop\ifdim\!angle>14.9999pt%         ** draw in major portion of ellipse 
    \!rotate(\!xxS,\!yyS)by(\!cos,\!sign\!sin)to(\!xxM,\!yyM) 
    \!rotate(\!xxM,\!yyM)by(\!cos,\!sign\!sin)to(\!xxE,\!yyE)
    \!xM=#1\!xxM  \advance\!xM\!xxloc  \!yM=#2\!yyM  \advance\!yM\!yyloc
    \!xE=#1\!xxE  \advance\!xE\!xxloc  \!yE=#2\!yyE  \advance\!yE\!yyloc
    \!qjoin (\!xM,\!yM) (\!xE,\!yE)
    \!xxS=\!xxE  \!yyS=\!yyE 
    \advance \!angle -15pt
  \repeat
  \ifdim\!angle>\!zpt%                   ** complete remaining arc, if any
    \!angle=100.53096\!angle%            ** convert angle to radians, divide
    \divide \!angle 360 %                **   by 2, and multiply by 32
    \!sinandcos\!angle\!!sin\!!cos%      ** get 32*sin & 32*cos
    \!rotate(\!xxS,\!yyS)by(\!!cos,\!sign\!!sin)to(\!xxM,\!yyM) 
    \!rotate(\!xxM,\!yyM)by(\!!cos,\!sign\!!sin)to(\!xxE,\!yyE)
    \!xM=#1\!xxM  \advance\!xM\!xxloc  \!yM=#2\!yyM  \advance\!yM\!yyloc
    \!xE=#1\!xxE  \advance\!xE\!xxloc  \!yE=#2\!yyE  \advance\!yE\!yyloc
    \!qjoin (\!xM,\!yM) (\!xE,\!yE)
  \fi
%
  \let\!M=\!MC%                          ** restore c/d mode
  \ignorespaces}%                        **   if appropriate
 
 
%  ** \!rotate(XREG,YREG)by(32cos,32sin)to(XXREG,YYREG)
%  ** rotates (XREG,YREG) by angle with specfied scaled cos & sin to
%  ** (XXREG,YYREG).  Uses \!dimenA & \!dimenB as scratch registers.
\def\!rotate(#1,#2)by(#3,#4)to(#5,#6){% 
  \!dimenA=#3#1\advance \!dimenA -#4#2%   ** Rcos(x+t)=Rcosx*cost - Rsinx*sint
  \!dimenB=#3#2\advance \!dimenB  #4#1%   ** Rsin(x+t)=Rsinx*cost + Rcosx*sint
  \divide \!dimenA 32  \divide \!dimenB 32 
  #5=\!dimenA  #6=\!dimenB
  \ignorespaces}
\def\!sin{4.17684}%                       ** 32*sin(pi/24) (pi/24=7.5deg)
\def\!cos{31.72624}%                      ** 32*cos(pi/24)
 
 
%  ** \!sinandcos{32*ANGLE in radians}{\SINCS}{\COSCS}
%  **   Computes the 32*sine and 32*cosine of a small ANGLE expressed in 
%  **   radians/32 and puts these values in the replacement texts of 
%  **   \SINCS and \COSCS
\def\!sinandcos#1#2#3{%
 \!dimenD=#1%                **  angle is expressed in radians/32: 1pt = 1/32rad
 \!dimenA=\!dimenD%          **  dimA will eventually contain 32sin(angle)in pts
 \!dimenB=32pt%              **  dimB will eventually contain 32cos(angle)in pts
 \!removept\!dimenD\!value%  **  get value of 32*angle, without "pt"
 \!dimenC=\!dimenD%          **  holds 32*angle**i/i! in pts
 \!dimenC=\!value\!dimenC \divide\!dimenC by 64 %   ** now 32*angle**2/2
 \advance\!dimenB by -\!dimenC%                     ** 32-32*angle**2/2
 \!dimenC=\!value\!dimenC \divide\!dimenC by 96 %   ** now 32*angle**3/3!
 \advance\!dimenA by -\!dimenC%                     ** now 32*(angle-angle**3/6)
 \!dimenC=\!value\!dimenC \divide\!dimenC by 128 %  ** now 32*angle**4/4!
 \advance\!dimenB by \!dimenC%
 \!removept\!dimenA#2%                              ** set 32*sin(angle)
 \!removept\!dimenB#3%                              ** set 32*cos(angle)
 \ignorespaces}


% *****************************************************************
% ***  RULES  (Draws rules, i.e., horizontal & vertical lines)  ***
% *****************************************************************

% **  User command:
% **    \putrule [<XDIMEN,YDIMEN>] from  XCOORD1 YCOORD1 
% **      to  XCOORD2 YCOORD2 

% **  Internal commands:
% **    \!puthline [<XDIMEN,YDIMEN>]    (h = horizontal)
% **      Set by dashpat to either: \!putsolidhline  or \!putdashedhline
% **    \!putvline [<XDIMEN,YDIMEN>]    (v = vertical)
% **      Either:  \!putsolidvline  or  \!putdashedvline


% **  \putrule [<XDIMEN,YDIMEN>] from XCOORD1 YCOORD1
% **    to XCOORD2 YCOORD2
% **  Draws a rule -- dashed or solid depending on the current dash pattern --
% **    from (X1,Y1) to (X2,Y2).  Uses TEK's  \hrule & \vrule & \leaders  
% **    constructions to handle horizontal & vertical lines efficiently both
% **    in terms of execution time and space in the DVI file.  
% **  See Subsection 4.1 of the manual.
\def\putrule#1from #2 #3 to #4 #5 {%
  \!xloc=\!M{#2}\!xunit  \!xxloc=\!M{#4}\!xunit%   
  \!yloc=\!M{#3}\!yunit  \!yyloc=\!M{#5}\!yunit%           
  \!dxpos=\!xxloc  \advance\!dxpos by -\!xloc
  \!dypos=\!yyloc  \advance\!dypos by -\!yloc
%
  \ifdim\!dypos=\!zpt
    \def\!!Line{\!puthline{#1}}\ignorespaces
  \else
    \ifdim\!dxpos=\!zpt
      \def\!!Line{\!putvline{#1}}\ignorespaces
    \else 
       \def\!!Line{}
    \fi
  \fi
  \let\!ML=\!M%           ** save current coord\dimen mode
  \!setdimenmode%         ** express locations in dimens
  \!!Line%
  \let\!M=\!ML%           ** restore previous c/d mode
  \ignorespaces}
 
 
% **  \!putsolidhline [<XDIMEN,YDIMEN>]
% **  Place horizontal solid line
\def\!putsolidhline#1{%
  \ifdim\!dxpos>\!zpt 
    \put{\!hline\!dxpos}#1[l] at {\!xloc} {\!yloc}
  \else 
    \put{\!hline{-\!dxpos}}#1[l] at {\!xxloc} {\!yyloc}
  \fi
  \ignorespaces}
 
% **  \!putsolidvline [shifted <XDIMEN,YDIMEN>]
% **  Place vertical solid line
\def\!putsolidvline#1{%
  \ifdim\!dypos>\!zpt 
    \put{\!vline\!dypos}#1[b] at {\!xloc} {\!yloc}
  \else 
    \put{\!vline{-\!dypos}}#1[b] at {\!xxloc} {\!yyloc}
  \fi
  \ignorespaces}
 
\def\!hline#1{\hbox to #1{\leaders \hrule height\linethickness\hfill}}
\def\!vline#1{\vbox to #1{\leaders \vrule width\linethickness\vfill}}
 
 
% **  \!putdashedhline [<XDIMEN,YDIMEN>]
% **  Place dashed horizontal line
\def\!putdashedhline#1{%
  \ifdim\!dxpos>\!zpt 
    \!DLsetup\!Flist\!dxpos
    \put{\hbox to \!totalleaderlength{\!hleaders}\!hpartialpattern\!Rtrunc}
      #1[l] at {\!xloc} {\!yloc} 
  \else 
    \!DLsetup\!Blist{-\!dxpos}
    \put{\!hpartialpattern\!Ltrunc\hbox to \!totalleaderlength{\!hleaders}}
      #1[r] at {\!xloc} {\!yloc} 
  \fi
  \ignorespaces}
 
% **  \!putdashedhline [<XDIMEN,YDIMEN>]
% **  Place dashed vertical line
\def\!putdashedvline#1{%
  \!dypos=-\!dypos%            ** vertical leaders go from top to bottom
  \ifdim\!dypos>\!zpt 
    \!DLsetup\!Flist\!dypos 
    \put{\vbox{\vbox to \!totalleaderlength{\!vleaders}
      \!vpartialpattern\!Rtrunc}}#1[t] at {\!xloc} {\!yloc} 
  \else 
    \!DLsetup\!Blist{-\!dypos}
    \put{\vbox{\!vpartialpattern\!Ltrunc
      \vbox to \!totalleaderlength{\!vleaders}}}#1[b] at {\!xloc} {\!yloc} 
  \fi
  \ignorespaces}
 

% **  The rest of the macros in this section are subroutines used by 
% **  \!putdashedhline and \!putdashedvline.
\def\!DLsetup#1#2{%            ** Dashed-Line set up
  \let\!RSlist=#1%             ** set !Rule-Skip list
  \!countB=#2%                 ** convert rule length to integer (number of sps)
  \!countA=\!leaderlength%     ** ditto, leaderlength
  \divide\!countB by \!countA% ** number of complete leader units
  \!totalleaderlength=\!countB\!leaderlength
  \!Rresiduallength=#2%
  \advance \!Rresiduallength by -\!totalleaderlength%  \** excess length
  \!Lresiduallength=\!leaderlength
  \advance \!Lresiduallength by -\!Rresiduallength
  \ignorespaces}
 
\def\!hleaders{%
  \def\!Rule##1{\vrule height\linethickness width##1}%
  \def\!Skip##1{\hskip##1}%
  \leaders\hbox{\!RSlist}\hfill}
 
\def\!hpartialpattern#1{%
  \!dimenA=\!zpt \!dimenB=\!zpt 
  \def\!Rule##1{#1{##1}\vrule height\linethickness width\!dimenD}%
  \def\!Skip##1{#1{##1}\hskip\!dimenD}%
  \!RSlist}
 
\def\!vleaders{%
  \def\!Rule##1{\hrule width\linethickness height##1}%
  \def\!Skip##1{\vskip##1}%
  \leaders\vbox{\!RSlist}\vfill}
 
\def\!vpartialpattern#1{%
  \!dimenA=\!zpt \!dimenB=\!zpt 
  \def\!Rule##1{#1{##1}\hrule width\linethickness height\!dimenD}%
  \def\!Skip##1{#1{##1}\vskip\!dimenD}%
  \!RSlist}
 
\def\!Rtrunc#1{\!trunc{#1}>\!Rresiduallength}
\def\!Ltrunc#1{\!trunc{#1}<\!Lresiduallength}
 
\def\!trunc#1#2#3{%          
  \!dimenA=\!dimenB         
  \advance\!dimenB by #1%
  \!dimenD=\!dimenB  \ifdim\!dimenD#2#3\!dimenD=#3\fi
  \!dimenC=\!dimenA  \ifdim\!dimenC#2#3\!dimenC=#3\fi
  \advance \!dimenD by -\!dimenC}
 

%  ****************************************************************
%  ***  LINEAR ARC  (Draws straight lines -- solid and dashed)  ***
%  ****************************************************************
 
%  **  User commands
%  **    \inboundscheckoff
%  **    \inboundscheckon
   
%  **  Internal commands 
%  **    \!start (XCOORD,YCOORD)
%  **    \!ljoin (XCOORD,YCOORD)
%  **    \!drawlinearsegment  --  set by \dashpat to either
%  **      \!linearsolid  or  \!lineardashed
%  **    \!advancedashing     --  set by \dashpat to either
%  **       \relax  or  \!!advancedashing
%  **    \!plotifinbounds     --  set by \inboundscheck off/on to either
%  **       \!plot  or  \!!plotifinbounds
%  **    \!initinboundscheck  --  set by \inboundscheck off/on to either
%  **       \relax  or  \!!initinboundscheck
 
 
%  \plotsymbolspacing  ** distance between consecutive plot positions
%  \!xS                ** starting x
%  \!yS                ** starting y
%  \!xE                ** ending   x
%  \!yE                ** ending   y
%  \!xdiff             ** x_end - x_start
%  \!ydiff             ** y_end - y_start
%  \!distacross        ** how far along curve next point to be plotted is
%  \!arclength         ** approximate length of arc for current interval
%  \!downlength        ** remaining length for "pen" to be down    
%  \!uplength          ** length for "pen" to be down    
%  \!intervalno        ** counts segments to curve
%  \totalarclength     ** cumulative distance along curve
%  \!npoints           ** approximately  (arc length / plotsymbolspacing)
 
%  **  Calls -- \!Pythag, \!divide, \!plot
 
 
%  **  \!start (XCOORD,YCOORD)
%  **  Sets initial point for linearly (or quadratically) interpolated curve
\def\!start (#1,#2){%
  \!plotxorigin=\!xorigin  \advance \!plotxorigin by \!plotsymbolxshift
  \!plotyorigin=\!yorigin  \advance \!plotyorigin by \!plotsymbolyshift
  \!xS=\!M{#1}\!xunit \!yS=\!M{#2}\!yunit
  \!rotateaboutpivot\!xS\!yS
  \!copylist\!UDlist\to\!!UDlist% **\!UDlist has the form \\{dimen1}\\{dimen2}..
%                                 ** Routine will draw dashed line with pen
%                                 ** down for dimen1, up for dimen2, ...
  \!getnextvalueof\!downlength\from\!!UDlist
  \!distacross=\!zpt%             ** 1st point goes at start of curve
  \!intervalno=0 %                ** initialize interval counter
  \global\totalarclength=\!zpt%   ** initialize distance traveled along curve
  \ignorespaces}
 

%  **  \!ljoin (XCOORD,YCOORD) 
%  **  Draws a straight line starting at the last point specified
%  **    by the most recent \!start, \!ljoin, or \!qjoin, and
%  **    ending at (XCOORD,YCOORD).
\def\!ljoin (#1,#2){%
  \advance\!intervalno by 1
  \!xE=\!M{#1}\!xunit \!yE=\!M{#2}\!yunit
  \!rotateaboutpivot\!xE\!yE
  \!xdiff=\!xE \advance \!xdiff by -\!xS%**  xdiff = xE - xS
  \!ydiff=\!yE \advance \!ydiff by -\!yS%**  ydiff = yE - yS
  \!Pythag\!xdiff\!ydiff\!arclength%     **  arclength = sqrt(xdiff**2+ydiff**2) 
  \global\advance \totalarclength by \!arclength%
  \!drawlinearsegment%   ** set by dashpat to \!linearsolid or \!lineardashed
  \!xS=\!xE \!yS=\!yE%   ** shift ending points to starting points
  \ignorespaces}
 
 
% **  The following routine is used to draw a "solid" line between (xS,yS)
% **  and (xE,yE).  Points are spaced nearly every  \plotsymbolspacing length
% **  along the line.  
\def\!linearsolid{%
  \!npoints=\!arclength
  \!countA=\plotsymbolspacing
  \divide\!npoints by \!countA%      ** now #pts =. arclength/plotsymbolspacing
  \ifnum \!npoints<1 
    \!npoints=1 
  \fi
  \divide\!xdiff by \!npoints
  \divide\!ydiff by \!npoints
  \!xpos=\!xS \!ypos=\!yS
%
  \loop\ifnum\!npoints>-1
    \!plotifinbounds
    \advance \!xpos by \!xdiff
    \advance \!ypos by \!ydiff
    \advance \!npoints by -1
  \repeat
  \ignorespaces}
 
 
% ** The following routine is used to draw a dashed line between (xS,yS)
% ** and (xE,yE). The dash pattern continues from the previous segment.
\def\!lineardashed{%
% **
  \ifdim\!distacross>\!arclength
    \advance \!distacross by -\!arclength  %nothing to plot in this interval
%
  \else
%
    \loop\ifdim\!distacross<\!arclength
%     ** plot point, interpolating linearly in x and y
      \!divide\!distacross\!arclength\!dimenA%  ** dimA = across/arclength
      \!removept\!dimenA\!t%  ** \!t holds value in dimA, without the "pt"
      \!xpos=\!t\!xdiff \advance \!xpos by \!xS
      \!ypos=\!t\!ydiff \advance \!ypos by \!yS
      \!plotifinbounds
      \advance\!distacross by \plotsymbolspacing
      \!advancedashing
    \repeat  
%
    \advance \!distacross by -\!arclength%    ** prepare for next interval 
  \fi
  \ignorespaces}
 
 
\def\!!advancedashing{%
  \advance\!downlength by -\plotsymbolspacing
  \ifdim \!downlength>\!zpt
  \else
    \advance\!distacross by \!downlength
    \!getnextvalueof\!uplength\from\!!UDlist
    \advance\!distacross by \!uplength
    \!getnextvalueof\!downlength\from\!!UDlist
  \fi}
 
 
% ** \inboundscheckoff & \inboundscheckon: See Subsection 5.5 of the manual.
\def\inboundscheckoff{%
  \def\!plotifinbounds{\!plot(\!xpos,\!ypos)}%
  \def\!initinboundscheck{\relax}\ignorespaces}
\def\inboundscheckon{%
  \def\!plotifinbounds{\!!plotifinbounds}%
  \def\!initinboundscheck{\!!initinboundscheck}%
  \!initinboundscheck\ignorespaces} 
\inboundscheckoff
 
% ** The following code plots the current point only if it falls in the
% ** current plotarea.  It doesn't matter if the coordinate system has
% ** changed since the plotarea was set up.  However, shifts of the plot
% ** are ignored (how the plotsymbol stands relative to its plot position is
% ** unknown anyway).
\def\!!plotifinbounds{%
  \ifdim \!xpos<\!checkleft
  \else
    \ifdim \!xpos>\!checkright
    \else
      \ifdim \!ypos<\!checkbot
      \else
         \ifdim \!ypos>\!checktop
         \else
           \!plot(\!xpos,\!ypos)
         \fi 
      \fi
    \fi
  \fi}
 
 
\def\!!initinboundscheck{%
  \!checkleft=\!arealloc     \advance\!checkleft by \!xorigin
  \!checkright=\!arearloc    \advance\!checkright by \!xorigin
  \!checkbot=\!areabloc      \advance\!checkbot by \!yorigin
  \!checktop=\!areatloc      \advance\!checktop by \!yorigin}
 

% *********************************
% *** LOGTEN  (Log_10 function) ***
% *********************************
%
% ** \!logten{X}
% ** Calculates log_10 of X.  X and LOG10(X) are in fixed point notation.
% **  X must be positive; it may have an optional `+' sign; any number
% **  of digits may be specified for X.  The absolute error in LOG10(X) is
% **  less than .0001 (probably < .00006).  That's about as good as you
% **  hope for, since TEX only operates to 5 figures after the decimal
% **  point anyway.

%  \!rootten=3.162278pt       **** These are values are set in ALLOCATIONS
%  \!tenAe=2.543275pt  (=A5)
%  \!tenAc=2.773839pt  (=A3)
%  \!tenAa=8.690286pt  (=A1)

\def\!logten#1#2{%
  \expandafter\!!logten#1\!nil
  \!removept\!dimenF#2%
  \ignorespaces}

\def\!!logten#1#2\!nil{%
  \if -#1%
    \!dimenF=\!zpt
    \def\!next{\ignorespaces}%
  \else
    \if +#1%
      \def\!next{\!!logten#2\!nil}%
    \else
      \if .#1%
        \def\!next{\!!logten0.#2\!nil}%
      \else
        \def\!next{\!!!logten#1#2..\!nil}%
      \fi
    \fi
  \fi
  \!next}

\def\!!!logten#1#2.#3.#4\!nil{%
  \!dimenF=1pt %                 ** DimF holds log10 original argument
  \if 0#1%                      
    \!!logshift#3pt %            ** Argument < 1
  \else %                        ** Argument >= 1
    \!logshift#2/%               ** Shift decimal pt as many places
    \!dimenE=#1.#2#3pt %         **   as there are figures in #2
  \fi %                          ** Now dimE holds revised X want log10 of
  \ifdim \!dimenE<\!rootten%          ** Transform X to XX between sqrt(10) 
    \multiply \!dimenE 10 %           **   and 10*sqrt(10)
    \advance  \!dimenF -1pt
  \fi
  \!dimenG=\!dimenE%                  ** dimG <- (XX + 10)
    \advance\!dimenG 10pt
  \advance\!dimenE -10pt %            ** dimE <- (XX - 10)
  \multiply\!dimenE 10 %              ** dimE = 10*(XX-10)
  \!divide\!dimenE\!dimenG\!dimenE%   ** Now dimE=10t==10*(XX-10)/(XX+10)
  \!removept\!dimenE\!t%              ** !t=10t, with "pt" removed
  \!dimenG=\!t\!dimenE%               ** dimG=100t**2
  \!removept\!dimenG\!tt%             ** !tt=100t**2, with "pt" removed
  \!dimenH=\!tt\!tenAe%               ** dimH=10*a5*(10t)**2 /100
    \divide\!dimenH 100
  \advance\!dimenH \!tenAc%           ** ditto + 10*a3
  \!dimenH=\!tt\!dimenH%              ** ditto * (10t)**2 /100
    \divide\!dimenH 100   
  \advance\!dimenH \!tenAa%           ** ditto + 10*a1
  \!dimenH=\!t\!dimenH%               ** ditto * 10t / 100
    \divide\!dimenH 100 %             ** Now dimH = log10(XX) - 1
  \advance\!dimenF \!dimenH}%         ** dimF = log10(X)

\def\!logshift#1{%
  \if #1/%
    \def\!next{\ignorespaces}%
  \else
    \advance\!dimenF 1pt 
    \def\!next{\!logshift}%
  \fi 
  \!next}
 
 \def\!!logshift#1{%
   \advance\!dimenF -1pt
   \if 0#1%
     \def\!next{\!!logshift}%
   \else
     \if p#1%
       \!dimenF=1pt
       \def\!next{\!dimenE=1p}%
     \else
       \def\!next{\!dimenE=#1.}%
     \fi
   \fi
   \!next}


% ***********************************************************
% *** PICTURES (Basic setups for PiCtures; \put commands) ***
% ***********************************************************
 
% **  User Commands:
% **    \beginpicture
% **    \endpicture    
% **    \endpicturesave <XREG,YREG>
% **    \setcoordinatesystem units <XUNIT,YUNIT> point at XREF YREF
% **    \put {OBJECT} [ORIENTATION] <XSHIFT,YSHIFT> at XCOORD YCOORD
% **    \multiput {OJBECT} [ORIENTATION] <XSHIFT,YSHIFT>) at
% **      XCOORD YCOORD
% **      *NUMBER_OF_TIMES DXCOORD DYCOORD  /
% **    \accountingon
% **    \accountingoff
% **    \stack [ORIENTATION] <LEADING> {LIST OF ITEMS}
% **    \lines [ORIENTATION] {LINES}
% **    \Lines [ORIENTATION] {LINES}
% **    \setdimensionmode
% **    \setcoordinatemode
% **    \Xdistance
% **    \Ydistance
  
% **  Internal commands:
% **    \!setputobject{OBJECT}{[ORIENTATION]<XSHIFT,YSHIFT>}
% **    \!dimenput{OBJECT}[ORIENTATION]<XSHIFT,YSHIFT>(XDIMEN,YDIMEN)
% **    \!setdimenmode
% **    \!setcoordmode
% **    \!ifdimenmode
% **    \!ifcoordmode
  
 
% **  \beginpicture
% **  \endpicture
% **  \endpicturesave <XREG,YREG>
% **    \beginpicture ... \endpicture  creates an hbox.  Objects are
% **    placed in this box using the \put command and the like (see below).
% **    The location of an object is specified in terms of coordinate system(s)
% **    established by \setcoordinatesystem.  Each coordinate system (there
% **    might be just one) specifies the length of 1 horizontal unit, the length
% **    of 1 vertical unit, and the coordinates of a "reference point".  The
% **    reference points of various coordinate systems will be in the same
% **    physical location.  The macros keep track of the size of the objects
% **    and their locations. The resulting hbox is the smallest hbox which
% **    encloses all the objects, and whose TEK reference point is the point
% **    on the left edge of the box closest vertically to the PICTEX reference
% **    point. Using \endpicturesave, you can (globally) save the distance TEK's
% **    reference point is to the right (respectively, up from) PICTEX's 
% **    reference point in the dimension register \XREG (respectively \YREG). 
% **    You can then \put the picture OBJECT into a larger picture so that its
% **    reference point is at (XCOORD,YCOORD) with the command
% **      \put {picture OBJECT} [Bl] <\XREG, \YREG> at  XCOORD YCOORD 

% **  \beginpicture : See Subsection 1.1 of the manual.
\def\beginpicture{%
  \setbox\!picbox=\hbox\bgroup%
  \!xleft=\maxdimen  
  \!xright=-\maxdimen
  \!ybot=\maxdimen
  \!ytop=-\maxdimen}
 
% **  \endpicture : See Subsection 1.1 of the manual.
\def\endpicture{%
  \ifdim\!xleft=\maxdimen%  ** check if nothing was put in picbox
    \!xleft=\!zpt \!xright=\!zpt \!ybot=\!zpt \!ytop=\!zpt 
  \fi
  \global\!Xleft=\!xleft \global\!Xright=\!xright
  \global\!Ybot=\!ybot \global\!Ytop=\!ytop
  \egroup%
  \ht\!picbox=\!Ytop  \dp\!picbox=-\!Ybot
  \ifdim\!Ybot>\!zpt
  \else 
    \ifdim\!Ytop<\!zpt
      \!Ybot=\!Ytop
    \else
      \!Ybot=\!zpt
    \fi
  \fi
  \hbox{\kern-\!Xleft\lower\!Ybot\box\!picbox\kern\!Xright}}
 
% **  \endpicturesave : See Subsection 8.4 of the manual.
\def\endpicturesave <#1,#2>{%
  \endpicture \global #1=\!Xleft \global #2=\!Ybot \ignorespaces}
 
 
% **   \setcoordinatesystem units <XUNIT,YUNIT> 
% **     point at XREF YREF  
% **   Each of `units <XUNIT,YUNIT>' and `point at XREF YREF' 
% **     are optional.
% **   Unit lengths must be given in dimensions (e.g., <10pt,1in>).
% **     Default unit lengths are 1pt, 1pt, or previous unit lengths.
% **   Reference point is specified in current units (e.g., 3 5 ). 
% **     Default reference point is 0 0 , or previous reference point.
% **   Unit lengths and reference points obey TEX's scoping rules.
% **   See Subsection 1.2 of the manual.
\def\setcoordinatesystem{%
  \!ifnextchar{u}{\!getlengths }
    {\!getlengths units <\!xunit,\!yunit>}}
\def\!getlengths units <#1,#2>{%
  \!xunit=#1\relax
  \!yunit=#2\relax
  \!ifcoordmode 
    \let\!SCnext=\!SCccheckforRP
  \else
    \let\!SCnext=\!SCdcheckforRP
  \fi
  \!SCnext}
\def\!SCccheckforRP{%
  \!ifnextchar{p}{\!cgetreference }
    {\!cgetreference point at {\!xref} {\!yref} }}
\def\!cgetreference point at #1 #2 {%
  \edef\!xref{#1}\edef\!yref{#2}%
  \!xorigin=\!xref\!xunit  \!yorigin=\!yref\!yunit  
  \!initinboundscheck % ** See linear.tex
  \ignorespaces}
\def\!SCdcheckforRP{%
  \!ifnextchar{p}{\!dgetreference}%
    {\ignorespaces}}
\def\!dgetreference point at #1 #2 {%
  \!xorigin=#1\relax  \!yorigin=#2\relax
  \ignorespaces}

 
%  ** \put {OBJECT} [XY] <XDIMEN,YDIMEN> at (XCOORD,YCOORD)
%  **   `[XY]' and `<XDIMEN,YDIMEN>' are optional.
%  **   First OBJECT is placed in an hbox (the "objectbox") and then a
%  **     "reference point" is assigned to the objectbox as follows:
%  **     [1] first, the reference point is taken to be the center of the box;
%  **     [2] next, centering is overridden by the specifications
%  **           X=l -- reference point along the left edge of the objectbox
%  **           X=r -- reference point along the right edge of the objectbox
%  **           Y=b -- reference point along the bottom edge of the objectbox
%  **           Y=B -- reference point along the Baseline of the objectbox
%  **           Y=t -- reference point along the top edge of the objectbox;
%  **     [3] finally the reference point is shifted left by XDIMEN, down
%  **           by YDIMEN  (both default to 0pt).
%  **   The objectbox is placed within PICBOX with its reference point at  
%  **     (XCOORD,YCOORD). 
%  **   If OBJECT is a saved box, say  box0, you have to write
%  **     \put{\box0}...   or  \put{\copy0}...
%  **   The objectbox is void after the put.
%  **   See Subsection 2.1 of the manual.
\long\def\put#1#2 at #3 #4 {%
  \!setputobject{#1}{#2}%
  \!xpos=\!M{#3}\!xunit  \!ypos=\!M{#4}\!yunit  
  \!rotateaboutpivot\!xpos\!ypos%
  \advance\!xpos -\!xorigin  \advance\!xpos -\!xshift
  \advance\!ypos -\!yorigin  \advance\!ypos -\!yshift
  \kern\!xpos\raise\!ypos\box\!putobject\kern-\!xpos%
  \!doaccounting\ignorespaces}
 
%  **   \multiput etc.  Like  \put.  The objectbox is not voided until the
%  **     termininating /, and is placed repeatedly with:
%  **     XCOORD YCOORD -- the objectbox is put down with its reference point
%  **       at (XCOORD,YCOORD);
%  **     *N DXCOORD DYCOORD -- each of N times the current
%  **       (xcoord,ycoord) is incremented by (DXCOORD,DYCOORD), and the
%  **       objectbox is put down with its reference point at (xcoord,ycoord)
%  **       (This specification has to follow an XCOORD YCOORD pair)
%  **     See Subsection 2.2 of the manual.
\long\def\multiput #1#2 at {%
  \!setputobject{#1}{#2}%
  \!ifnextchar"{\!putfromfile}{\!multiput}}
\def\!putfromfile"#1"{%
  \expandafter\!multiput \input #1 /}
\def\!multiput{%
  \futurelet\!nextchar\!!multiput}
\def\!!multiput{%
  \if *\!nextchar
    \def\!nextput{\!alsoby}%
  \else
    \if /\!nextchar
      \def\!nextput{\!finishmultiput}%
    \else
      \def\!nextput{\!alsoat}%
    \fi
  \fi
  \!nextput}
\def\!finishmultiput/{%
  \setbox\!putobject=\hbox{}%
  \ignorespaces}
 
%  **   \!alsoat XCOORD YCOORD 
%  **     The objectbox is put down with reference point at XCOORD,YCOORD
\def\!alsoat#1 #2 {%
  \!xpos=\!M{#1}\!xunit  \!ypos=\!M{#2}\!yunit  
  \!rotateaboutpivot\!xpos\!ypos%
  \advance\!xpos -\!xorigin  \advance\!xpos -\!xshift
  \advance\!ypos -\!yorigin  \advance\!ypos -\!yshift
  \kern\!xpos\raise\!ypos\copy\!putobject\kern-\!xpos%
  \!doaccounting
  \!multiput}
 
% **   \!alsoby*N DXCOORD DYCOORD
% **     N times, the current (XCOORD,YCOORD) is advanced by (DXCOORD,DYCOORD),
% **     and the current (shifted, oriented) OBJECT is put down.
\def\!alsoby*#1 #2 #3 {%
  \!dxpos=\!M{#2}\!xunit \!dypos=\!M{#3}\!yunit 
  \!rotateonly\!dxpos\!dypos
  \!ntemp=#1%
  \!!loop\ifnum\!ntemp>0
    \advance\!xpos by \!dxpos  \advance\!ypos by \!dypos
    \kern\!xpos\raise\!ypos\copy\!putobject\kern-\!xpos%
    \advance\!ntemp by -1
  \repeat
  \!doaccounting 
  \!multiput}
 
% **  \accountingoff : Suspends PiCTeX's accounting of the aggregate
% **    size of the picture box.
% **  \accounting on : Reinstates accounting.
% **  See Subsection 8.2 of the manual.
\def\accountingon{\def\!doaccounting{\!!doaccounting}\ignorespaces}
\def\accountingoff{\def\!doaccounting{}\ignorespaces}
\accountingon
\def\!!doaccounting{%
  \!xtemp=\!xpos  
  \!ytemp=\!ypos
  \ifdim\!xtemp<\!xleft 
     \!xleft=\!xtemp 
  \fi
  \advance\!xtemp by  \!wd 
  \ifdim\!xright<\!xtemp 
    \!xright=\!xtemp
  \fi
  \advance\!ytemp by -\!dp
  \ifdim\!ytemp<\!ybot  
    \!ybot=\!ytemp
  \fi
  \advance\!ytemp by  \!dp
  \advance\!ytemp by  \!ht 
  \ifdim\!ytemp>\!ytop  
    \!ytop=\!ytemp  
  \fi}
 
\long\def\!setputobject#1#2{%
  \setbox\!putobject=\hbox{#1}%
  \!ht=\ht\!putobject  \!dp=\dp\!putobject  \!wd=\wd\!putobject
  \wd\!putobject=\!zpt
  \!xshift=.5\!wd   \!yshift=.5\!ht   \advance\!yshift by -.5\!dp
  \edef\!putorientation{#2}%
  \expandafter\!SPOreadA\!putorientation[]\!nil%
  \expandafter\!SPOreadB\!putorientation<\!zpt,\!zpt>\!nil\ignorespaces}
 
\def\!SPOreadA#1[#2]#3\!nil{\!etfor\!orientation:=#2\do\!SPOreviseshift}
 
\def\!SPOreadB#1<#2,#3>#4\!nil{\advance\!xshift by -#2\advance\!yshift by -#3}
 
\def\!SPOreviseshift{%
  \if l\!orientation 
    \!xshift=\!zpt
  \else 
    \if r\!orientation 
      \!xshift=\!wd
    \else 
      \if b\!orientation
        \!yshift=-\!dp
      \else 
        \if B\!orientation 
          \!yshift=\!zpt
        \else 
          \if t\!orientation 
            \!yshift=\!ht
          \fi 
        \fi
      \fi
    \fi
  \fi}
 
 
%  **  \!dimenput{OBJECT} <XDIMEN,YDIMEN> [XY] (XLOC,YLOC)
%  **    This is an internal put routine, similar to \put, except that
%  **    XLOC=distance right from reference point, YLOC=distance up from
%  **    reference point. XLOC and YLOC are dimensions, so this routine
%  **    is completely independent of the current coordinate system. 
%  **    This routine does NOT do ROTATIONS.
\long\def\!dimenput#1#2(#3,#4){%
  \!setputobject{#1}{#2}%
  \!xpos=#3\advance\!xpos by -\!xshift
  \!ypos=#4\advance\!ypos by -\!yshift
  \kern\!xpos\raise\!ypos\box\!putobject\kern-\!xpos%
  \!doaccounting\ignorespaces}
 
 
%  ** The following macros permit the picture drawing routines to be used 
%  ** either in the default "coordinate mode", or in "dimension mode".
%  **   In coordinate mode  \!M(1.5,\!xunit)    expands to  1.5\!xunit
%  **   In dimension  mode  \!M(1.5pt,\!xunit)  expands to  1.5pt
%  ** Dimension mode is useful in coding macros.
%  ** Any special purpose picture macro that sets dimension mode should 
%  ** reset coordinate mode before completion.
%  ** See Subsection 9.2 of the manual.
\def\!setdimenmode{%
  \let\!M=\!M!!\ignorespaces}
\def\!setcoordmode{%
  \let\!M=\!M!\ignorespaces}
\def\!ifcoordmode{%
  \ifx \!M \!M!}
\def\!ifdimenmode{%
  \ifx \!M \!M!!}
\def\!M!#1#2{#1#2} 
\def\!M!!#1#2{#1}
\!setcoordmode
\let\setdimensionmode=\!setdimenmode
\let\setcoordinatemode=\!setcoordmode

%  ** \Xdistance{XCOORD}, \Ydistance{YCOORD}  are the horizontal and
%  **   vertical distances from the origin (0,0) to the point
%  **   (XCOORD,YCOORD)  in the current coordinate system.
%  ** See Subsection 9.2 of the manual.
\def\Xdistance#1{%
  \!M{#1}\!xunit
  \ignorespaces}
\def\Ydistance#1{%
  \!M{#1}\!yunit
  \ignorespaces}

% ** The following macros -- \stack, \line, and \Lines -- are useful for
% **   annotating PiCtures. They can be used outside the \beginpicture ...
% **   \endpicture environment.

% ** \stack [POSITIONING] <LEADING> {VALUESLIST}
% ** Builds a vertical stack of the values in VALUESLIST. Values in
% ** VALUESLIST are separated by commas.  In the resulting stack, values are
% ** centered by default, and positioned flush left (right) if 
% ** POSITIONING = l (r).  Values are separated vertically by LEADING,
% ** which defaults to \stackleading.
% ** See Subsection 2.3 of the manual.
\def\stack{%
  \!ifnextchar[{\!stack}{\!stack[c]}}
\def\!stack[#1]{%
  \let\!lglue=\hfill \let\!rglue=\hfill
  \expandafter\let\csname !#1glue\endcsname=\relax
  \!ifnextchar<{\!!stack}{\!!stack<\stackleading>}}
\def\!!stack<#1>#2{%
  \vbox{\def\!valueslist{}\!ecfor\!value:=#2\do{%
    \expandafter\!rightappend\!value\withCS{\\}\to\!valueslist}%
    \!lop\!valueslist\to\!value
    \let\\=\cr\lineskiplimit=\maxdimen\lineskip=#1%
    \baselineskip=-1000pt\halign{\!lglue##\!rglue\cr \!value\!valueslist\cr}}%
  \ignorespaces}

% ** \lines [POSITIONING] {LINES}
% ** Builds a vertical array of the lines in LINES. Each line in LINES
% ** is terminated by a \cr.  In the resulting array, lines are
% ** centered by default, and positioned flush left (right) if 
% ** POSITIONING = l (r).  The lines in the array are subject to TeX's
% ** usual spacing rules: in particular the baselines are ordinarily an equal
% ** distance apart. The baseline of the array is the baseline of the
% ** the bottom line.
% ** See Subsection 2.3 of the manual.
\def\lines{%
  \!ifnextchar[{\!lines}{\!lines[c]}}
\def\!lines[#1]#2{%
  \let\!lglue=\hfill \let\!rglue=\hfill
  \expandafter\let\csname !#1glue\endcsname=\relax
  \vbox{\halign{\!lglue##\!rglue\cr #2\crcr}}%
  \ignorespaces}

% ** \Lines [POSITIONING] {LINES}
% ** Like \lines, but the baseline of the array is the baseline of the
% ** top line.  See Subsection 2.3 of the manual.
\def\Lines{%
  \!ifnextchar[{\!Lines}{\!Lines[c]}}
\def\!Lines[#1]#2{%
  \let\!lglue=\hfill \let\!rglue=\hfill
  \expandafter\let\csname !#1glue\endcsname=\relax
  \vtop{\halign{\!lglue##\!rglue\cr #2\crcr}}%
  \ignorespaces}


% *********************************************
% *** PLOTTING (Things to do with plotting) ***
% *********************************************
 
% **  User commands
% **    \setplotsymbol ({PLOTSYMBOL} [ORIENTATION] <XSHIFT,YSHIFT>)
% **    \savelinesandcurves on "FILE_NAME"
% **    \dontsavelinesandcurves
% **    \writesavefile {MESSAGE}
% **    \replot {FILE_NAME}
 
% **  Internal command
% **    \!plot(XDIMEN,YDIMEN)
 
% **  \setplotsymbol ({PLOTSYMBOL} [ ] < , >)
% **  Save PLOTSYMBOL away in an hbox for use with curve plotting routines
% **  See Subsection 5.2 of the manual.
\def\setplotsymbol(#1#2){%
  \!setputobject{#1}{#2}
  \setbox\!plotsymbol=\box\!putobject%
  \!plotsymbolxshift=\!xshift 
  \!plotsymbolyshift=\!yshift 
  \ignorespaces}
 
\setplotsymbol({\fiverm .})%       ** initialize plotsymbol
 
 
% **  \!plot is either \!!plot (when no lines and curves are being saved) or
% **                   \!!!plot (when   lines and curves are being saved)
 
% **  \!!plot(XDIMEN,YDIMEN)
% **  Places the current plotsymbol a horizontal distance=XDIMEN-xorigin 
% **    and a vertical distance=YDIMEN-yorigin from the current
% **    reference point.  
\def\!!plot(#1,#2){%
  \!dimenA=-\!plotxorigin \advance \!dimenA by #1%    ** over
  \!dimenB=-\!plotyorigin \advance \!dimenB by #2%    ** up
  \kern\!dimenA\raise\!dimenB\copy\!plotsymbol\kern-\!dimenA%
  \ignorespaces}
 
% **  \!!!plot(XDIMEN,YDIMEN)
% **  Like \!!plot, but also saves the plot location in units of 
% **    scaled point, on file `replotfile'
\def\!!!plot(#1,#2){%
  \!dimenA=-\!plotxorigin \advance \!dimenA by #1%    ** over
  \!dimenB=-\!plotyorigin \advance \!dimenB by #2%    ** up
  \kern\!dimenA\raise\!dimenB\copy\!plotsymbol\kern-\!dimenA%
  \!countE=\!dimenA
  \!countF=\!dimenB
  \immediate\write\!replotfile{\the\!countE,\the\!countF.}%
  \ignorespaces}


% ** \savelinesandcurves on "FILE_NAME"
% **   Switch to save locations used for plotting lines and curves
% **   (No advantage in saving locations for solid lines; however
% **   replotting curve locations speeds things up by a factor of about 4. 
% ** \dontsavelinesandcurves
% **   Terminates \savelinesandcurves. The default.
% ** See Subsection 5.6 of the manual.
\def\savelinesandcurves on "#1" {%
  \immediate\closeout\!replotfile
  \immediate\openout\!replotfile=#1%
  \let\!plot=\!!!plot}

\def\dontsavelinesandcurves {%
  \let\!plot=\!!plot}
\dontsavelinesandcurves

% ** \writesavefile {MESSAGE}
% ** The message is preceded by a "%", so that it won't interfere
% ** with replotting.
% ** See Subsection 5.6 of the manual.
{\catcode`\%=11\xdef\!Commentsignal{%}}
\def\writesavefile#1 {%
  \immediate\write\!replotfile{\!Commentsignal #1}%
  \ignorespaces}

% ** \replot "FILE_NAME"
% **   Replots the locations saved earlier under \savelinesandcurves
% **   on "FILE_NAME"
% ** See Subsection 5.6 of the manual.
\def\replot"#1" {%
  \expandafter\!replot\input #1 /}
\def\!replot#1,#2. {%
  \!dimenA=#1sp
  \kern\!dimenA\raise#2sp\copy\!plotsymbol\kern-\!dimenA
  \futurelet\!nextchar\!!replot}
\def\!!replot{%
  \if /\!nextchar 
    \def\!next{\!finish}%
  \else
    \def\!next{\!replot}%
  \fi
  \!next}
% **************************************************
% ***  PYTHAGORAS  (Euclidean distance function) ***
% **************************************************

% ** User command:
% **   \placehypotenuse for <dimension1> and <dimension2> in <register> 

% ** Internal command:
% **   \!Pythag{X}{Y}{Z}
% **     Input X,Y are dimensions, or dimension registers.
% **     Output Z == sqrt(X**2+Y**2) must be a dimension register.
% **     Assumes that |X|+|Y| < 2048pt (about 28in).
 
% ** Without loss of generality, suppose  x>0, y>0.  Put s = x+y,
% **   z = sqrt(x**2+y**2). Then  z = s*f,  where  f = sqrt(t**2 + (1-t)**2)
% **   = sqrt((1+tau**2)/2), where  t = x/s  and  tau = 2(t-1/2) .
 
% ** Uses the \!divide macro (which uses registers \!dimenA--\!dimenD.
% ** Uses the \!removept macro   (e.g., 123.45pt --> 123.45)
% ** Uses registers \!dimenE--\!dimenI.
\def\!Pythag#1#2#3{%
  \!dimenE=#1\relax                                     
  \ifdim\!dimenE<\!zpt 
    \!dimenE=-\!dimenE 
  \fi%                                            ** dimE = |x|
  \!dimenF=#2\relax
  \ifdim\!dimenF<\!zpt 
    \!dimenF=-\!dimenF 
  \fi%                                            ** dimF = |y|
  \advance \!dimenF by \!dimenE%                  ** dimF = s = |x|+|y|
  \ifdim\!dimenF=\!zpt 
    \!dimenG=\!zpt%                               ** dimG = z = sqrt(x**2+y**2)
  \else 
    \!divide{8\!dimenE}\!dimenF\!dimenE%          ** now dimE = 8t = (8|x|)/s
    \advance\!dimenE by -4pt%                     ** 8tau = (8t-4)*2
      \!dimenE=2\!dimenE%                         **   (tau = 2*t - 1)
    \!removept\!dimenE\!!t%                       ** 8tau, without "pt"
    \!dimenE=\!!t\!dimenE%                        ** (8tau)**2, in pts
    \advance\!dimenE by 64pt%                     ** u = [64 + (8tau)**2]/2
    \divide \!dimenE by 2%                        **   [u = (8f)**2]
    \!dimenH=7pt%                                 ** initial guess g at sqrt(u)
    \!!Pythag\!!Pythag\!!Pythag%                  ** 3 iterations give sqrt(u)
    \!removept\!dimenH\!!t%                       ** 8f=sqrt(u), without "pt"
    \!dimenG=\!!t\!dimenF%                        ** z = (8f)*s/8
    \divide\!dimenG by 8
  \fi
  #3=\!dimenG
  \ignorespaces}

\def\!!Pythag{%                                   ** Newton-Raphson for sqrt
  \!divide\!dimenE\!dimenH\!dimenI%               ** v = u/g
  \advance\!dimenH by \!dimenI%                   ** g <-- (g + u/g)/2
    \divide\!dimenH by 2}

% **  \placehypotenuse for <XI> and <ETA> in <ZETA>
% **  See Subsection 9.3 of the manual.
\def\placehypotenuse for <#1> and <#2> in <#3> {%
  \!Pythag{#1}{#2}{#3}}


% **********************************************
% *** QUADRATIC ARC  (Draws a quadratic arc) ***
% **********************************************
 
% **  Internal command
% **    \!qjoin (XCOORD1,YCOORD1) (XCOORD2,YCOORD2)
 
% **  \!qjoin (XCOORD1,YCOORD1) (XCOORD2,YCOORD2)
% **  Draws an arc starting at the (last) point specified by the most recent
% **  \!qjoin, or \!ljoin, or \!start  and passing through (X_1,Y_1), (X_2,Y_2).
% **  Uses quadratic interpolation in both  x  and  y:  
% **    x(t), 0 <= t <= 1, interpolates  x_0, x_1, x_2  at  t=0, .5, 1
% **    y(t), 0 <= t <= 1, interpolates  y_0, y_1, y_2  at  t=0, .5, 1
 
\def\!qjoin (#1,#2) (#3,#4){%
  \advance\!intervalno by 1
  \!ifcoordmode
    \edef\!xmidpt{#1}\edef\!ymidpt{#2}%
  \else
    \!dimenA=#1\relax \edef\!xmidpt{\the\!dimenA}%
    \!dimenA=#2\relax \edef\!xmidpt{\the\!dimenA}%
  \fi
  \!xM=\!M{#1}\!xunit  \!yM=\!M{#2}\!yunit   \!rotateaboutpivot\!xM\!yM
  \!xE=\!M{#3}\!xunit  \!yE=\!M{#4}\!yunit   \!rotateaboutpivot\!xE\!yE
%
% ** Find coefficients for x(t)=a_x + b_x*t + c_x*t**2
  \!dimenA=\!xM  \advance \!dimenA by -\!xS%   ** dimA = I = xM - xS
  \!dimenB=\!xE  \advance \!dimenB by -\!xM%   ** dimB = II = xE-xM
  \!xB=3\!dimenA \advance \!xB by -\!dimenB%   ** b=3I-II
  \!xC=2\!dimenB \advance \!xC by -2\!dimenA%  ** c=2(II-I)
%
% ** Find coefficients for y(t)=y_x + b_y*t + c_y*t**2
  \!dimenA=\!yM  \advance \!dimenA by -\!yS%   
  \!dimenB=\!yE  \advance \!dimenB by -\!yM%  
  \!yB=3\!dimenA \advance \!yB by -\!dimenB%  
  \!yC=2\!dimenB \advance \!yC by -2\!dimenA% 
%
% ** Use Simpson's rule to calculate arc length over [0,1/2]:
% **   arc length = 1/2[1/6 f(0) + 4/6 f(1/4) + 1/6 f(1/2)]
% ** with f(t) = sqrt(x'(t)**2 + y'(t)**2).
  \!xprime=\!xB  \!yprime=\!yB%          ** x'(t) = b + 2ct
  \!dxprime=.5\!xC  \!dyprime=.5\!yC%    ** dt=1/4 ==> dx'(t) = c/2
  \!getf \!midarclength=\!dimenA
  \!getf \advance \!midarclength by 4\!dimenA
  \!getf \advance \!midarclength by \!dimenA
  \divide \!midarclength by 12
%
% ** Get arc length over [0,1].
  \!arclength=\!dimenA
  \!getf \advance \!arclength by 4\!dimenA
  \!getf \advance \!arclength by \!dimenA
  \divide \!arclength by 12%             ** Now have arc length over [1/2,1]
  \advance \!arclength by \!midarclength
  \global\advance \totalarclength by \!arclength
%
%
% ** Check to see if there's anything to plot in this interval
  \ifdim\!distacross>\!arclength 
    \advance \!distacross by -\!arclength%   ** nothing 
%
  \else
    \!initinverseinterp%  ** initialize for inverse interpolation on arc length
    \loop\ifdim\!distacross<\!arclength%     ** loop over points on arc 
      \!inverseinterp%    ** find  t  such that arc length[0,t] = distacross,
%                         **   using inverse quadratic interpolation
%                         ** now evaluate x(t)=(c*t + b)*t + a
      \!xpos=\!t\!xC \advance\!xpos by \!xB
        \!xpos=\!t\!xpos \advance \!xpos by \!xS
%                                             ** evaluate y(t)
      \!ypos=\!t\!yC \advance\!ypos by \!yB
        \!ypos=\!t\!ypos \advance \!ypos by \!yS
      \!plotifinbounds%                       ** plot point if in bounds
      \advance\!distacross \plotsymbolspacing%** advance arc length for next pt
      \!advancedashing%                       ** see "linear"
    \repeat  
%
    \advance \!distacross by -\!arclength%    ** prepare for next interval 
  \fi
%
  \!xS=\!xE%              ** shift ending points to starting points
  \!yS=\!yE
  \ignorespaces}
 
 
% ** \!getf -- Calculates sqrt(x'(t)**2 + y'(t)**2) and advances
% **   x'(t) and y'(t)
\def\!getf{\!Pythag\!xprime\!yprime\!dimenA%
  \advance\!xprime by \!dxprime
  \advance\!yprime by \!dyprime}
 
 
% ** \!initinverseinterp -- initializes for inverse quadratic interpolation
% ** of arc length provided  1/3 < midarclength/arclength < 2/3; otherwise
% ** initializes for inverse linear interpolation.
\def\!initinverseinterp{%
  \ifdim\!arclength>\!zpt
    \!divide{8\!midarclength}\!arclength\!dimenE% ** dimE=8w=8r/s, where  r 
%                                               **  = midarclength, s=arclength
% **  Test for  w  out of range:  w<1/3  or w>2/3
    \ifdim\!dimenE<\!wmin \!setinverselinear
    \else 
      \ifdim\!dimenE>\!wmax \!setinverselinear
      \else%                                    ** w  in range: initialize
        \def\!inverseinterp{\!inversequad}\ignorespaces
%
% **     Calculate the coefficients  \!beta  and  \!gamma  of the quadratic
% **                    t = \!beta*v + \!gamma*v**2
% **     taking the values  t=0, 1/2, 1  at  v=0, w==r/s, 1  respectively:
% **        \!beta = (1/2 - w**2)/[w(1-w)] 
% **        \!gamma = 1 - beta.
%
         \!removept\!dimenE\!Ew%           **  8w, without "pt"
         \!dimenF=-\!Ew\!dimenE%           **  -(8w)**2
         \advance\!dimenF by 32pt%         **  32 - (8w)**2
         \!dimenG=8pt 
         \advance\!dimenG by -\!dimenE%    **  8 - 8w
         \!dimenG=\!Ew\!dimenG%            **  (8w)*(8-8w)
         \!divide\!dimenF\!dimenG\!beta%   **  beta = (32-(8w)**2)/(8w(8-8w))
%                                          **       = (1/2 - w**2)/(w(1-w))
         \!gamma=1pt
         \advance \!gamma by -\!beta%      **  gamma = 1-beta
      \fi%       ** end of the \ifdim\!dimenE>\!wmax
    \fi%         ** end of the \ifdim\!dimenE<\!wmin
  \fi%           ** end of the \ifdim\!arclength>\!zpt
  \ignorespaces}
 
 
% ** For 0 <= t <= 1, let AL(t) = arclength[0,t]/arclength[0,1]; note
% ** AL(0)=0, AL(1/2)=midarclength/arclength, AL(1)=1.  This routine
% ** calculates an approximation to AL^{-1}(distance across/arclength),
% ** using the assumption that AL^{-1} is quadratic.  Specifically, 
% ** it finds  t  such that
% **    AL^{-1}(v) =. t = v*(\!beta + \!gamma*v)
% ** where  \!beta  and  \!gamma  are set by \!initinv, and where
% ** v=distance across/arclength
\def\!inversequad{%
  \!divide\!distacross\!arclength\!dimenG%   ** dimG = v = distacross/arclength
  \!removept\!dimenG\!v%                     ** v, without "pt"
  \!dimenG=\!v\!gamma%                       ** gamma*v
  \advance\!dimenG by \!beta%                ** beta + gamma*v
  \!dimenG=\!v\!dimenG%                      ** t = v*(beta + gamma*v)
  \!removept\!dimenG\!t}%                    ** t, without "pt"
 
 
% ** When  w <= 1/3  or  w >= 2/3, the following routine writes (using
% ** plain TEK's \wlog command) a warning message on the user's log file,
% ** and initializes for inverse linear interpolation on arc length.
\def\!setinverselinear{%
  \def\!inverseinterp{\!inverselinear}%
  \divide\!dimenE by 8 \!removept\!dimenE\!t
  \!countC=\!intervalno \multiply \!countC 2
  \!countB=\!countC     \advance \!countB -1
  \!countA=\!countB     \advance \!countA -1
  \wlog{\the\!countB th point (\!xmidpt,\!ymidpt) being plotted 
    doesn't lie in the}%
  \wlog{ middle third of the arc between the \the\!countA th 
    and \the\!countC th points:}%
  \wlog{ [arc length \the\!countA\space to \the\!countB]/[arc length 
    \the \!countA\space to \the\!countC]=\!t.}%
  \ignorespaces}
 
% **  Inverse linear interpolation
\def\!inverselinear{% 
  \!divide\!distacross\!arclength\!dimenG
  \!removept\!dimenG\!t}


% **************************************
% **  ROTATIONS  (Handles rotations) ***
% **************************************
 
% ** User commands
% **   \startrotation [by COS_OF_ANGLE SIN_OF_ANGLE] [about XPIVOT YPIVOT]
% **   \stoprotation

% **   \startrotation [by COS_OF_ANGLE SIN_OF_ANGLE] [about XPIVOT YPIVOT]
% ** Future (XCOORD,YCOORD)'s will be rotated about (XPIVOT,YPIVOT) 
% ** by the angle with the give COS and SIN. Both fields are optional.
% ** [COS,SIN] defaults to previous value, or (1,0).
% ** (XPIVOT,YPIVOT) defaults to previous value, or (0,0)
% ** You can't change the coordinate system in the scope of a rotation.
% ** See Subsection 9.1 of the manual.
\def\startrotation{%
  \let\!rotateaboutpivot=\!!rotateaboutpivot
  \let\!rotateonly=\!!rotateonly
  \!ifnextchar{b}{\!getsincos }%
    {\!getsincos by {\!cosrotationangle} {\!sinrotationangle} }}
\def\!getsincos by #1 #2 {%
  \edef\!cosrotationangle{#1}%
  \edef\!sinrotationangle{#2}%
  \!ifcoordmode 
    \let\!ROnext=\!ccheckforpivot
  \else
    \let\!ROnext=\!dcheckforpivot
  \fi
  \!ROnext}
\def\!ccheckforpivot{%
  \!ifnextchar{a}{\!cgetpivot}%
    {\!cgetpivot about {\!xpivotcoord} {\!ypivotcoord} }}
\def\!cgetpivot about #1 #2 {%
  \edef\!xpivotcoord{#1}%
  \edef\!ypivotcoord{#2}%
  \!xpivot=#1\!xunit  \!ypivot=#2\!yunit
  \ignorespaces}
\def\!dcheckforpivot{%
  \!ifnextchar{a}{\!dgetpivot}{\ignorespaces}}
\def\!dgetpivot about #1 #2 {%
  \!xpivot=#1\relax  \!ypivot=#2\relax
  \ignorespaces}
  

% ** Following terminates rotation.
% ** See Subsection 9.1 of the manual.
\def\stoprotation{%
  \let\!rotateaboutpivot=\!!!rotateaboutpivot
  \let\!rotateonly=\!!!rotateonly
  \ignorespaces}
 
% ** !!rotateaboutpivot{XREG}{YREG}
% ** XREG <-- xpvt + cos(angle)*(XREG-xpvt) - sin(angle)*(YREG-ypvt)
% ** YREG <-- ypvt + cos(angle)*(YREG-ypvt) + sin(angle)*(XREG-xpvt)
% ** XREG,YREG are dimension registers. Can't be \!dimenA to \!dimenD
\def\!!rotateaboutpivot#1#2{%
  \!dimenA=#1\relax  \advance\!dimenA -\!xpivot
  \!dimenB=#2\relax  \advance\!dimenB -\!ypivot
  \!dimenC=\!cosrotationangle\!dimenA
    \advance \!dimenC -\!sinrotationangle\!dimenB
  \!dimenD=\!cosrotationangle\!dimenB
    \advance \!dimenD  \!sinrotationangle\!dimenA
  \advance\!dimenC \!xpivot  \advance\!dimenD \!ypivot
  #1=\!dimenC  #2=\!dimenD
  \ignorespaces}

% ** \!!rotateonly{XREG}{YREG}
% ** Like \!!rotateaboutpivot, but with a pivot of  (0,0)
\def\!!rotateonly#1#2{%
  \!dimenA=#1\relax  \!dimenB=#2\relax 
  \!dimenC=\!cosrotationangle\!dimenA
    \advance \!dimenC -\!rotsign\!sinrotationangle\!dimenB
  \!dimenD=\!cosrotationangle\!dimenB
    \advance \!dimenD  \!rotsign\!sinrotationangle\!dimenA
  #1=\!dimenC  #2=\!dimenD
  \ignorespaces}
\def\!rotsign{}
\def\!!!rotateaboutpivot#1#2{\relax}
\def\!!!rotateonly#1#2{\relax}
\stoprotation

\def\!reverserotateonly#1#2{%
  \def\!rotsign{-}%
  \!rotateonly{#1}{#2}%
  \def\!rotsign{}%
  \ignorespaces}


% **********************************
% *** SHADING  (Handles shading) ***
% **********************************

% **  User commands
% **    \setshadegrid [span <SPAN>] [point at XSHADE YSHADE] 
% **    \setshadesymbol [<LS, RS, BS, TS>] ({SHADESYMBOL}
% **      <XDIMEN,YDIMEN> [ORIENTATION])

% **  Internal commands:
% **    \!startvshade  (xS,ybS,ytS)
% **    \!starthshade  (yS,xlS,xrS)
% **    \!lshade [<LS,RS,BS,TS>]
% **       ** when shading vertically:
% **       [the region from (xS,ybS,ytS) to] (xE,ybE,ytE)
% **       ** when shading horizontally:
% **       [the region from (yS,xlS,xrS) to] (yE,xlE,xrE)
% **    \!qshade [<LS,RS,BS,TS>]
% **       ** when shading vertically:
% **       [the region from (xS,ybS,ytS) to] (xM,ybM,ytM)  (xE,ybE,ytE)
% **       ** when shading horizontally:
% **       [the region from (yS,xlS,xrS) to] (yM,xlM,xrM)  (yE,xlE,xrE)
% **    \!lattice{ANCHOR}{SPAN}{LOCATION}{INDEX}{LATTICE LOCATION}
% **    \!override{NOMINAL DIMEN}{REPLACEMENT DIMEN}{DIMEN}


% **  The shading routine can operate either in a "vertical mode" or a
% **  "horizontal mode".  In vertical mode, the region to be shaded is specified
% **  in the form
% **                 {(x,y): xl <= x <= xr  &  yb(x) <= y <= yt(x)}
% **  where  yb  and  yt  are functions of  x.  In horizontal mode, the region
% **  is specified in the form
% **                 {(x,y): yb <= y <= yt  &  xl(y) <= x <= xr(y)}.
% **  The functions  yb  and  yt  may be either both linear or both quadratic;
% **  similarly for  xl  and  xr.  A region with say, piecewise quadratic bottom
% **  and top boundaries, can be shaded by consecutive (vertical) \!qshades,
% **  proceeding from left to right.  Similarly, a region with piecewise     
% **  quadratic left and right boundaries can be shaded by consecutive
% **  (horizontal) \!qshades, proceeding from bottom to top.  More complex
% **  regions can be shaded by partitioning them into appropriate subregions,
% **  and shading those.
    
% **  Shading is accomplished by placing a user-selected shading symbol at
% **  those points of a regular grid which fall within the region to be
% **  shaded.  This region can be "shrunk" so that a largish shading symbol
% **  will not extend outside it.  Shrinking is accomplished by specifying
% **  shrinkages for the left, right, bottom, and top boundaries, in a manner
% **  discussed further below.

% **  \shades and \!joins MUST NOT be intermingled.  Finish drawing a curve
% **  before starting to shade a region, and finish shading a region before
% **  starting to draw a curve.


% **  \setshadegrid [span <SPAN>] [point at XSHADE YSHADE] 
% **  The shading symbol is placed down on the points of a grid centered
% **  at the coordinate point (XSHADE,YSHADE).  The grid points are of the
% **  form (j*SPAN,k*SPAN), with  j+k  even.  SPAN is specified
% **  as a dimension.
% **  (XSHADE,YSHADE) defaults to previous (XSHADE,YSHADE) (or (0,0) if none)
% **  SPAN defaults to previous span (or 5pt if none)
% **  See Subsection 7.2 of the manual.
\def\setshadegrid{%
  \!ifnextchar{s}{\!getspan }
    {\!getspan span <\!dshade>}}
\def\!getspan span <#1>{%
  \!dshade=#1\relax
  \!ifcoordmode 
    \let\!GRnext=\!GRccheckforAP
  \else
    \let\!GRnext=\!GRdcheckforAP
  \fi
  \!GRnext}
\def\!GRccheckforAP{%
  \!ifnextchar{p}{\!cgetanchor }
    {\!cgetanchor point at {\!xshadesave} {\!yshadesave} }}
\def\!cgetanchor point at #1 #2 {%
  \edef\!xshadesave{#1}\edef\!yshadesave{#2}%
  \!xshade=\!xshadesave\!xunit  \!yshade=\!yshadesave\!yunit
  \ignorespaces}
\def\!GRdcheckforAP{%
  \!ifnextchar{p}{\!dgetanchor}%
    {\ignorespaces}}
\def\!dgetanchor point at #1 #2 {%
  \!xshade=#1\relax  \!yshade=#2\relax
  \ignorespaces}

% **  \setshadesymbol  [<LS, RS, BS, TS>] ({SHADESYMBOL}
% **    <XDIMEN,YDIMEN> [ORIENTATION])
% **  Saves SHADESYMBOL away in an hbox for use with shading routines.
% **  A shade symbol will not be plotted if its plot position comes within
% **    distance LS of the left boundary,  RS of the right boundary,  TS of the
% **    top boundary,  BS of the bottom boundary.  These parameters have 
% **    default values that should work in most cases (see below).
% **    To override a default value, specify the replacement value
% **    in the appropriate subfield of the shrinkages field.
% **    0pt may be coded as  "z" (without the quotes).  To accept a
% **    default value, leave the field empty.  Thus
% **      [,z,,5pt]  sets  LS=default, RS=0pt, BS=default, TS=5pt .
% **    Skipping the shrinkages field accepts all the defaults.
% **  See Subsection 7.1 of the manual.
\def\setshadesymbol{%
  \!ifnextchar<{\!setshadesymbol}{\!setshadesymbol<,,,> }}

\def\!setshadesymbol <#1,#2,#3,#4> (#5#6){%
% **  set the shadesymbol
  \!setputobject{#5}{#6}%                        
  \setbox\!shadesymbol=\box\!putobject%
  \!shadesymbolxshift=\!xshift \!shadesymbolyshift=\!yshift
%
% **  set the shrinkages
  \!dimenA=\!xshift \advance\!dimenA \!smidge% ** default LS = xshift - smidge
  \!override\!dimenA{#1}\!lshrinkage%         
  \!dimenA=\!wd \advance \!dimenA -\!xshift%   ** default RS = width - xshift
    \advance\!dimenA \!smidge%                                  - smidge
    \!override\!dimenA{#2}\!rshrinkage
  \!dimenA=\!dp \advance \!dimenA \!yshift%    ** default BS = depth + yshift
    \advance\!dimenA \!smidge%                                  - smidge
    \!override\!dimenA{#3}\!bshrinkage
  \!dimenA=\!ht \advance \!dimenA -\!yshift%   ** default TS = height - yshift
    \advance\!dimenA \!smidge%                                  - smidge
    \!override\!dimenA{#4}\!tshrinkage
  \ignorespaces}
\def\!smidge{-.2pt}%

% ** \!override{NOMINAL DIMEN}{REPLACEMENT DIMEN}{DIMEN}
% ** Overrides the NOMINAL DIMEN by the REPLACEMENT DIMEN to produce DIMEN,
% ** according to the following rules:
% **   REPLACEMENT DIMEN empty: DIMEN <-- NOMINAL DIMEN
% **   REPLACEMENT DIMEN z:     DIMEN <-- 0pt
% **   otherwise:               DIMEN <-- REPLACEMENT DIMEN
% ** DIMEN must be a dimension register
\def\!override#1#2#3{%
  \edef\!!override{#2}% 
  \ifx \!!override\empty
    #3=#1\relax
  \else
    \if z\!!override
      #3=\!zpt
    \else
      \ifx \!!override\!blankz
        #3=\!zpt
      \else
        #3=#2\relax
      \fi
    \fi
  \fi
  \ignorespaces}
\def\!blankz{ z}

\setshadesymbol ({\fiverm .})%       ** initialize plotsymbol
%                                    ** \fivesy ^^B  is a small cross


% ** \!startvshade [at] (xS,ybS,ytS)
% ** Initiates vertical shading mode
\def\!startvshade#1(#2,#3,#4){%
  \let\!!xunit=\!xunit%
  \let\!!yunit=\!yunit%
  \let\!!xshade=\!xshade%
  \let\!!yshade=\!yshade%
  \def\!getshrinkages{\!vgetshrinkages}%
  \let\!setshadelocation=\!vsetshadelocation%
  \!xS=\!M{#2}\!!xunit
  \!ybS=\!M{#3}\!!yunit
  \!ytS=\!M{#4}\!!yunit
  \!shadexorigin=\!xorigin  \advance \!shadexorigin \!shadesymbolxshift
  \!shadeyorigin=\!yorigin  \advance \!shadeyorigin \!shadesymbolyshift
  \ignorespaces}
 
% ** \!starthshade [at] (yS,xlS,xrS)
% ** Initiates horizontal shading mode
\def\!starthshade#1(#2,#3,#4){%
  \let\!!xunit=\!yunit%
  \let\!!yunit=\!xunit%
  \let\!!xshade=\!yshade%
  \let\!!yshade=\!xshade%
  \def\!getshrinkages{\!hgetshrinkages}%
  \let\!setshadelocation=\!hsetshadelocation%
  \!xS=\!M{#2}\!!xunit
  \!ybS=\!M{#3}\!!yunit
  \!ytS=\!M{#4}\!!yunit
  \!shadexorigin=\!xorigin  \advance \!shadexorigin \!shadesymbolxshift
  \!shadeyorigin=\!yorigin  \advance \!shadeyorigin \!shadesymbolyshift
  \ignorespaces}


% **  \!lattice{ANCHOR}{SPAN}{LOCATION}{INDEX}{LATTICE LOCATION}
% **  Consider the lattice with points  ANCHOR + j*SPAN. This routine determines
% **  the index  k  of the smallest lattice point >= LOCATION, and sets
% **  LATTICE LOCATION = ANCHOR + k*SPAN.
% **  INDEX is assumed to be a count register, LATTICE LOCATION a dimen reg.
\def\!lattice#1#2#3#4#5{%
  \!dimenA=#1%                        ** dimA = ANCHOR
  \!dimenB=#2%                        ** dimB = SPAN  (assumed > 0pt)
  \!countB=\!dimenB%                  ** ctB  = SPAN, as a count
%
% ** Determine index of smallest lattice point >= LOCATION
  \!dimenC=#3%                        ** dimC = LOCATION
  \advance\!dimenC -\!dimenA%         ** now dimC = LOCATION-ANCHOR
  \!countA=\!dimenC%                  ** ctA = above, as a count
  \divide\!countA \!countB%           ** now ctA = desired index, if dimC <= 0
  \ifdim\!dimenC>\!zpt
    \!dimenD=\!countA\!dimenB%        ** (tentative k)*span
    \ifdim\!dimenD<\!dimenC%          ** if this is false, ctA = desired index
      \advance\!countA 1 %            ** if true, have to add 1
    \fi
  \fi
%
  \!dimenC=\!countA\!dimenB%          ** lattice location = anchor + ctA*span
    \advance\!dimenC \!dimenA
  #4=\!countA%                        ** the desired index
  #5=\!dimenC%                        ** corresponding lattice location
  \ignorespaces}


% ** \!qshade [with shrinkages] [[LS,RS,BS,TS]]
% ***** during vertical shading:
% **    [the region from (xS,ybS,ytS) to] (xM,ybM,ytM) [and] (xE,ybE,ytE)
% ** Shades the region {(x,y): xS <= x <= xE, yb(x) <= y <= yt(x)}, where 
% **   yb is the quadratic thru (xS,ybS) & (xM,ybM) & (xE,ybE)
% **   yt is the quadratic thru (xS,ytS) & (xM,ybM) & (xE,ytE)
% ** xS,ybS,ytS are either given by \!startvshade or carried over
% **   as the ending values of the immediately preceding \!qshade.
% ** For the interpretation of LS, RS, BS, & TS, see \setshadesymbol. The
% **   values set there can be overridden, for the course of this \!qshade
% **   only, in the same manner as overrides are specified for
% **   \setshadesymbol.
% ***** during horizontal shading:
% **    [the region from (yS,xlS,xrS) to] (yM,xlM,xrM) [and] (yE,xlE,xrE)
\def\!qshade#1(#2,#3,#4)#5(#6,#7,#8){%
  \!xM=\!M{#2}\!!xunit
  \!ybM=\!M{#3}\!!yunit
  \!ytM=\!M{#4}\!!yunit
  \!xE=\!M{#6}\!!xunit
  \!ybE=\!M{#7}\!!yunit
  \!ytE=\!M{#8}\!!yunit
  \!getcoeffs\!xS\!ybS\!xM\!ybM\!xE\!ybE\!ybB\!ybC%**Get coefficients B & C for
  \!getcoeffs\!xS\!ytS\!xM\!ytM\!xE\!ytE\!ytB\!ytC%**y=y0 + B(x-X0) + C(x-X0)**2
  \def\!getylimits{\!qgetylimits}%
  \!shade{#1}\ignorespaces}
 
% ** \!lshade ... (xE,ybE,ytE)
% ** This is like \!qshade, but the top and bottom boundaries are linear,
% ** rather than quadratic.
\def\!lshade#1(#2,#3,#4){%
  \!xE=\!M{#2}\!!xunit
  \!ybE=\!M{#3}\!!yunit
  \!ytE=\!M{#4}\!!yunit
  \!dimenE=\!xE  \advance \!dimenE -\!xS%   ** xE-xS
  \!dimenC=\!ytE \advance \!dimenC -\!ytS%  ** ytE-ytS
  \!divide\!dimenC\!dimenE\!ytB%            ** ytB = (ytE-ytS)/(xE-xS)
  \!dimenC=\!ybE \advance \!dimenC -\!ybS%  ** ybE-ybS
  \!divide\!dimenC\!dimenE\!ybB%            ** ybB = (ybE-ybS)/(xE-xS)
  \def\!getylimits{\!lgetylimits}%
  \!shade{#1}\ignorespaces}
 
% **  \!getcoeffs{X0}{Y0}{X1}{Y1}{X2}{Y2}{B}{C}
% **  Finds  B  and  C  such that the quadratic  y = Y0 + B(x-X0) + C(x-X0)**2
% **  passes through (X1,Y1) and (X2,Y2):  when X0=0=Y0, the formulas are:
% **                   B = S1 - X1*C,   C = (S2-S1)/X2
% **  with
% **                 S1 = Y1/X1,   S2 = (Y2-Y1)/(X2-X1).
\def\!getcoeffs#1#2#3#4#5#6#7#8{% 
  \!dimenC=#4\advance \!dimenC -#2%            ** dimC=Y1-Y0
  \!dimenE=#3\advance \!dimenE -#1%            ** dimE=X1-X0
  \!divide\!dimenC\!dimenE\!dimenF%            ** dimF=S1
  \!dimenC=#6\advance \!dimenC -#4%            ** dimC=Y2-Y1
  \!dimenH=#5\advance \!dimenH -#3%            ** dimH=X2-X1
  \!divide\!dimenC\!dimenH\!dimenG%            ** dimG=S2
  \advance\!dimenG -\!dimenF%                  ** dimG=S2-S1
  \advance \!dimenH \!dimenE%                  ** dimH=X2-X0
  \!divide\!dimenG\!dimenH#8%                  ** C=(S2-S1)/(X2-X0)
  \!removept#8\!t%                             ** C, without "pt"
  #7=-\!t\!dimenE%                             ** -C*(X1-X0)
  \advance #7\!dimenF%                         ** B=S1-C*(X1-X0)
  \ignorespaces}
 
 
\def\!shade#1{%
% ** Get LS,RS,BS,TS for this panel
  \!getshrinkages#1<,,,>\!nil% %       ** now effective LS=dimE, RS=dimF,
%                                      **   BS=dimG, TS=dimH
  \advance \!dimenE \!xS%              ** now dimE=xS+LS
  \!lattice\!!xshade\!dshade\!dimenE%  ** set parity=index of left-mst x-lattice
    \!parity\!xpos%                    **   point >= xS+LS, xpos=its location
  \!dimenF=-\!dimenF%                  ** set dimF=xE-RS
    \advance\!dimenF \!xE
%               
  \!loop\!not{\ifdim\!xpos>\!dimenF}%  ** loop over x-lattice points <= xE-RS
    \!shadecolumn%                 
    \advance\!xpos \!dshade%           ** move over to next column
    \advance\!parity 1%                ** increase index of x-point
  \repeat
%
  \!xS=\!xE%                           ** shift ending values to starting values
  \!ybS=\!ybE
  \!ytS=\!ytE
  \ignorespaces}
 
 
\def\!vgetshrinkages#1<#2,#3,#4,#5>#6\!nil{%
  \!override\!lshrinkage{#2}\!dimenE
  \!override\!rshrinkage{#3}\!dimenF
  \!override\!bshrinkage{#4}\!dimenG
  \!override\!tshrinkage{#5}\!dimenH
  \ignorespaces}
\def\!hgetshrinkages#1<#2,#3,#4,#5>#6\!nil{%
  \!override\!lshrinkage{#2}\!dimenG
  \!override\!rshrinkage{#3}\!dimenH
  \!override\!bshrinkage{#4}\!dimenE
  \!override\!tshrinkage{#5}\!dimenF
  \ignorespaces}
 
 
\def\!shadecolumn{%
  \!dxpos=\!xpos
  \advance\!dxpos -\!xS%            ** dx = x - xS
  \!removept\!dxpos\!dx%            ** ditto, without "pt"
  \!getylimits%                     ** get top and bottom y-values
  \advance\!ytpos -\!dimenH%        ** less TS
  \advance\!ybpos \!dimenG%         ** plus BS
  \!yloc=\!!yshade%                 ** get anchor point for this column
  \ifodd\!parity 
     \advance\!yloc \!dshade
  \fi
  \!lattice\!yloc{2\!dshade}\!ybpos%
    \!countA\!ypos%                 ** ypos=smallest y point for this column
  \!dimenA=-\!shadexorigin \advance \!dimenA \!xpos%      ** over
  \loop\!not{\ifdim\!ypos>\!ytpos}% ** loop over ypos <= yt(t)
    \!setshadelocation%             ** vmode: xloc=xpos, yloc=ypos 
%                                   ** hmode: xloc=ypos, yloc=xpos 
    \!rotateaboutpivot\!xloc\!yloc%
    \!dimenA=-\!shadexorigin \advance \!dimenA \!xloc%    ** over
    \!dimenB=-\!shadeyorigin \advance \!dimenB \!yloc%    ** up
    \kern\!dimenA \raise\!dimenB\copy\!shadesymbol \kern-\!dimenA
    \advance\!ypos 2\!dshade
  \repeat
  \ignorespaces}
 
\def\!qgetylimits{%
  \!dimenA=\!dx\!ytC              
  \advance\!dimenA \!ytB%         ** yt(t)=ytS + dx*(Bt + dx*Ct)
  \!ytpos=\!dx\!dimenA
  \advance\!ytpos \!ytS
  \!dimenA=\!dx\!ybC              
  \advance\!dimenA \!ybB%         ** yb(t)=ybS + dx*(Bb + dx*Cb)
  \!ybpos=\!dx\!dimenA
  \advance\!ybpos \!ybS}
 
\def\!lgetylimits{%
  \!ytpos=\!dx\!ytB%              ** yt(t)=ytS + dx*Bt
  \advance\!ytpos \!ytS
  \!ybpos=\!dx\!ybB%              ** yb(t)=ybS + dx*Bb
  \advance\!ybpos \!ybS}
 
\def\!vsetshadelocation{%         ** vmode: xloc=xpos, yloc=ypos 
  \!xloc=\!xpos
  \!yloc=\!ypos}
\def\!hsetshadelocation{%         ** hmode: xloc=ypos, yloc=xpos 
  \!xloc=\!ypos
  \!yloc=\!xpos}


% **************************************
% *** TICKS  (Draws ticks on graphs) ***
% **************************************

% ** User commands
% **   \ticksout
% **   \ticksin
% **   \gridlines
% **   \nogridlines
% **   \loggedticks
% **   \unloggesticks
% ** See Subsection 3.4 of the manual

% ** The following is an option of the \axis command
% **   ticks 
% **     [in] [out] 
% **     [long] [short] [length <LENGTH>] 
% **     [width <WIDTH>]
% **     [andacross] [butnotacross] 
% **     [logged] [unlogged] 
% **     [unlabeled] [numbered] [withvalues VALUE1 VALUE2 ... VALUEk / ]
% **     [quantity Q] [at LOC1 LOC2 ... LOCk / ] [from LOC1 to LOC2 by
% **       LOC_INCREMENT]
% ** See Subsection 3.2 of the manual for the rules.

% ** The various options of the  tick  field are processed by the
% ** \!nextkeyword  command defined below.
% ** For example, `\!nextkeyword short '  expands to  `\!ticksshort',
% ** while `\!nextkeyword withvalues' expands to `\!tickswithvalues'.

\def\!axisticks {%
  \def\!nextkeyword##1 {%
    \expandafter\ifx\csname !ticks##1\endcsname \relax
      \def\!next{\!fixkeyword{##1}}%
    \else
      \def\!next{\csname !ticks##1\endcsname}%
    \fi
    \!next}%
  \!axissetup
    \def\!axissetup{\relax}%
  \edef\!ticksinoutsign{\!ticksinoutSign}%
  \!ticklength=\longticklength
  \!tickwidth=\linethickness
  \!gridlinestatus
  \!setticktransform
  \!maketick
  \!tickcase=0
  \def\!LTlist{}%
  \!nextkeyword}

\def\ticksout{%
  \def\!ticksinoutSign{+}}
\def\ticksin{%
  \def\!ticksinoutSign{-}}
\ticksout

\def\gridlines{%
  \def\!gridlinestatus{\!gridlinestootrue}}
\def\nogridlines{%
  \def\!gridlinestatus{\!gridlinestoofalse}}
\nogridlines

\def\loggedticks{%
  \def\!setticktransform{\let\!ticktransform=\!logten}}
\def\unloggedticks{%
  \def\!setticktransform{\let\!ticktransform=\!donothing}}
\def\!donothing#1#2{\def#2{#1}}
\unloggedticks

% ** \!ticks/ : terminates read of tick options
\expandafter\def\csname !ticks/\endcsname{%
  \!not {\ifx \!LTlist\empty}
    \!placetickvalues
  \fi
  \def\!tickvalueslist{}%
  \def\!LTlist{}%
  \expandafter\csname !axis/\endcsname}

\def\!maketick{%
  \setbox\!boxA=\hbox{%
    \beginpicture
      \!setdimenmode
      \setcoordinatesystem point at {\!zpt} {\!zpt}   
      \linethickness=\!tickwidth
      \ifdim\!ticklength>\!zpt
        \putrule from {\!zpt} {\!zpt} to
          {\!ticksinoutsign\!tickxsign\!ticklength}
          {\!ticksinoutsign\!tickysign\!ticklength}
      \fi
      \if!gridlinestoo
        \putrule from {\!zpt} {\!zpt} to
          {-\!tickxsign\!xaxislength} {-\!tickysign\!yaxislength}
      \fi
    \endpicturesave <\!Xsave,\!Ysave>}%
    \wd\!boxA=\!zpt}
  
\def\!ticksin{%
  \def\!ticksinoutsign{-}%
  \!maketick
  \!nextkeyword}

\def\!ticksout{%
  \def\!ticksinoutsign{+}%
  \!maketick
  \!nextkeyword}

\def\!tickslength<#1> {%
  \!ticklength=#1\relax
  \!maketick
  \!nextkeyword}

\def\!tickslong{%
  \!tickslength<\longticklength> }

\def\!ticksshort{%
  \!tickslength<\shortticklength> }

\def\!tickswidth<#1> {%
  \!tickwidth=#1\relax
  \!maketick
  \!nextkeyword}

\def\!ticksandacross{%
  \!gridlinestootrue
  \!maketick
  \!nextkeyword}

\def\!ticksbutnotacross{%
  \!gridlinestoofalse
  \!maketick
  \!nextkeyword}

\def\!tickslogged{%
  \let\!ticktransform=\!logten
  \!nextkeyword}

\def\!ticksunlogged{%
  \let\!ticktransform=\!donothing
  \!nextkeyword}

\def\!ticksunlabeled{%
  \!tickcase=0
  \!nextkeyword}

\def\!ticksnumbered{%
  \!tickcase=1
  \!nextkeyword}

\def\!tickswithvalues#1/ {%
  \edef\!tickvalueslist{#1! /}%
  \!tickcase=2
  \!nextkeyword}

\def\!ticksquantity#1 {%
  \ifnum #1>1
    \!updatetickoffset
    \!countA=#1\relax
    \advance \!countA -1
    \!ticklocationincr=\!axisLength
      \divide \!ticklocationincr \!countA
    \!ticklocation=\!axisstart
    \loop \!not{\ifdim \!ticklocation>\!axisend}
      \!placetick\!ticklocation
      \ifcase\!tickcase
          \relax %  Case 0: no labels
        \or
          \relax %  Case 1: numbered -- not available here
        \or
          \expandafter\!gettickvaluefrom\!tickvalueslist
          \edef\!tickfield{{\the\!ticklocation}{\!value}}%
          \expandafter\!listaddon\expandafter{\!tickfield}\!LTlist%
      \fi
      \advance \!ticklocation \!ticklocationincr
    \repeat
  \fi
  \!nextkeyword}

\def\!ticksat#1 {%
  \!updatetickoffset
  \edef\!Loc{#1}%
  \if /\!Loc
    \def\next{\!nextkeyword}%
  \else
    \!ticksincommon
    \def\next{\!ticksat}%
  \fi
  \next}    
      
\def\!ticksfrom#1 to #2 by #3 {%
  \!updatetickoffset
  \edef\!arg{#3}%
  \expandafter\!separate\!arg\!nil
  \!scalefactor=1
  \expandafter\!countfigures\!arg/
  \edef\!arg{#1}%
  \!scaleup\!arg by\!scalefactor to\!countE
  \edef\!arg{#2}%
  \!scaleup\!arg by\!scalefactor to\!countF
  \edef\!arg{#3}%
  \!scaleup\!arg by\!scalefactor to\!countG
  \loop \!not{\ifnum\!countE>\!countF}
    \ifnum\!scalefactor=1
      \edef\!Loc{\the\!countE}%
    \else
      \!scaledown\!countE by\!scalefactor to\!Loc
    \fi
    \!ticksincommon
    \advance \!countE \!countG
  \repeat
  \!nextkeyword}

\def\!updatetickoffset{%
  \!dimenA=\!ticksinoutsign\!ticklength
  \ifdim \!dimenA>\!offset
    \!offset=\!dimenA
  \fi}

\def\!placetick#1{%
  \if!xswitch
    \!xpos=#1\relax
    \!ypos=\!axisylevel
  \else
    \!xpos=\!axisxlevel
    \!ypos=#1\relax
  \fi
  \advance\!xpos \!Xsave
  \advance\!ypos \!Ysave
  \kern\!xpos\raise\!ypos\copy\!boxA\kern-\!xpos
  \ignorespaces}

\def\!gettickvaluefrom#1 #2 /{%
  \edef\!value{#1}%
  \edef\!tickvalueslist{#2 /}%
  \ifx \!tickvalueslist\!endtickvaluelist
    \!tickcase=0
  \fi}
\def\!endtickvaluelist{! /}

\def\!ticksincommon{%
  \!ticktransform\!Loc\!t
  \!ticklocation=\!t\!!unit
  \advance\!ticklocation -\!!origin
  \!placetick\!ticklocation
  \ifcase\!tickcase
    \relax % Case 0: no labels
  \or %      Case 1: numbered
    \ifdim\!ticklocation<-\!!origin
      \edef\!Loc{$\!Loc$}%
    \fi
    \edef\!tickfield{{\the\!ticklocation}{\!Loc}}%
    \expandafter\!listaddon\expandafter{\!tickfield}\!LTlist%
  \or %      Case 2: labeled
    \expandafter\!gettickvaluefrom\!tickvalueslist
    \edef\!tickfield{{\the\!ticklocation}{\!value}}%
    \expandafter\!listaddon\expandafter{\!tickfield}\!LTlist%
  \fi}

\def\!separate#1\!nil{%
  \!ifnextchar{-}{\!!separate}{\!!!separate}#1\!nil}
\def\!!separate-#1\!nil{%
  \def\!sign{-}%
  \!!!!separate#1..\!nil}
\def\!!!separate#1\!nil{%
  \def\!sign{+}%
  \!!!!separate#1..\!nil}
\def\!!!!separate#1.#2.#3\!nil{%
  \def\!arg{#1}%
  \ifx\!arg\!empty
    \!countA=0
  \else
    \!countA=\!arg
  \fi
  \def\!arg{#2}%
  \ifx\!arg\!empty
    \!countB=0
  \else
    \!countB=\!arg
  \fi}
 
\def\!countfigures#1{%
  \if #1/%
    \def\!next{\ignorespaces}%
  \else
    \multiply\!scalefactor 10
    \def\!next{\!countfigures}%
  \fi
  \!next}

\def\!scaleup#1by#2to#3{%
  \expandafter\!separate#1\!nil
  \multiply\!countA #2\relax
  \advance\!countA \!countB
  \if -\!sign
    \!countA=-\!countA
  \fi
  #3=\!countA
  \ignorespaces}

\def\!scaledown#1by#2to#3{%
  \!countA=#1\relax%                          ** get original #
  \ifnum \!countA<0 %                         ** take abs value,
    \def\!sign{-}%                            **   remember sign
    \!countA=-\!countA
  \else
    \def\!sign{}%
  \fi
  \!countB=\!countA%                          ** copy |#|
  \divide\!countB #2\relax%                   ** integer part (|#|/sf)
  \!countC=\!countB%                          ** get sf * (|#|/sf)
    \multiply\!countC #2\relax
  \advance \!countA -\!countC%                ** ctA is now remainder
  \edef#3{\!sign\the\!countB.}%               ** +- integerpart.
  \!countC=\!countA %                         ** Tack on proper number
  \ifnum\!countC=0 %                          **   of zeros after .
    \!countC=1
  \fi
  \multiply\!countC 10
  \!loop \ifnum #2>\!countC
    \edef#3{#3\!zero}%
    \multiply\!countC 10
  \repeat
  \edef#3{#3\the\!countA}%                    ** Add on rest of remainder
  \ignorespaces}

\def\!placetickvalues{%
  \advance\!offset \tickstovaluesleading
  \if!xswitch
    \setbox\!boxA=\hbox{%
      \def\\##1##2{%
        \!dimenput {##2} [B] (##1,\!axisylevel)}%
      \beginpicture 
        \!LTlist
      \endpicturesave <\!Xsave,\!Ysave>}%
    \!dimenA=\!axisylevel
      \advance\!dimenA -\!Ysave
      \advance\!dimenA \!tickysign\!offset
      \if -\!tickysign
        \advance\!dimenA -\ht\!boxA
      \else
        \advance\!dimenA  \dp\!boxA
      \fi
    \advance\!offset \ht\!boxA 
      \advance\!offset \dp\!boxA
    \!dimenput {\box\!boxA} [Bl] <\!Xsave,\!Ysave> (\!zpt,\!dimenA)
  \else
    \setbox\!boxA=\hbox{%
      \def\\##1##2{%
        \!dimenput {##2} [r] (\!axisxlevel,##1)}%
      \beginpicture 
        \!LTlist
      \endpicturesave <\!Xsave,\!Ysave>}%
    \!dimenA=\!axisxlevel
      \advance\!dimenA -\!Xsave
      \advance\!dimenA \!tickxsign\!offset
      \if -\!tickxsign
        \advance\!dimenA -\wd\!boxA
      \fi
    \advance\!offset \wd\!boxA
    \!dimenput {\box\!boxA} [Bl] <\!Xsave,\!Ysave> (\!dimenA,\!zpt)
  \fi}


\normalgraphs
\catcode`!=12 %  *****  THIS MUST NEVER BE OMITTED
%
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% 
% New Complex and Quaternion-Hyperbolic Reflection Groups
%
% Daniel Allcock

% load the style file
\def\FOOTNOTE{\footnote*}
%

\author{Daniel Allcock\FOOTNOTE{Supported by National Science
Foundation Graduate and Postdoctoral Fellowships.}}

\title{New complex and quaternion-hyperbolic reflection groups}
%

\email{allcock@math.utah.edu}

\homepage{http://www.math.utah.edu/$\sim$allcock}

\note{Dedicated to my father John Allcock, 1940--1991}

\date{9 September 1997}
%
% Commenced 20 February 1996

\address{Department of Mathematics,
University of Utah, Salt Lake City, UT 84112, USA}

\subject{22E40 (11F06, 11E39, 53C35)}

% small enough font to dodge overfull hbox
\font\titlefont=cmb10 scaled \magstep2
\plaintitlepage
%

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%dashes
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\def\emdash{---}

%
%----------Options-(comment-or-uncomment-as-desired)-------------
%
\magnification=1100
%
%    for doublespacing
%\baselineskip=24pt
%
\symbolictags

%    eliminates the nasty overfullbox bars:
\def\hideboxes{\overfullrule=0pt}
%\hideboxes 

%-----------------beginning-of-document---------------------------
%

% Definitions for cross-references
%
% Do not tamper with the following line.
% ---begin deftags---
\deftag{1}{sec-introduction}
\deftag{2}{sec-lattice-background}
\deftag{2.1}{eq-def-of-reflection}
\deftag{3}{sec-lattice-reference}
\deftag{3.1}{thm-barnes-wall-properties}
\deftag{4}{sec-hyperbolic-space}
\deftag{4.1}{eq-half-space-coordinates}
\deftag{4.2}{eq-def-of-translation}
\deftag{4.3}{eq-product-of-translations}
\deftag{4.4}{eq-inverse-of-translation}
\deftag{4.5}{eq-commutator-of-translations}
\deftag{4.6}{eq-conjugate-of-translation}
\deftag{5}{sec-reflections}
\deftag{5.1}{thm-rho-stabilizer-finite-index}
\deftag{5.1}{eq-got-translation}
\deftag{5.2}{thm-what-long-roots-do}
\deftag{5.2}{eq-coordinates-of-r}
\deftag{5.3}{eq-ip-with-r'}
\deftag{5.4}{eq-height-of-v'}
\deftag{5.3}{thm-what-short-roots-do}
\deftag{5.5}{eq-coordinates-of-r-2}
\deftag{5.6}{eq-ip-with-r'-2}
\deftag{5.7}{eq-height-of-v'-2}
\deftag{5.8}{eq-foo3}
\deftag{5.9}{eq-relation-between-xi's}
\deftag{5.10}{eq-foo4}
\deftag{5.11}{eq-relation-between-xi's-2}
\deftag{6}{sec-reflection-groups}
\deftag{6.1}{thm-cov-radius-implies-reflective}
\deftag{6.1}{tab-table-of-balls}
\deftag{6.2}{thm-well-covered-implies-reflective}
\deftag{6.3}{thm-13-reflection-groups}
\deftag{6.1}{fig-coxeter-diagram}
\deftag{6.4}{thm-rho-stabilizer-finite-index-2}
\deftag{6.5}{thm-braiding-reflections}
\deftag{6.6}{thm-eI2,1-eI3,1-eI4,1}
\deftag{6.7}{thm-qI2,1-qI3,1}
\deftag{6.8}{thm-linear-independence-descends}
\deftag{6.9}{thm-coxeter-todd-well-covered}
\deftag{6.10}{thm-eI7,1}
\deftag{6.11}{thm-barnes-wall-well-covered}
\deftag{6.12}{thm-affine-diagram-automorphisms}
\deftag{6.1}{eq-dimension-of-fixed-space}
\deftag{6.13}{thm-leech-holes-in-barnes-wall}
\deftag{6.14}{thm-qI5,1}
\deftag{7}{sec-selfdual-classification}
\deftag{7.1}{thm-indefinite-selfdual-classification}
\deftag{7.2}{thm-definite-selfdual-classification}
\deftag{8}{sec-comparison}
\deftag{8.1}{thm-projective-reflections-are-honest}
\deftag{8.2}{thm-classify-honest-reflections}
\deftag{8.3}{thm-primitive-reflections-in-my-groups}
\deftag{8.4}{thm-cone-angles}
\deftag{8.5}{thm-primitive-reflections-in-their-groups}
\deftag{8.6}{thm-my-groups-are-new}
\deftag{9}{sec-conclusion}
\deftag{9.1}{thm-conjecture}
% ---end deftags---
% Do not tamper with the previous line or the following line.
% ---begin defcites---
\defcite{1}{allcock:oh2}
\defcite{2}{allcock:cubic-surf}
\defcite{3}{bachoc:selfdual-Hurwitz-lattices}
\defcite{4}{bachoc:8-dimensional-Hurwitz-lattices}
\defcite{5}{reb:thesis}
\defcite{6}{reb:lzl}
\defcite{7}{reb:like-leech}
\defcite{8}{borel:arithmetic-groups}
\defcite{9}{jhc:26dim}
\defcite{10}{jhc:leech-radius}
\defcite{11}{jhc:laminated-lattices}
\defcite{12}{jhc:vinberg-gps}
\defcite{13}{jhc:Coxeter-Todd}
\defcite{14}{splag}
\defcite{15}{coxeter:quotients-of-braid-groups}
\defcite{16}{cox:gens-and-rlns}
\defcite{17}{deligne:monodromy-of-hypergeometric-functions}
\defcite{18}{feit:unimodular-Eisenstein}
\defcite{19}{milnor:bilinear-forms}
\defcite{20}{mostow:remarkable-polyhedra}
\defcite{21}{mostow:picard-lattices-from-half-inegral-conditions}
\defcite{22}{mostow:monodromy-groups-on-the-complex-n-ball}
\defcite{23}{thurston:shapes-of-polyhedra}
\defcite{24}{vin:groups-of-quad-forms}
\defcite{25}{vin:unimodular-quad-forms}
\defcite{26}{vin:I18.1-and-I19.1}
\defcite{27}{raw:quaternionic-leech}
\defcite{28}{raw:complex-leech}
% ---end defcites---
% Do not tamper with the preceding line.

\def\w{\omega}
\def\wbar{\bar{\w}}

%
\def\quaternionic{{\bbb H}}
\def\K{{\bbb K\kern.05em}}	
\def\O{{\bbb O}}
\def\eisen {\cale} % Eisenstein integers
\def\gauss {\calg} % Gaussian integers
\def\Hurwitz{\calh}
\def\ring{\calr}

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%
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%

%
\def\H{\quaternionic}

\def\reflec{\mathop{\rm Reflec}\nolimits} % group generated by reflections 
\def\shr{\mathop{\reflec}_0\nolimits} % group generated by
					 % reflections in short
					 % roots 
\def\isom{\mathop{\rm Isom}\nolimits}
\def\GL{\mathop{\rm GL}\nolimits}
\def\SL{\mathop{\rm SL}\nolimits}
\def\height{{\rm ht\,}}
\def\diag{\hbox{diag}\,}
\def\spanof#1{\langle#1\rangle}
\def\cpnm{\C P^{n-1}}
\def\kpnp{\K P^{n+1}}
\def\khnp{\K H^{n+1}}
\def\dkhnp{\partial\khnp}
\def\drhnp{\partial\rhnp}
\def\hyperbolic{H}
\def\rh {\R\hyperbolic^}
\def\ch {\C\hyperbolic^}	
\def\qh{\quaternionic\hyperbolic^} 
\def\kh{\K\hyperbolic^}
\def\qhn{\qh n}
\def\chn{\ch n}
\def\rhn{\rh n}
\def\chnp {\ch{n+1}} 
\def\rhnp {\rh{n+1}} 
\def\del{\partial}
\def\drh {\del\rh}
\def\I#1,#2{I_{#1,#2}}
\def\II#1,#2{I\negspace I_{#1,#2}}
\def\oh{\O\hyperbolic^}

\def\trianglegroup#1#2#3{(#1,#2,#3)}
\def\squaregroup#1#2#3#4{\proofbox(#1,#2,#3,#4)}
\def \pu #1,#2{PU(#1,#2)}
\def\ip#1#2{\langle#1{|}#2\rangle}
\def\Eisenstein {\eisen}
\def\Gaussian{\gauss}
\def\cct{\Lambda_6^\Eisenstein} % complex Coxeter-Todd lattice
\def\rct{K_{12}} % real form of Coxeter-Todd lattice 
\def\rbwl{BW_{16}} % real form of barnes-wall lattice
\def\qbwl{\Lambda_4^\Hurwitz} % quaternionic Barnes-Wall lattice 
\def\leech{\Lambda_{24}}
\def\qleech{\leech^\Hurwitz} % quaternionic leech lattice
\def\geeight{E_8^\Gaussian}	% Gaussian E8 lattice 
\def\qeeight{E_8^\Hurwitz}	%quaternionic E8 lattice

\def\eI#1,#2{\I#1,#2^\eisen}   % Lorentzian lattices of
				 % various types
\def\eII#1,#2{\II#1,#2^\eisen}
\def\qI#1,#2{\I#1,#2^\Hurwitz}
\def\qII#1,#2{\II#1,#2^\Hurwitz}
\def\gI#1,#2{\I#1,#2^\gauss}
\def\gII#1,#2{\II#1,#2^\gauss}
\def\zI#1,#2{\I#1,#2^\Z}
\def\rI#1,#2{\I#1,#2^\ring}
\def\rII#1,#2{\II#1,#2^\ring}
\def\zI#1,#2{\I#1,#2^\Z}
\def\zII#1,#2{\II#1,#2^\Z}

\def\vec#1#2{(#1_1,\ldots,#1_{#2})}
\def\vector#1#2{\vec{#1}{#2}}

% imaginary constants
\def\i{{\rm i}}
\def\j{{\rm j}}
\def\k{{\rm k}}

\def\romp#1{({\it\romannumeral#1$\,'$\/})}
\def\Cbar{\bar C}
\def\G{\Gamma}
\def\negspace{\!}

\abstract
We consider the automorphism groups of various Lorentzian
lattices over the Eisenstein, Gaussian, and Hurwitz integers,
and in some of them we find reflection groups of finite index.
These provide explicit constructions of new finite-covolume
reflection groups acting on complex and quaternionic hyperbolic
spaces of high dimensions.  Specifically, we provide groups
acting on $\chn$ for all $n<6$ and $n=7$, and on $\qhn$ for
$n=1,2,3,$ and $5$. We compare our groups with those discovered
by Deligne and Mostow and by Thurston, and show that our most
interesting examples are new.  For many of these Lorentzian lattices we
show that the entire symmetry group is generated by reflections,
and obtain a description of the group in terms of the
combinatorics of a lower-dimensional positive-definite lattice.
The techniques needed for our lower-dimensional examples are
elementary; to construct our best examples we also use certain
facts about the Leech lattice.  We conjecture that Lorentzian
lattices provide examples of hyperbolic reflection groups in
dimensions even higher than those considered here, and mention
connections to moduli of cubic surfaces. By studying
orbits of norm $0$ vectors in certain selfdual Lorentzian lattices we
provide a new and geometric proof of the classifications of
selfdual Eisenstein lattices of dimension $\leq6$ and of selfdual
Hurwitz lattices of dimension $\leq4$.

\section{\Tag{sec-introduction}. Introduction}

In this paper we carry out complex and quaternionic analogues of
some of Vinberg's extensive study of reflection groups on real
hyperbolic space. In \cite{vin:groups-of-quad-forms} and
\cite{vin:unimodular-quad-forms} he investigated the 
symmetry groups of the integral quadratic forms
$\diag[-1,+1,\ldots,+1]$, or equivalently the Lorentzian
lattices $\I n,1$. He was able to describe these groups quite
concretely for $n\leq17$, and extensions of his work by Vinberg
and Kaplinskaja \cite{vin:I18.1-and-I19.1} and Borcherds
\cite{reb:lzl} provide concrete 
descriptions for all $n\leq23$. In particular, the subgroup of
$\aut\I n,1$ generated by reflections has finite index just when
$n\leq19$.

In our work, we study the symmetry groups of Lorentzian lattices
over the rings $\Gaussian$ and $\Eisenstein$ of Gaussian and
Eisenstein integers and the ring $\Hurwitz$ of Hurwitz integers
(a discrete subring of the skew field $\H$ of quaternions). Most
of the paper is devoted to the most natural of such lattices,
the selfdual ones. The symmetry groups of these lattices provide
a large number of discrete groups generated by reflections and
acting with finite-volume quotient on on the hyperbolic spaces
$\ch n$ and $\qh n$. We construct a total of 18 such groups,
including groups acting on $\ch7$ and $\qh5$. 
At least one of our groups has been discovered before, in the
work of Mostow and Deligne
\cite{deligne:monodromy-of-hypergeometric-functions}, Mostow
\cite{mostow:picard-lattices-from-half-inegral-conditions} and
Thurston 
\cite{thurston:shapes-of-polyhedra}, but we show that our
``largest'' examples are new. 
To the
author's knowledge,  quaternion-hyperbolic reflection groups not
been studied before.

Our results and techniques have already found important
application in work of the author, J. Carlson and D. Toledo
\cite{allcock:cubic-surf} on  moduli spaces of smooth cubic surfaces over
$\C$. Namely, the (coarse) moduli space of such surfaces is
isomorphic to the quotient of $\ch4$ by a certain reflection
group (studied here), minus the images of the mirrors of certain
reflections. Furthermore, the usual (fine) moduli space of
marked cubic surfaces may be
realized as a 
quotient of $\ch4$ by a congruence subgroup of this
reflection group.
\goodbreak

The techniques used by Vinberg and others for the real
hyperbolic case rely heavily on the fact that if a discrete
group $G$ is generated by reflections of $\rh n$, then the
mirrors of the reflections of $G$ chop $\rh n$ into pieces; each
piece may be taken as a fundamental domain for $G$. Work with
complex or quaternionic reflection groups is much more
complicated, since hyperplanes have real codimension 2 or 4, and
so the mirrors  
fail to chop hyperbolic space into pieces. Our solution to
this problem is to avoid fundamental domains altogether.
Each of our groups is defined as the
subgroup $\reflec L$ of $\aut L$ generated by reflections, where
$L$ is a Lorentzian lattice over $\Gaussian$, $\Eisenstein$ or
$\Hurwitz$. (A Lorentzian lattice is a free module equipped with
a Hermitian form of signature $-+\cdots+$.) 
Since $\aut L$ is an arithmetically defined group,
to show that the quotient of $\ch n$ or $\qh n$ by $\reflec L$ has finite
volume, it suffices to show that $\reflec L$ has finite index in
$\aut L$. In this case we say that $L$ is reflective. Our basic
strategy for proving a suitable lattice $L$ to be reflective is to prove first that
$\reflec L$ acts with only finitely many orbits on the vectors
of $L$ of norm $0$, and second that the stabilizer in $\reflec
L$ of one such vector has finite index in the stabilizer in
$\aut L$. That is, we work mostly arithmetically, avoiding use of
such tools as the bisectors introduced by Mostow for his study
\cite{mostow:remarkable-polyhedra} of 
reflection groups on $\ch2$ (including some nonarithmetic ones).

However, there are certain steps in our constructions where
geometric issues play a key role. We express each of our
Lorentzian lattices $L$ in the form $\Lambda\oplus\II1,1$, where
$\Lambda$ is positive-definite and $\II1,1$ is a certain
2-dimensional lattice, the ``hyperbolic cell''. It turns out
that this description of $L$ allows one to easily write down a
large collection of reflections of $L$, parameterized by (a
central extension of) the lattice $\Lambda$. It turns out that
if $\Lambda$ satisfies several properties, such as providing a
good covering of  Euclidean space by balls, then one can
automatically deduce that $L$ 
is reflective. This implication is the content of
Theorem~\tag{thm-well-covered-implies-reflective}. The rest of
Section~\tag{sec-reflection-groups} is devoted to the
application of this theorem (and related ideas) in the
study of various examples. In particular, we prove that each of
the selfdual lattices
$$
\vbox{\halign{$#$\hfil&\qquad$#$\hfil\cr
\eI n,1&n=1,2,3,4,7,\cr
\gII n,1&n=1,5,\cr
\qI n,1&n=1,2,3,5\cr}}
$$
is reflective. (These lattices are defined in
Section~\tag{sec-lattice-reference} and characterized in
Theorem~\tag{thm-indefinite-selfdual-classification}.) For some
of these, we obtain more detailed information. In particular, we
prove that $\reflec\eI n,1=\aut\eI n,1$ for $n=2$, $3$, $4$ or
$7$ and that $\reflec\qI n,1$ has index at most $4$ in $\aut\qI
n,1$ for $n=2$, $3$ or 
$5$. The group $\reflec\eI4,1$ is particularly interesting
because the quotient of $\ch4$ by it is a partial
compactification of the coarse moduli space of smooth cubic
surfaces (see \cite{allcock:cubic-surf}). We also give explicit
descriptions of the reflection groups of $\eI1,1$, 
$\gII1,1$ and $\qI1,1$ as subgroups of certain Coxeter groups, acting on
$\ch1\isomorphism \rh2$ and  $\qh1\isomorphism\rh4$. 

We also note that the geometric idea we use, namely that good
coverings of Euclidean space can lead to hyperbolic reflection
groups, applies even when $\C$ and $\H$ are replaced by the
nonassociative field $\O$ of octaves. In \cite{allcock:oh2} we construct
two octave reflection groups acting on $\oh2$ and one acting
on $\oh1\isomorphism \rh8$, and interpret these groups as the
stabilizers of ``lattices'' over a certain discrete subring of
$\O$. 

We provide background information on lattices in
Section~\tag{sec-lattice-background} and examples of them in
Section~\tag{sec-lattice-reference}; the latter should be
referred to only as needed. Section~\tag{sec-hyperbolic-space}
establishes our conventions regarding hyperbolic geometry. In
Section~\tag{sec-reflections} we relate certain geometric
properties of a positive-definite lattice $\Lambda$ to the
reflection group of $\Lambda\oplus\II1,1$ and lay other
foundations for Section~\tag{sec-reflection-groups}, where we
construct all of our examples. In Sections~\tag{sec-reflections}
and \tag{sec-reflection-groups}, 
statements of many results are  complicated by the fact
that while lattices over the three rings $\Gaussian$,
$\Eisenstein$ and $\Hurwitz$ may be treated in parallel, the
exact results one can obtain depend slightly on the ring under
consideration. The reader may focus on just one of these rings
(we suggest $\Eisenstein$) and still fully understand all of the
ideas presented. In Section~\tag{sec-selfdual-classification} we
explain the correspondence between primitive isotropic
sublattices of $\I n+1,1$ and positive-definite selfdual
lattices of dimension $n$. We use this correspondence to provide
a quick 
geometric proof of the classification of selfdual
lattices over $\Eisenstein$ and $\Hurwitz$ in dimensions
$\leq 6$ and $\leq 4$, respectively. The only examples besides the trivial
lattices are the Coxeter-Todd lattice $\cct$ and a quaternionic
form $\qbwl$ of the Barnes-Wall lattice.  
In Section~\tag{sec-comparison} we identify properties of three of our groups,
namely $\reflec L$ for $L=\eI7,1$, $\eI4,1$ and $\qII5,1$, that
distinguish them from the 94 groups constructed in
\cite{deligne:monodromy-of-hypergeometric-functions},
\cite{mostow:picard-lattices-from-half-inegral-conditions} and
\cite{thurston:shapes-of-polyhedra}. We also sketch a proof that
$\reflec\eI3,1$ 
appears on the lists given there.
Finally, in
Section~\tag{sec-conclusion} we make a few closing comments and
conjecture that the Lorentzian lattices $\I n,1$ over
$\Eisenstein$, $\Gaussian$ and $\Hurwitz$ are reflective
for $n$ considerably larger the cases considered
here.

The easiest route  to a new reflection group
is our treatment of $\gII5,1=\geeight\oplus\gII1,1$, which acts
on $\ch5$. The proof that $\gII5,1$ is reflective requires only
Theorem~\tag{thm-rho-stabilizer-finite-index}, the Gaussian case
of Theorem~\tag{thm-what-long-roots-do}, and the relevant parts
of Theorem~\tag{thm-well-covered-implies-reflective} and
Corollary~\tag{thm-13-reflection-groups}.  
There are
two ``tracks'' through  Section~\tag{sec-reflection-groups}, the first leading to a
large number of low-dimensional examples (including $\gII5,1$)
and the second leading to a detailed study of the lattices
$\eI7,1$ and $\qI5,1$. See the comments there for more
information. Here we  mention only that these
two lattices succumb to our techniques because of several
special properties of the Coxeter-Todd
and Barnes-Wall lattices. In particular, the
automorphism group of each is generated by reflections, and
(suitable scaled) each has a very nice embedding in the
Leech lattice $\leech$. Our basic approach in this paper was
inspired by Conway's remarkable description \cite{jhc:26dim} of
the isometry group of the $\Z$-lattice
$\II25,1=\leech\oplus\II1,1$ in terms of 
the combinatorics of $\leech$, so it is  pleasing to see
$\leech$ playing a role here as well.

This paper is derived in part from the author's Ph.D. thesis at
Berkeley; he would like to thank his dissertation advisor,
R. Borcherds, for his interest and suggestions\emdash in particular
for suggesting that the quaternionic Barnes-Wall lattice would
provide a reflection group on $\qh5$.

%\section {\oldTag{sec-26dim}. Inspiration: The 
%26-dimensional Lorentzian lattice}

%We begin with a naive  description of the
%group of symmetries of the Lorentzian lattice $\II25,1$, 
%which was  first investigated
%by Conway in \oldcite{jhc:26dim}. The Leech lattice
%$\leech$ is a lattice in 24-dimensional Euclidean space, that
%is a discrete subgroup $\Z^{24}$ pairs of elements of which have
%integral inner product. Of its
%many remarkable properties
%we need only two: balls centered at its points exactly cover
%$\R^{24}$, and the
%distance between two of its elements is one of $\sqrt4$,
%$\sqrt6$, $\sqrt8$,\dots. (For background on $\leech$, see 
%\oldcite{jhc:dot0} and \oldcite{reb:leech-lattice}.)

%One may construct an interesting reflection  group acting on the
%hyperbolic space $\rh{25}$ as follows: consider the upper-half
%space model for $\rh{25}$, in which a hyperplane appears as
%a vertical half-plane or as a hemisphere centered at a point of
%$\R^{24}\subseteq \del\rh{25}$. We construct a family of hyperplanes in
%$\rh{25}$ by thinking of $\leech$ as a subset of $\R^{24}$ and
%centering a hemisphere of radius $\sqrt{2}$
%at each element of $\leech$. Then the region of $\rh{25}$
%above these hyperplanes is a fundamental domain for the
%group generated by 
%reflections in these hyperplanes. 
%To verify this, all we need to do is to check
%that our mirrors intersect  at angles which are
%integer submultiples of $\pi$. The required computations are
%trivial\emdash hyperplanes corresponding to neighboring points of
%$\leech$ meet at right angles, and those corresponding to points
%at distance $\sqrt6$ 
%meet at $\pi/3$; no other pair of hyperplanes meets. 
%The fundamental domain has infinite volume, but it is 
%obviously invariant under all of the symmetries 
%of $\leech$, including translations.
%Adjoining these symmetries to
%the reflection group, we obtain a group $G$ acting on
%$\rh{25}$ with finite covolume. 
%$\rh{25}/G$ has a cusp at infinity, and since the hemispheres
%exactly cover $\R^{24}$, $\rh{25}/G$ also has one cusp for  each orbit of
%deep hole of $\leech$. (A deep hole is a point of $\R^{24}$ at
%maximal distance from $\leech$.) Under the translations of
%$\leech$ there are only finitely many orbits of deep holes, so
%$\rh{25}/G$ has finite volume.

%$G$ may be identified with  the group of the Lorentzian lattice $\II25,1$
%as  follows.  $\II25,1$ may be defined to be 
%$\leech \oplus \II1,1$, where $\II1,1$ is
%the quadratic form given by the inner product matrix
%$$
%\II1,1=
%\pmatrix{0 & 1 \cr 1 & 0\cr}\;.
%$$
%We write elements of $\II25,1$ as triples
%$(\lambda;\mu,\nu)$ with $\lambda\in\leech$ and 
%$\mu, \nu\in \Z$. The inner product of any two elements
%of $\II25,1$ is an integer,
%so reflection in any norm~$2$ vector of $\II25,1$ preserves 
%$\II25,1$.   For each
%$\lambda\in\leech$, the vector $(\lambda;1,1-\lambda^2/2)$ lies
%in $\II25,1$ and has norm~2. Choosing the upper-half-space model
%for $\rh{25}$ with $(0;0,1)$ as the point at infinity, the
%reflections in these vectors are the reflections  of our original
%description. (The rest of $G$
%is generated by $\aut\leech$ acting on $\leech$ in the obvious way; together
%with translations fixing $(0;0,1)$; these are given by
%Eq.~\eqtag{eq-def-of-translation}, with $z=0$.)

%Conway's  analysis of $\aut\II25,1$ relied on the simple
%coordinatization by $\leech$ of this set of  norm $2$ vectors. In this paper
%we advance this idea by studying Lorentzian lattices of the form
%$\Lambda\oplus\II1,1$, over rings 
%other than $\Z$. Just as knowledge of the covering radius of
%$\leech$ allowed Conway to deduce that reflections (almost)
%generate $\aut\II25,1$, we will see that the covering radius of
%$\Lambda$ plays a role in our study of $\aut (\Lambda\oplus\II1,1)$. 

\section{\Tag{sec-lattice-background}. Lattices}

We denote by $\ring$ any one of the rings $\eisen$, $\gauss$,
and $\Hurwitz$\emdash the Eisenstein, Gaussian, and Hurwitz
integers. That is, $\gauss=\Z[\i ]$ and $\eisen=\Z[\w]$, where
$\w=(-1+\sqrt{-3})/2$ is a primitive cube root of unity.  The
ring $\Hurwitz$ is the integral span of its $24$ units $\pm1$,
$\pm \i $, $\pm \j $, $\pm \k $ and $(\pm1\pm \i \pm \j \pm \k )/2$ in the
skew field $\H$ of quaternions. We write $\K$ for the field
($\C$ or $\H$) naturally containing $\ring$.  Conjugation
$x\mapsto\bar x$ denotes complex or quaternionic conjugation, as
appropriate.  For any element $x$ of $\K$, we write $\re
x=(x+\bar x)/2$ and $\im x=(x-\bar x)/2$ for the real and
imaginary parts of $x$, and say that $x$ is imaginary if $\re
x=0$. If $X\sset\K$ then we write $\im X$ for the set of  imaginary
elements of $X$. For any $x\in\K$, $x\bar x$ is a positive real
number, and the absolute value $|x|$ of
$x$ is defined to be $(x\bar x)^{1/2}$.  It is convenient
to define the element $\theta=\w-\bar\w=\sqrt{-3}$ of
$\eisen$. We will sometimes also consider $\w$ and $\theta$ as
elements of $\Hurwitz$, via the embedding
$\eisen\rightarrow\Hurwitz$ defined  by
$\w\mapsto(-1+\i +\j +\k )/2$ or equally well by $\theta\mapsto \i +\j +\k $.
 
A {lattice} $\Lambda$ over $\ring$ is a free (right) module over
$\ring$ equipped with a Hermitian form, which is to say a
$\Z$-bilinear pairing (the inner product) $\ip{\cdot}{\cdot}:
\Lambda\times \Lambda\rightarrow \K$ such that
$$
\ip{x}{y}=\overline{\ip{y}{x}}\qquad\hbox{and}\qquad
\ip{x}{y\a}=\ip{x}{y}\a
$$
for all $\a\in\ring$. A Hermitian form on a (right) vector space
over $\K$ is defined similarly.
Section~\tag{sec-lattice-reference} defines a number of
interesting lattices and lists some of their
properties. Sometimes we indicate that a lattice $\Lambda$ is an
$\ring$-lattice by writing $\Lambda^\ring$ or somesuch.

If $S\sset \Lambda$ then we denote by $S^\perp$ its orthogonal
complement: those elements of $\Lambda$ whose inner products
with all elements of $S$ vanish.  We say that $\Lambda$ is
integral if for all $x,y\in \Lambda$, the inner product
$\ip{x}{y}$ lies in $\ring$, and that $\Lambda$ is nonsingular if
$\Lambda^\perp=\{0\}$.  All lattices we consider will
be integral and nonsingular unless otherwise specified.  The
dual $\Lambda^*$ of $\Lambda$ is the set of all $\ring$-linear maps
from $\Lambda$ to $\ring$.  An integral lattice $\Lambda$ is
called {selfdual} if the natural map from $\Lambda$ to
$\Lambda^*$ is onto. A selfdual lattice is sometimes called
``unimodular'', because the matrix of inner products of any
basis for $\Lambda$ has determinant $\pm1$; we use ``selfdual'' to avoid
discussing determinants of quaternionic matrices.

The {norm} of a vector $v\in V$ is defined to be
$v^2=\ip{v}{v}$; some authors call this the squared norm of $v$.
%Since $v^2$ may be negative under indefinite Hermitian
%forms taking its square root would be artificial.  
We say that $v$ is isotropic, or null, if $v^2=0$. A lattice is
isotropic (or null) if each of its elements is. A lattice is
called even if each of its elements has even norm and odd if it
is not even. A sublattice
$\Lambda'$ of $\Lambda$ is called primitive if
$\Lambda'=\Lambda\cap(\Lambda'\tensor\R)$. A vector $v$ of
$\Lambda$ is called primitive if $v=w\a$ for $w\in \Lambda$ and
$\a\in\ring$ implies that $\a$ is a unit. Because the rings
$\Gaussian$, $\Eisenstein$ and $\Hurwitz$ are principal ideal
domains, $v$ is primitive if and only if its $\ring$-span is
primitive as a sublattice. We will sometimes write $\spanof{v}$
for the $\ring$-span of $v$. In the context of Lorentzian
lattices (see below), any isotropic sublattice has dimension
$\le1$. When we refer to null sublattices of Lorentzian lattices
we implicitly restrict attention to those of dimension
$1$\emdash that is, we exclude from consideration the
zero-dimensional lattice. In Lorentzian lattices, the concepts
of primitive null vectors and primitive null lattices almost
coincide.

We sometimes define an $\ring$-lattice by describing a Hermitian
form on $\ring^n$. We do this by giving an $n\times n$ matrix
$(\phi_{ij})$ with entries in $\ring$ such that
$\overline{\phi_{ij}}=\phi_{ji}$. Then the Hermitian form is
given by
$$
\ip{\vector{x}{n}}{\vector{y}{n}}=\sum_{i,j=1}^{n}\bar{x}_i\phi_{ij}y_j\;.
$$
We may also view a lattice as a subset of a vector space $V$
over the field $\K$\emdash simply take $V$ to be the (right) vector
space $\Lambda\tensor\R$ over $\K=\R\tensor\ring$. The Hermitian
form on $\Lambda$ gives rise to one on $V$.  If $\Lambda$ is
nonsingular then $\Lambda^*$ may be identified with the set of
vectors in $V$ having $\ring$-integral inner product with each
element of $\Lambda$.

Every nonsingular Hermitian form on a vector space $V$ over $\K$
is equivalent under $\GL(V)$ to one given by a diagonal matrix,
with each diagonal entry being $\pm1$.  The signature of a form
$\Phi$ is the number of $+1$'s minus the number of $-1$'s. The
signature of $\Phi$ characterizes $\Phi$ up to equivalence under
$\GL(V)$.  We write $\K^{n,m}$ for the vector space $\K^{n+m}$
equipped with a Hermitian form of signature $n-m$. The
isometry group of $\K^{n,m}$ is the unitary group
$U(n,m;\K)$.  The term ``Lorentzian'' is applied to various
concepts in the study of real Minkowski space $\R^{n,1}$. By
analogy with this we call an $n$-dimensional lattice Lorentzian
if its signature is $n-2$.

If $\Lambda$ is positive-definite then $\Lambda\tensor\R$ is a
copy of Euclidean space, under the metric
$d(x,y)=\sqrt{(x-y)^2}$. Points of $\Lambda\tensor\R$ at maximal
distance from $\Lambda$ are called deep holes of $\Lambda$. The
maximal distance is called the covering radius of $\Lambda$,
because closed balls of that radius placed at lattice points
exactly cover $\Lambda\tensor\R$. The lattice points nearest a
deep hole are called the vertices of the hole.  The covering
radii of the $\Z$-lattices $\im\Gaussian$, $\im\Eisenstein$ and
$\im\Hurwitz$ are $1/2$, $3^{1/2}\negspace/2$ and $3^{1/2}\negspace/2$,
respectively. The first two are obvious and the last follows
because $\im\Hurwitz$ is the 3-dimensional cubic lattice spanned
by $\i $, $\j $ and $\k $.  Any two deep holes of $\im\ring$ are
equivalent under translation by some element of $\im\ring$.

Suppose that $V$ is a $\K$-vector space, $\xi\in\K$ is a root of
unity and $v\in V$ has nonzero norm.  We define
$\xi$-reflection in $v$ to be the map
$$
v\mapsto v- r(1-\xi){\ip{r}{v}\over r^2}\;.
\eqno\eqTag{eq-def-of-reflection}
$$
This is an automorphism of $V$ as a right vector space equipped
with Hermitian form $\ip{\cdot}{\cdot}$; it fixes $r^\perp$
pointwise and carries $r$ to $r\xi$.  ({\it Warning:\/} if
$\K=\H$ then although the reflection acts by right scalar
multiplication on $r$, it does not act this way on all of the
$\H$-span of $r$. This is due to the noncommutativity of
multiplication in $\H$.)  Unless otherwise specified, we will use the term
``reflection'' to mean ``reflection in a vector of positive
norm''. Under the conventions of
Section~\tag{sec-hyperbolic-space}, 
$(-1)$-reflections in negative norm vectors act on hyperbolic space as
inversions in points, rather than by reflections in
hyperplanes. This is why we focus on positive-norm vectors.
We call $r^\perp$ the 
mirror of the reflection.  Reflections of order 2, 3,$\,\ldots$
are sometimes called biflections, triflections, etc.  A
$\xi$-reflection is a biflection just if $\xi=-1$; in this case
we recover the classical notion of a reflection.

Suppose $L$ is an integral lattice.  If $v\in L$ has norm $1$
(resp. $2$) then we say that $v$ is a short (resp. long) root of
$L$.  Inspection of Eq.~\eqtag{eq-def-of-reflection} reveals that if
$\xi$ is a unit of $\ring$ then $\xi$-reflection in any short
root of $L$ preserves $L$.  Furthermore, biflections in long
roots of $L$ also preserve $L$.  We define the reflection group
$\reflec L$ to be the subgroup of $\aut L$ generated by
reflections (in positive-norm vectors), and we say that $L$ is
reflective if $\reflec L$ 
has finite index in $\aut L$.  In general, a group generated by
reflections is called a reflection group. Since $\aut L$ is an
arithmetically defined subgroup of the semisimple real Lie group
$U(L\tensor\R;\K)$, a theorem of Borel and Harish-Chandra
\cite{borel:arithmetic-groups} implies that it has finite
covolume therein. It follows that $L$ is reflective if and only
if $\reflec L$ also has finite 
covolume. We define $\shr L$ to
be the subgroup of $\reflec L$ generated by reflections in the
short roots of $L$.  It may happen that $\reflec L$  contains
reflections other than those in its roots, but we will not use
such reflections except briefly in the proof of
Theorem~\tag{thm-rho-stabilizer-finite-index}.

\section{\Tag{sec-lattice-reference}. Reference: examples of lattices}
 
This section contains background information on the various
specific complex and
quaternionic lattices we use; it should be
referred to only as necessary. We briefly define each
lattice as a module over $\ring$, list a few important
properties, and give references to the literature. The main
source is \ecite{splag}{Chap.~4}.  All lattices described
here are integral.  When lattices are described as subsets of
$\K^n$, it should be understood that  the Hermitian form is
$\ip{\vector{x}{n}}{\vector{y}{n}}=\sum\bar{x}_iy_i$.

The simplest lattice is $\ring^n$, which is obviously selfdual.
Its symmetry group contains the left-multiplication by each
diagonal matrix all of whose diagonal entries are units of $\ring$. It
is easy to see that this is the entire group preserving each of
the scalar classes of short roots, and thus must contain the
reflections in these roots. The group generated by these
reflections has the same order as the group of diagonal
matrices, so the groups  coincide. (The reason the argument is this
complicated is that the diagonal matrices act on the left,
whereas reflections are defined in terms of
right-multiplication. This is only important if
$\ring=\Hurwitz$.) Adjoining to this group the permutations of
coordinates, which are generated by biflections in long roots such
as $(1,-1,0,\ldots,0)$, we see that $\aut\ring^n$ is a
reflection group.

If $\Lambda$ is a lattice, then its {real form} is the
$\Z$-module $\Lambda$ equipped with the inner product
$(x,y)=\re\ip{x}{y}$.  Here are three forms of the $E_8$ root
lattice:

$$\displaylines{ E_8={1\over2}\set{\vec{x}{8}\in\Z^8 }{
x_i\equiv x_j \pmod{2},\;
\sum x_i\in 4\Z}\;, \cr
E_8^\gauss = {1\over1+\i }\set{\vec{x}{4}\in\gauss^4 }{ x_i\equiv
x_j
\pmod{1+\i },\; \sum x_i\in 2\gauss}\;, \cr
\qeeight = \set{(x_1,x_2)\in\Hurwitz^2 }{
x_1+x_2\in(1+\i )\Hurwitz}\;.
\cr}$$
It is straightforward to identify the real forms of these
$\ring$-lattices with each other; each has covering radius 1 and
minimal norm 2.  Often the dimension of a lattice is indicated
by a subscript. Unfortunately, as for $E_8$, this sometimes
refers to its dimension as a $\Z$-lattice and sometimes to its
dimension as an
$\ring$-lattice. There seems to be no universal solution to this
notational problem.

Another set of useful even Gaussian lattices are
$$
D_{2n}^\gauss=\set{\vector{x}{n}\in\gauss^n}{
\sum x_i\congruent 0\pmod{1+\i }}\;, 
$$
whose real forms are the $D_{2n}$
root lattices.
The $D_4$ lattice is also the real form of $\Hurwitz$,
scaled up by a factor of $2^{1/2}$. The covering radius of $D_{2n}$
is 
$(n/2)^{1/2}$. 

The Eisenstein lattice 
$$ 
D_3(\sqrt{-3})=\set{(x,y,z)\in\Eisenstein^3}{x+y+z\congruent0\pmod\theta}
$$
was introduced by 
Feit \cite{feit:unimodular-Eisenstein}.
It has $54$ long roots and $72$ vectors  of norm
$3$; biflections in the former and triflections in the latter
preserve the lattice. Its covering radius is
$1$; this can be seen as follows. 
According to \ecite{splag}{p. 126}, the real form of the lattice
$\set{(x,y,z)\in\Eisenstein^3}{x\congruent y\congruent z\ (\mod \theta)}$
is the $E_6$ root lattice scaled up by $(3/2)^{1/2}$. This
identification can be used to show that the real form of
$D_3(\sqrt{-3})$ is the real form of $E_6^*$ scaled up by
$(3/2)^{1/2}$, where $E_6^*$ is the dual (over $\Z$) of $E_6$.
By \ecite{splag}{p. 127}, the
covering radius of $E_6^*$ is $(2/3)^{1/2}$, so the
covering radius of $D_3(\sqrt{-3})$ is $1$.

The Coxeter-Todd lattice $\cct$ is a selfdual
$\Eisenstein$-lattice with minimal norm 2. It is discussed at length in
\cite{jhc:Coxeter-Todd};
we quote just one of the definitions given there.
$$
\cct={1\over\theta}\set{\vec{x}{6}\in\eisen }{
x_i\equiv x_j
\pmod{\theta\eisen},\; \sum x_i\in3\eisen}\;.
$$
%Its covering radius is $\sqrt{4/3}$, but the proofs of 
%Theorems~\tag{thm-eI7,1} and \tag{thm-qI5,1} show that the Leech
%holes of $\cct$ are more important for our purposes than the
%deep holes.
It automorphism group is the  finite
complex reflection group $6\cdot U_4(3){:}2$, and $\cct$
shares many
interesting properties with $E_8$ and the Leech lattice $\leech$. We refer to 
\cite{jhc:Coxeter-Todd} for details.

The quaternionic Barnes-Wall lattice is
$$
\qbwl={1\over 1+\i }\set{\vector{x}{4}\in\Hurwitz }{
x_i\congruent x_j \pmod{(1+\i )\Hurwitz},\;
\sum x_i\in2\Hurwitz}\;.
$$
We may recognize the real form of $2^{1/2}\qbwl$ by identifying
the vector 
$$
(a_1+b_1\i +c_1\j +d_1\k ,\ldots,a_4+b_4\i +c_4\j +d_4\k )
$$
with the vector in $\R^{16}$ whose coordinates we arrange in the
square array
% all this faffing about is to make sure that the array is
% square no matter which fonts are used
%
% user parameters
\newdimen\vpadding
\newdimen\hpadding
\vpadding=0pt
\hpadding=2pt
%
\newbox\boxa
\newbox\boxb
\newbox\boxc
\newbox\boxd
\setbox\boxa=\hbox{$\displaystyle\matrix{a_1&a_2\cr d_1&c_2\cr}$}
\setbox\boxb=\hbox{$\displaystyle\matrix{a_3&a_4\cr d_3&c_4\cr}$}
\setbox\boxc=\hbox{$\displaystyle\matrix{b_1&d_2\cr c_1&b_2\cr}$}
\setbox\boxd=\hbox{$\displaystyle\matrix{b_3&d_4\cr c_3&b_4\cr}$}
\newdimen\maxx
\newdimen\maxy
\newdimen\maxdim
\maxx=\wd\boxa
\ifnum\maxx<\wd\boxb \maxx=\wd\boxb\fi
\ifnum\maxx<\wd\boxc \maxx=\wd\boxc\fi
\ifnum\maxx<\wd\boxd \maxx=\wd\boxd\fi
\advance\maxx by \hpadding
\maxy=\ht\boxa
\ifnum\maxy<\ht\boxb \maxy=\ht\boxb\fi
\ifnum\maxy<\ht\boxc \maxy=\ht\boxc\fi
\ifnum\maxy<\ht\boxd \maxy=\ht\boxd\fi
\advance\maxy by \vpadding
\ifnum\maxx<\maxy \maxdim=\maxy \else \maxdim=\maxx\fi
\def\pad#1{\hbox to\maxdim{\hfil\vbox to\maxdim{\vfil #1\vfil}\hfil}}
$$
{4\over\sqrt8}\;
\vcenter{\halign{\vrule\pad{\box#}\vrule&\pad{\box#}\vrule\cr
\noalign{\hrule}%
\boxa&\boxb\cr
\noalign{\hrule}%
\boxc&\boxd\cr
\noalign{\hrule}}}
$$
where the inner product is the usual one on $\R^{16}$. This
array may be taken to be (say) the left 4 columns of the
$4\times6$ array of the MOG
description \ecite{splag}{Chaps.~4, 11} of the Leech lattice
$\leech$, and then the real form of $2^{1/2}\qbwl$ is visibly
the real Barnes-Wall lattice $\rbwl$ 
\ecite{splag}{Chap.~4}. 

\beginproclaim Theorem
\Tag{thm-barnes-wall-properties}.
The  lattice $\qbwl$ is selfdual and spanned by its
minimal vectors, which have norm
$2$. Its automorphism group is generated by the biflections in
its minimal vectors. Each class of $\qbwl$ modulo $\qbwl(1+\i )$ is
represented by a vector of norm at most $3$. 
The deep holes of $\qbwl$ coincide with the set 
$\set{\lambda(1+\i
)^{-1}}{\lambda\in\qbwl,\lambda^2\congruent1(2)}$.\endproclaim  

\beginproof{Proof:}
Proofs of all claims except the  last appear in
\ecite{bachoc:selfdual-Hurwitz-lattices}{Sect.~4.6}.  
Most of the rest of the work has been done for us by Conway and Sloane
\ecite{jhc:laminated-lattices}{Sect.~5}. They
showed that the deep
holes of $\rbwl$ nearest $0$ are the halves of certain
vectors $v\in\rbwl$ of norm 12, and further that 
such $v$ are not congruent modulo 2 to minimal vectors of $\rbwl$.
(They write $\Lambda_{16}$ for  $\rbwl$.)
After rescaling, we find that the deep holes of $\qbwl$ nearest
$0$ are the
halves of certain elements $v$ of norm $6$ in $\qbwl$. Since
each such $v$ has even norm 
and is not congruent modulo $2=(1-\i )(1+\i )$ to any root,
it must map to  $0$ in $\qbwl/\qbwl(1+\i )$. Therefore
$v=\lambda(1+\i )$ for some $\lambda$ of norm $3$ in $\qbwl$ and so the
deep holes nearest $0$ have the form
$v/2=\lambda(1+\i )/2=\lambda \i (1+\i)^{-1}$. 

The deep holes of $\qbwl$ are the translates
by lattice vectors of the deep holes nearest zero. That is, the
set of deep holes coincides with the set
$$ 
\set{\lambda(1+\i )^{-1}}{\hbox{$\lambda\in\qbwl$ is congruent modulo
$1+\i $ to a norm $3$ lattice vector}}\;.
$$
The norms of any two lattice vectors that are
congruent modulo $1+\i $ have the same parity. Since each lattice
vector is congruent to some vector of norm $0$, $2$ or $3$, the
set above coincides with the one  in the statement of the theorem.
%
\endproof

%
% INTERESTING BUT TANGENTIAL FACTS ABOUT BARNES-WALL LATTICE 
%
%Note that $\aut\rbwl$ is much larger
%than $\aut\qbwl$, having  order
%$2^{21}3^55^27=89\,181\,388\,800$ and structure $2^{1+8}_+\cdot
%O_8^+(2)$.  
%Embedding our array of
%coordinates in the MOG array also identifies $2^{1/2}\cdot\qbwl$
%as an $\Hurwitz$-sublattice of
%the quaternionic Leech lattice $\qleech$, which is described in 
%\oldcite{raw:quaternionic-leech}. The automorphism group of
%$\qleech$ is $2\cdot G_2(4)$, and inspection of the list of maximal
%subgroups of $G_2(4)$ in \oldcite{ATLAS} or
%\oldcite{raw:quaternionic-leech} shows that $O_6^-(2)$ is not
%involved in $G_2(4)$, and hence fairly few automorphisms of
%$\qbwl$ extend to $\qleech$. This contrasts with the real case;
%every automorphism of $\rbwl$ extends to $\leech$ (p. 131, \oldcite{splag}). 
%
%Finally, it is amusing to note that $\qbwl$ may be defined as
%the sublattice of $\Hurwitz\tensor\Hurwitz$ spanned by the
%vectors of the forms $(\hbox{norm }1)\tensor(\hbox{norm }2)$ and
%$(\hbox{norm }2)\tensor(\hbox{norm }1)$, where
%$\Hurwitz\tensor\Hurwitz$ is equipped with the inner product
%$$
%\ip{a\tensor b}{c\tensor d}=\re(\bar ac)\bar bd\;.
%$$

Now we describe some indefinite selfdual lattices. 
The lattice  $\rI n,m$
is the $\ring$-module $\ring^{n+m}$ equipped with the
inner product given by the diagonal matrix 
$$ 
\diag[+1,\ldots,+1,-1,\ldots,-1] 
$$
with $n$ (resp. $m$) $+1$'s (resp. $(-1)$'s). 
The lattice $\rII1,1$ is the module $\ring^2$ with 
inner product matrix $\smallmatrix0110$. If $\ring=\Eisenstein$
or $\Hurwitz$ then $\rII1,1\isomorphism \rI1,1$. If
$\ring=\Gaussian$ then this lattice is even, whereas $\gI1,1$ is
odd. We define the Gaussian lattices $\gII 4m+n,n$ to be the
lattices 
$$ 
\gII 4m+n,n=\geeight\oplus\cdots\oplus\geeight\oplus
	\gII1,1\oplus\cdots\oplus\gII1,1\;, 
$$
where there are $m$ summands $\geeight$ and $n$ summands
$\gII1,1$. These lattices are even and selfdual.
By
Theorem~\tag{thm-indefinite-selfdual-classification},
every indefinite selfdual lattice over $\ring$ appears among the
examples just given. In particular,
$\cct\oplus\eII1,1\isomorphism\eI7,1$ and
$\qbwl\oplus\qII1,1\isomorphism\qI5,1$.

%
% An nice clean proof of all the facts about Barnes-Wall quoted
% from Bachoc; I 
% came up with this proof before learning of her work.
%
%!!Theorem 
%\oldTag{thm-Barnes-Wall-lattice-selfdual}.
%$\qbwl$ is an integral selfdual $\Hurwitz$-lattice with no
%elements of norm 1.
%!!

%\beginproof{Proof:}
%It is easy to see that $\qbwl$ contains no elements of norm 1.
%The vectors
%$$\eqalign{
%v_1 & =  (\i+1,0,0,0), \cr
%v_2 & =  (1,-1,0,0), \cr
%v_3 & =  (0,1,-1,0),\hbox{ and}\cr
%v_4 & =  (1+\i,1+\i,1+\i,1+\i)/2
%\cr}$$
%form a basis for $\qbwl$: if $x\in\qbwl$, then by adding a
%multiple of $v_4$, we may suppose that $x_i\in\Hurwitz$. Then,
%by adding multiples of $v_4-\i v_4=(1,1,1,1)$, $v_3$, and $v_2$,
%we may suppose that $x_4=x_3=x_2=0$. But then since 
%$\sum x_i\in(1+i)\Hurwitz$, we see that $x$ is a multiple of $v_1$.
%$\qbwl$ is integral because $\ip{v_i}{v_j}\in\Hurwitz$ for all
%$i$ and $j$. To see that $\qbwl$ is selfdual, observe that the
%span of the $v_i$ over $\gauss$ is a copy of $E_8^\gauss$, which
%being selfdual over $\gauss$ contains a basis dual to the $v_i$.
%%invent%\QED
%\endproof

%% here's some notation I might change.
%\def\U{U}			% +/-{1,i,j,k}
%\def\scalact#1,#2,#3,#4{[#1,#2,#3,#4]}	% left multiplies some
%					% coordinates by some scalars
%%\def\scalars{\Sigma}		% the group of such maps
%%\def\monom{\scalars.S_4}	% the monomial group
%\def\monom{\Sigma_4}
%\def\qbwln#1{{\qbwl}^{(#1)}}  %vectors in \qbwl with norm #1

%We now observe some symmetries of $\qbwl$. We will use the ATLAS
%notation \oldcite{ATLAS} for groups, and we write $\U$ for the
%subgroup $\{\pm1,\pm \i,\pm \j,\pm \k\}$ of index 3 in the units of
%$\Hurwitz$. The permutations of coordinates obviously lie in
%$\aut\qbwl$, and are in fact generated by reflections in vectors
%such as $(1,-1,0,0)$. Reflection in $v_1$ acts by negating
%vectors'  first coordinates, and if $u\in\U$ then reflection in
%$(1,u,0,0)$
%followed by a transposition of coordinates provides
%the transformation $\scalact -u,-\bar u,1,1$, where
%$\scalact u_1,u_2,u_3,{u_4}$ denotes the map
%$(x_1,\ldots,x_4)\mapsto(u_1x_1,\ldots,u_4x_4)$. 
%The generators found so far generate a group containing 
%all $\scalact u_1,u_2,u_3,{u_4}$
%with $u_1u_2u_3u_4=\pm1$; together with the permutations, they
%generate the group $\monom$ of order $8^3\,2{\cdot}4!=24\,576$. 
%Left multiplication by $\w$
%provides another symmetry, but we will not need this.

%We write $\qbwln n$ for the set of elements of $\qbwl$ of norm $n$.
%It is straightforward to enumerate the elements of $\qbwln n$
%for any given $n$. Here are the results for $n=2$ and $n=3$;
%they are listed by giving a typical representative under the
%action of $\monom$.

%\calign{\indent$#$\qquad\hfil&$#$\hfil&\quad=\hfil$#$\cr
%\noalign{Norm=2:}
%(1+\i,0,0,0)\w^n&3.4.8&96\rlap{\hbox{ vectors}}\cr
%(1,1,0,0)\w^n&3\,{4\choose2}\,8^2&1\,152\cr
%(1,1,1,1)(\i+1)\w^n/2&3.8^3.2&3\,072\cr
%&&\overline{4320}\rlap{\hbox{ total}}\cr
%\noalign{Norm \=3:}
%(1,\w,\bar\w,0)&4!\,8^3&12\,288\rlap{\hbox{ vectors}}\cr
%(1+\i,1+\i,1+\i,1-\i+\ell)\w^n/2&3.4.4.8^3.2&49\,152\cr
%\noalign{\indent\qquad(4 orbits; $\ell=\pm2\j$ or $\pm2\k$)}
%&&\overline{61\,440}\rlap{\hbox{ total}}\cr
%}

%!!Theorem 
%\oldTag{thm-Barnes-Wall-lattice-group}.
%$\aut\qbwl$ has order $2^{13}3^45=3\,317\,760$, structure
%$2^{1+6}_-.O^-_6(2)$, and acts transitively on $\qbwln2$ and
%on $\qbwln3$. Modulo $\qbwl(1+i)$, the vectors of $\qbwl$ fall into
%$135$ classes each represented by $32$ roots, and $120$ classes each
%represented by $512$ elements of $\qbwln3$, together with the
%class represented by $0$.
%!!

%\beginproof{Proof:}
%Consider the quotient map
%$q:\Hurwitz\rightarrow\Hurwitz/(1+\i)\Hurwitz\isomorphism\F_4$. We
%denote the image of $\w$ in $\F_4$ again by the symbol $\w$, so
%$\F_4=\{0,1,\w,\bar\w\}$, with $\w^3=1$. Note that quaternion
%conjugation in $\Hurwitz$ maps to the field automorphism of
%$\F_4$. We set $V=\qbwl/\qbwl(1+\i)$ and also denote by $q$ the
%natural map $\qbwl\rightarrow V$. $V$ is a
%vector space over  $\F_4$;
%the action of  $\aut\qbwl$ on $V$ preserves the nonsingular unitary form
%$\ip{q(v)}{q(w)}=q(\ip{v}{w})$. Elementary geometry
%shows that if 2 roots of $\qbwl$ are congruent modulo $(1+\i)$
%then either they are orthogonal or they are proportional by an element of
%$\U$. Therefore at least 4320/32=135 points of $V$ are images of
%roots. All nonsingular unitary forms on a 4-dimensional vector space over
%$\F_4$ are equivalent, and have only 135 isotropic points
%(besides 0). Therefore the roots must fall into sets of 32, as
%claimed. We call such a set a {frame}. Furthermore, the
%reflections in these roots map to the 45 transvections along the
%isotropic vectors of $V$; these form a conjugacy class in and
%hence generate the simple group $U_4(2)$. This group acts
%transitively on the nonzero isotropic points of $V$, so
%$\reflec\qbwl$ acts transitively on  frames.

%One frame is the orbit under $\monom$ of $(1+\i,0,0,0)$. Modulo
%$\monom$, any transformation fixing this frame may be taken to
%fix $(1,0,0,0)$, $(0,1,0,0)$, and $(0,0,1,0)$. But then
%$(0,0,0,1)$ must map to itself or to its negative, and both
%possibilities can be achieved by an element of $\monom$. Since
%$\monom$ is generated by reflections, this simultaneously
%identifies $\monom$ as the stabilizer of a frame and shows that
%$\aut\qbwl$ is a reflection group of order
%$135{|\monom|}=135\cdot8^32\cdot4!$. 

%We now identify the kernel $K$ of the action of $\aut\qbwl$ on
%$V$. Inspection reveals that simultaneous negation of any two
%coordinates fixes (modulo $1+\i$) each of the vectors $v_1,\ldots,v_4$ given in
%the proof of Theorem~\tag{thm-Barnes-Wall-lattice-selfdual},
%and so we conclude that the product of
%reflections in any two elements of a frame lies in
%$K$. (Another way to see this is to observe that a transvection
%of $U_4(2)$ is an involution.) 
%Simple computation now shows that  
%the permutations of coordinates with cycle shape
%$2^2$ lie in $K$, as do all maps
%$u\scalact\pm1,\pm1,\pm1,{\pm1}$ with $u\in\U$ and an even
%number of $-1$'s. These transformations generate a group of
%order $2^2\cdot8\cdot2^2=2^7={|\aut\qbwl|}/{|U_4(2)|}$ which
%must be all of $K$. The  central involution
%$\scalact-1,-1,-1,{-1}$ lies in the center of $K$ and modulo it
%all elements of $K$ 
%commute. Inspecting the quadratic form induced on the $\F2$
%vector space $K/\scalact-1,-1,-1,{-1}$ by the central extension,
%we see that $K$ is an 
%extraspecial 2-group of type $2^{1+6}_-$. Since
%$U_4(2)\isomorphism O_6^-(2)$, $\aut\qbwl$ has the structure claimed.

%Every vector  of $\qbwln3$ maps to a nonisotropic element of
%$V$. Since $U_4(2)$ acts transitively on these 120 elements,
%each is represented by the same number of vectors in $\qbwln3$,
%namely 61440/120=512. Furthermore, to demonstrate transitivity
%of $\aut\qbwl$ of $\qbwln3$ it suffices to show transitively
%on any such set of 512. The vectors $(1,\w,\bar\w,0)$,
%$(1,\i\w,\j\bar\w,0)$, $(1,\j\w,\k\bar\w,0)$, and
%$(1,\k\w,\i\bar\w,0)$ are all congruent modulo $(1+\i)$ and their
%orbits under $K$ are disjoint. Since no nontrivial element of $K$
%fixes any of them, their $4\cdot2^7=512$ images under $K$ form
%one such set. These are all obviously equivalent under $\monom$.
%%invent%\QED
%\endproof

\section{\Tag{sec-hyperbolic-space}. Hyperbolic space}

The
hyperbolic space
$\khnp$ ($n\geq0$) is defined as the image in projective
space $\kpnp$ of the
set of vectors of negative norm in $\K^{n+1,1}$; its boundary
$\dkhnp$ is the image of the (nonzero) null vectors. We
write elements of $\K^{n+1,1}$ in the
form $(\lambda;\mu,\nu)$ with $\lambda\in\K^{n,0}$ and
$\mu,\nu\in\K$, with inner product  
$$
\ip{(\lambda_1;\mu_1,\nu_1)}{(\lambda_2;\mu_2,\nu_2)}=
\ip{\lambda_1}{\lambda_2} + \bar{\mu}_1\nu_2 + \bar{\nu}_1\mu_2\;.
$$
This corresponds to a decomposition
$\K^{n+1,1}\isomorphism\K^{n,0}\oplus\smallmatrix{0}{1}{1}{0}$. 
We will sometimes refer to points in projective space by naming
vectors in the underlying vector space.

It is convenient to distinguish  the
isotropic vector $(0;0,1)$ and give it the name $\rho$.  
Every point of $\khnp\cup\dkhnp$ except
$\rho$ has a unique preimage in $\K^{n+1,1}$ with inner product
$1$ with $\rho$, and so we may make the identifications
$$
\eqalign{
\khnp=\{(\lambda;1,z):\lambda\in\K^n,
\lambda^2+2\re(z)>0\}\;.\cr
\dkhnp\setminus\{\rho\}=\{(\lambda;1,z):\lambda\in\K^n,
\lambda^2+2\re(z)=0\}\;.\cr
}\eqno\eqTag{eq-half-space-coordinates}
$$ 
We define the height of a vector $v\in\K^{n+1,1}$ to be
$\height v=\ip{\rho}{v}$. For $v=(\lambda;\mu,\nu)$ we simply have
$\height v=\mu$. For vectors of any fixed norm, the height function
measures how far away from $\rho$ the corresponding points in
projective space are; the smaller the height, the
closer to $\rho$. We will sometimes say that for vectors $v$ and
$v'$ the height of $v'$ is less than that of $v$. By this we
will mean that $|\height v'|<|\height v|$.

We say that a vector  $(\lambda;\mu,\nu)$ with $\mu\neq0$ lies
over $\lambda\mu^{-1}\in\K^{n}$. It is obvious that all the
scalar multiples of any given vector of nonzero height lie over
the same point of $\K^n$, so we may think of points in
projective space (except for those in $\rho^\perp$) as lying
over elements of $\K^n$. The geometric content of this
definition is that the lines in $\kpnp$ passing through $\rho$
and meeting $\khnp$ are in one-to-one correspondence with 
$\K^n$. The points in the line associated to
$\lambda\in\K^n$ are just the scalar multiples of those of the
form $(\lambda;1,z)$ with $z\in\K$. Special cases are given in
Eq.~\eqtag{eq-half-space-coordinates}. In particular, the family of
isotropic vectors  of height one lying over $\lambda$ are
parameterized by the elements of $\im\K$. This description of
$\dkhnp\setminus\{\rho\}$ as a bundle over $\K^n$ with fiber
$\im\K$ will help us relate the properties of lattices in $\K^n$
to properties of groups acting on $\khnp$.

The subgroup  of $U(n+1,1;\K)$ fixing $\rho$ contains
transformations $T_{x,z}$  (with $x\in\K^n$,
$z\in\im\K$) defined by
$$\eqalignno{
\rho & \mapsto  \rho  \cr 
\llap{$T_{x,z}$:\qquad}
(0;1,0) & \mapsto  (x;1,z-x^2/2)&\eqTag{eq-def-of-translation}  \cr
(\lambda;0,0) & \mapsto  (\lambda;0,-\ip{x}{\lambda}) 
\rlap{\qquad for each  $\lambda\in\K^n$.} 
\cr}$$
(The map is defined in terms of some unspecified but
fixed inner product on $\K^n$.)
We call these maps  translations. If we regard
elements 
of $\K^{n+1,1}$
as column vectors then  $T_{x,z}$ acts by multiplication on the
left by the matrix
$$
\pmatrix{
I_n 	& x 		& 0 	\cr 
0 	& 1		& 0	\cr
-x^*	& z-x^2/2	& 1	\cr}\;.
$$
(We have  written  $x^*$
for  the linear function $y\mapsto\ip{x}{y}$ 
on $\K^{n,0}$ defined by $x$.) We have the relations
$$\displaylines{
T_{x,z}\circ T_{x',z'} = 
T_{x+x',z+z'+\im\ip{x'}{x}} 
\hfil\llap{\eqTag{eq-product-of-translations}}\hfilneg \cr
T_{x,z}^{-1} = T_{-x,-z}
\hfil\llap{\eqTag{eq-inverse-of-translation}}\hfilneg \cr 
T_{x,z}^{-1}\circ T_{x',z'}^{-1}\circ T_{x,z}\circ T_{x',z'} =
T_{0,2\im\ip{x'}{x}}\;,
\hfil\llap{\eqTag{eq-commutator-of-translations}}\hfilneg
\cr}$$
which are most easily verified in the order listed.
These relations make it
clear that  the
translations form a group and that its
center and commutator subgroup coincide and consist of the
$T_{0,z}$. We call elements of this subgroup
central translations. The translations form 
a (complex or quaternionic) Heisenberg group which  acts
freely and  transitively
on $\dkhnp\setminus\{\rho\}$. 
%In Section~\tag{sec-26dim} our arrangement
%of mirrors was invariant under a group $\leech\isomorphism\Z^{24}$;
%these symmetries are translations in the current sense. 
If $v\in\K^{n+1,1}$ lies over $\lambda\in\K^n$ then $T_{x,z}(v)$
lies over $\lambda+x$. That is, the translations act in the
natural way (by translations) on the points of $\K^n$
over which vectors in 
$\K^{n+1,1}$ lie. 

We note that these constructions all make sense with $\K=\R$,
and in fact simplify. Since $\im\R=0$, the translations form
an abelian group, which is just the obvious set of translations
in the usual upper half-space model for $\rhnp$. The 
obvious projection map from the upper half-space to $\R^n$ 
carries points of $\rhnp$ to the points of $\R^n$ over which
they lie, in the sense defined above. This is the source of the
terminology. 

The simultaneous stabilizer of $(0;1,0)$ and $(0;0,1)$
is the  unitary group $U(n,0;\K)$, which fixes pointwise the
second summand in the decomposition
$\K^{n+1,1}=\K^{n,0}\oplus\K^{1,1}$. If $S$ is an
element of this unitary group then matrix computations reveal 
$$
S\circ T_{x,z}\circ S^{-1}=T_{Sx,z}\;.
\eqno\eqTag{eq-conjugate-of-translation}
$$
This is useful in its own right and also
shows that the group of translations is  normal in the full
stabilizer of   
$\rho$.

\section{\Tag{sec-reflections}. Reflections in Lorentzian
lattices}

The Lorentzian lattices we will consider all have the form
$\Lambda\oplus\II1,1$, where $\Lambda$ is a
positive-definite $\ring$-lattice and $\II1,1$ is 
the 2-dimensional selfdual lattice defined
by the matrix
$\II1,1=\smallmatrix0110$.
In general we will write $L$ for a Lorentzian lattice
$\Lambda\oplus\II1,1$, where $\Lambda$ and even $\ring$ may be
left unspecified, except that $\Lambda$ will always be
positive-definite.  
We write elements of $L=\Lambda\oplus\II1,1$ in the form
$(\lambda;\mu,\nu)$ with $\lambda\in \Lambda$ and
$\mu,\nu\in\ring$. This embeds $L$ in the description of
$\K^{n+1,1}$ given in Section~\tag{sec-hyperbolic-space} and
allows us to transfer 
to $L$ several important concepts defined there. In particular,
$\rho=(0;0,1)$ is an element of $L$ and we define the height of
elements of $L$ as before. For $v\in L$ of nonzero height we can
speak of the point of $\Lambda\tensor\R$ over which $v$ lies
(which need not be an element of $\Lambda$).

There are two basic ideas in this section. First, that this
description of $L$ provides a  way to write down a large
collection of reflections of $L$, essentially parameterized by
the elements of a discrete Heisenberg group of translations. The
second idea is that if \rom1 $r$ is a root of $L$, 
\rom2 $v$ is a null vector in $\K^{n+1,1}$, \rom3 neither $r$
nor $v$ has height zero, and \rom4 the points of $\K^n$ over
which $r$ and $v$ lie are sufficiently close, then by applying a
reflection of $L$ one can reduce the height of $v$. (This
reflection might be in some root other than $r$.)

Both of these ideas can be found, in the simpler setting of real
hyperbolic space,  in Conway's study \cite{jhc:26dim} of the
automorphism group of the Lorentzian $\Z$-lattice
$\II25,1=\leech\oplus\II1,1$. Here $\leech$ is the famous Leech
lattice, and Conway found a set of reflections permuted freely by a
group of translations naturally isomorphic to the additive group
of $\leech$. By using facts about the covering radius of
$\leech$ and using the second idea described above, he was
able to prove that these reflections generate the entire
reflection group of $\II25,1$.

The major complication in transferring this approach to
our setting is that the discrete group of translations is no
longer a copy of $\Lambda$ but a central extension of $\Lambda$
by $\im\ring$. This  issue  dramatically complicates the
precise formulation (Theorems~\tag{thm-what-long-roots-do} and
\tag{thm-what-short-roots-do}) of the second main idea.
For example, it is complicated to state exactly what happens
when one can't {\it quite\/} reduce the height of $v\in\K^{n+1,1}$ by
using a reflection.

We begin by finding the translations in $\aut L$ and showing that
under simple conditions, $\reflec L$ contains a large number of
them. The translation $T_{x,z}$
preserves $L$ just if $x\in \Lambda$ and $z-x^2/2\in\ring$. If
$\ring=\Eisenstein$ or $\Hurwitz$ then for any given $x\in \Lambda$
we may  choose $z\in\im\K$ such that $T_{x,z}\in\aut L$, by
taking $z=0$ or $\theta/2$ according as $x^2$ is even or
odd. If $\ring=\Gaussian$ then such a $z$ exists if and only if
$x^2$ is even;  $z$ may then be taken to be zero.  
The different rings behave differently because
$\Eisenstein$ and $\Hurwitz$ contain elements with half-integral
real parts, while $\Gaussian$ does not.  
All the central translations $T_{0,z}$ with
$z\in\ring$  lie in $\aut L$\emdash they fix $\Lambda$ pointwise and act by
isometries on the $\II1,1$ summand.
The assertions of the next theorem are precise
formulations of the idea that if $\aut\Lambda$ contains many 
reflections then $\reflec L$ contains many translations.

\beginproclaim Theorem
\Tag{thm-rho-stabilizer-finite-index}.
Let $L=\Lambda\oplus\II1,1$ for some  positive-definite
$\ring$-lattice $\Lambda$. Define
$$\displaylines{
\Lambda_0=\set{x\in \Lambda}{T_{x,z}\in\reflec L
\hbox{\rm\  for some }z\in\im\K}
\hbox{ and}\cr
\cals=\{z\in\im\ring \mid T_{0,z}\in\reflec L\}\;. 
\cr}$$
%
\item{\rom1}
If no element of $\Lambda$ is fixed by every reflection of
$\Lambda$ then $\Lambda_0$ has finite index in $\Lambda$.
\item{\rom2}
If $r$ is a root of $\Lambda$ (a long root, if
$\ring=\Gaussian$) having 
inner product $1$ with some element of $\Lambda$, then
$r\in \Lambda_0$.
\item{\rom3}
$\cals$ contains the integral span of
the elements of 
the form $2\im\ip{x}{y}$ with $x,y\in \Lambda_0$.
\item{\rom4}
Under the hypothesis of \rom1, the stabilizer of $\rho$  in
$\reflec L$ has finite index in the stabilizer  in
$\aut L$.\par\endproclaim 

\remark{Remarks:}
By Eq.~\eqtag{eq-product-of-translations}
and \eqtag{eq-inverse-of-translation}, $\Lambda_0$ is closed
under addition and negation, 
so  assertion
\rom1 makes sense.
We will not need  an analogue of \rom2 for short roots in
Gaussian lattices.
In Theorem~\tag{thm-rho-stabilizer-finite-index-2}, similar but
stronger hypotheses are used to 
obtain similar but stronger conclusions.


\beginproof{Proof:}
Let $R$ be a $\xi$-reflection of $\Lambda$ with mirror $M$. We
regard $R$ as acting on $L$, fixing the summand $\II1,1$ 
pointwise. If $T_{x,z}\in\aut L$ then  
$T_{x,z}^{-1}\circ R\circ T_{x,z}\in\reflec L$. By
Eqs.~\eqtag{eq-inverse-of-translation}, 
\eqtag{eq-conjugate-of-translation} and
\eqtag{eq-product-of-translations}, 
$$ 
T_{x,z}^{-1}\circ R\circ T_{x,z}\circ R^{-1}=
T_{-x,-z}\circ T_{Rx,z} = T_{Rx-x,-\im\ip{Rx}{x}}\;,
$$
proving that
$$
Rx-x\in \Lambda_0 
\eqno\eqTag{eq-got-translation}
$$
for all $x\in \Lambda$ and all reflections $R$ of $\Lambda$.
The geometric picture behind this computation is
that both $M$ and its translate by $T_{x,z}^{-1}$ pass 
through $\rho$ and are parallel there. 
It is not surprising that one can construct translations out of
reflections in pairs of  parallel mirrors.

\rom1 
We will show first that $\Lambda_0$ contains an
$\ring$-sublattice orthogonal to $M$. Suppose $\xi$ is an $n$th
root of unity. Then the self-map ($n$ times the projection to
$M$) of $\Lambda$ given by $x\mapsto\sum_{a=1}^nR^a(x)$ is an
$\ring$-lattice endomorphism. It rank as a map on $\Lambda$
is the same as its rank as a map on the
$\K$-vector space $\Lambda\tensor\R$. Since $R$ is a reflection,
this rank is
$\dimension \Lambda-1$, and we see that $\Lambda$ contains a
1-dimensional $\ring$-sublattice orthogonal to $M$. If $x$
is a nonzero element of this lattice
then $Rx-x$ is also, and 
Eq.~\eqtag{eq-got-translation} shows that
$Rx-x$ lies in $\Lambda_0$. This shows that $\Lambda_0$ contains a
1-dimensional $\ring$-sublattice  orthogonal to
$M$. By hypothesis, no vector of $\Lambda$ is orthogonal to
every mirror of $\Lambda$, so $\Lambda_0$  has finite index
in $\Lambda$.

\rom2
If $r^2=1$ then we only need to prove anything in the cases
$\ring=\Eisenstein$ or $\Hurwitz$.  
Take $x=r$ and  $R$ to be the $(-\w)$-reflection in
$r$. Then Eq.~\eqtag{eq-got-translation} shows that $\Lambda_0$
contains $Rx-x=r(-\w)-r=r\wbar$. 
Applying the same argument to $r\w$ we see that $r\in
\Lambda_0$. 

If $r^2=2$ then in Eq.~\eqtag{eq-got-translation} we take $R$ to be the
biflection in $r$ and by hypothesis we may take $x\in \Lambda$
with $\ip{x}{r}=1$. We find 
that $Rx-x$ is a multiple of $r$ because it lies in the
$(-1)$-eigenspace of $R$. Since it has inner product $-2$ with
$r$, we have $Rx-x=-r$, which  implies   $r\in \Lambda_0$.

\rom3 Follows immediately from
Eq.~\eqtag{eq-commutator-of-translations} by taking commutators of 
translations of $\reflec L$.

\rom4
The null vectors  of height $1$ in $L$ are exactly those vectors
$(\lambda;1,z)$ with $\lambda\in \Lambda$, $z\in\ring$ and $\re
z= -\lambda^2/2$; the translations in $\aut L$ permute them
transitively. Since the simultaneous stabilizer of $\rho$ and
one of these, say $(0;1,0)$, is the finite group $\aut
\Lambda$,  it 
suffices to prove that the group of translations in $\reflec L$
has finite index in the group of those in $\aut L$. This follows
from the facts that $\Lambda_0$ has finite index in $\Lambda$
and $\cals$ has finite index in $\im\ring$. The former fact is
\rom1 and the latter follows from \rom1 and \rom3.
%
\endproof

It is straightforward to
enumerate the roots of $L$ of any given height $h$; 
For $h=1$ one finds that these are the vectors 
$$\eqalign{
\hbox{Norm 2:}\qquad& (\lambda;1,z),\qquad  \re z=(2-\lambda^2)/2\cr
\hbox{Norm 1:}\qquad& (\lambda;1,z),\qquad  \re z=(1-\lambda^2)/2\;,
\cr}$$
with $\lambda\in \Lambda$ and $z\in\ring$.
If $\ring=\eisen\hbox{ or }\Hurwitz$, then 
height $1$ 
roots of both norms lie over each $\lambda\in \Lambda$, and  the 
translations of $L$ act simply  transitively on each set. If
$\ring=\gauss$ then height one  roots lie over each
$\lambda\in \Lambda$: long roots over $\lambda$ of even norm and
short roots over $\lambda$ of odd norm.
Again, the translations  act simply transitively on each
set. This is another manifestation of the fact that
$\Eisenstein$ and $\Hurwitz$ but not $\Gaussian$ have elements
with half-integer real part.
One may also enumerate roots of larger heights\emdash for example, if $\Lambda$
is an $\eisen$-lattice, then there are short (resp. long) roots of $L$
of height $\theta$ 
over each $\lambda\theta^{-1}\in \Lambda\theta^{-1}$ with 
$\lambda^2\congruent 1$ (resp. $2$) modulo $3$.
For more information see Table~\tag{tab-table-of-balls} and the
discussion concerning it.

Now we will develop the second main idea of this section, by
investigating the effects of reflections in roots of small
height $h$. We first deal with long roots and then with short
ones. The results are complicated to state because the exact results
one can obtain vary slightly with the choice of $\ring$ and
$h$. The essential ideas are all present in the
$\ring=\Eisenstein$, $h=1$ case.

\beginproclaim Theorem
\Tag{thm-what-long-roots-do}.
Suppose $\Lambda$ is a positive-definite $\ring$-lattice and $L=
\Lambda\oplus\II1,1$. Let $h$ be
$$
\eqalign{
\hbox{\rlap{$1$}%
\phantom{\hbox{$1$ or $\theta$}}}\qquad\qquad
&\hbox{if $\ring=\Gaussian$ or $\Hurwitz$, or}\cr
\hbox{$1$ or $\theta$}\qquad\qquad
&\hbox{if $\ring=\eisen$.}
\cr}
$$
Suppose $r$ is a long root of $L$ of height $h$, lying over
$\lambda h^{-1}$ for $\lambda\in \Lambda$. Let $v\in L\tensor\R$
be isotropic, have height one and lie over
$\ell\in\Lambda\tensor\R$. Set $D^2=(\ell-\lambda h^{-1})^2$
and suppose $D^2\leq1/|h|^2$ (or $D^2\leq 3^{1/2}$ if
$\ring=\Gaussian$). Then there exists a long root $r'$ of $L$ of
height $h$ also lying over $\lambda h^{-1}$ such that either
%
\item{\rom1}
biflection in $r'$ carries $v$ to a vector of height smaller
than that of $v$,
\par\noindent
or else one of the following holds:
\item{\rom2}
$\ring=\Gaussian$, $D^2=3^{1/2}$ and $\ip{r'}{v}=1-3^{1/2}/2+\i /2$.
\item{\rom3}
$\ring=\Eisenstein$ or $\Hurwitz$, $D^2=1/|h|^2$ and
$\ip{r'}{v}=h^{-1}(\w+1)$. \par\endproclaim 

\beginproof{Proof:}
Since $v$ has height $1$ and norm $0$ and 
lies over $\ell$, we
know that for some $w\in\im\K$ we have
$$ 
v=(\ell;1,w-\ell^2\negspace/2)\;.
$$
Similarly, we deduce that 
$$ 
r=\(\lambda;h,z_0+{2-\lambda^2\over2|h|^2}h\) 
\eqno\eqTag{eq-coordinates-of-r}
$$
for some $z_0\in\K$ such that $\re(\bar{h}z_0)=0$. Every other
long root $r'$ of $L$ with height $h$ that lies over
$\lambda h^{-1}$ has the form $r'=r+(0;0,z)$ for some $z\in\ring$
satisfying $\re(\bar hz)=0$.  We will obtain the theorem by choosing $z$
carefully.

We have
$$\eqalignno{
\ip{r'}{v}
&=\ip{\lambda}{\ell}+\bar h(w-\ell^2/2)+
	\({(2-\lambda^2)\bar h\over2|h|^2}+\bar{z}_0+\bar z\)\cr
&=h^{-1}\[h\ip{\lambda}{\ell}+|h|^2w-{|h|^2\ell^2\over2}
	+{2-\lambda^2\over 2} + h\bar{z}_0 +h\bar z\]\cr
&=h^{-1}\[1-{|h|^2\over2}\Bigl(\ell^2-2\ip{\lambda h^{-1}}{\ell}
	+(\lambda h^{-1})^2\Bigr) + |h|^2w+h\bar{z}_0+h\bar z\]\cr
&=h^{-1}\[1-{|h|^2\over2}\Bigl(\ell^2-\ip{\lambda h^{-1}}{\ell}
	-\ip{\ell}{\lambda h^{-1}}
	+(\lambda h^{-1})^2\Bigr)\right.\cr
	&\qquad\qquad\qquad
	\left.-{|h|^2\over2}\Bigl(-\ip{\lambda h^{-1}}{\ell}+
	\ip{\ell}{\lambda h^{-1}}\Bigr)
	+ |h|^2w+h\bar{z}_0+h\bar z\]\cr
&=h^{-1}\[\(1-{|h|^2\over2}D^2\) + 
	|h|^2\im\ip{\lambda h^{-1}}{\ell} + |h|^2 w +
	h\bar{z}_0+h\bar z\]\cr
&=h^{-1}[a+B]&\eqTag{eq-ip-with-r'}\;,\cr}
$$
where $a=1-|h|^2D^2/2$ is the real part of the term in brackets 
and $B$ is the 
imaginary part. (Note that $h\bar{z}_0$
and $h\bar z$ are imaginary because for all $x,y\in\K$, 
$\re(x\bar y)=\re(\bar yx)=\re(\overline{\bar yx})=\re(\bar xy)$
and we know that $\bar hz_0$ and $\bar h z$ are imaginary.)
The important thing to observe here is that $a$ contains
information about $D^2$, which is bounded by hypothesis, and $B$
contains a term $h\bar z$, over which we have some influence.

Now let $v'$ be the image of $v$ under biflection in $r'$. Since
$v'=v-r'\ip{r'}{v}$, we have
$$\eqalignno{
\ip{\rho}{v'}
&=\ip{\rho}{v}-\ip{\rho}{r'}\ip{r'}{v}\cr
&=1-hh^{-1}(a+B)\cr
&=|h|^2D^2/2-B\cr
&=a'-B\;,&\eqTag{eq-height-of-v'}
\cr}$$
where $a'=1-a=|h|^2D^2/2$ is real and $B$ is as above.
We may take $z$ to be any element of 
$\ring$
satisfying $\re(\bar h z)=0$;  this condition may be
written $z\in\ring\cap (h\cdot\im\K)$. Since the definition of $B$
involves the term $h\bar z$ we see that by changing our choice
of $z$ we may change $B$ by any element of
$$\eqalign{ 
h\cdot\overline{\ring\cap(h\cdot\im\K)}
&=h\cdot(\ring\cap((\im\K)\cdot\bar h))\cr
&=(h\ring)\cap(h(\im\K)\bar h)\cr
&=(h\ring)\cap\im\K\cr
&=\im(h\ring)\;.\cr}
$$
We will now consider the possibilities for $\ring$ separately.

If $\ring=\Gaussian$ then we only need to prove anything for the
case $h=1$; then $\im(h\Gaussian)=\im\Gaussian$, so
by making an appropriate choice of $z$ we may suppose that 
$B=b\i $ with $b\in(-1/2,1/2]$. By hypothesis,
$a'\leq 3^{1/2}/2$. Therefore
$|\height(v')|^2=(a')^2+|B|^2$ is less than $1=|\height v|^2$
(so that conclusion \rom1 applies) unless $a'=3^{1/2}/2$ and
$b=1/2$. In this case, $D^2=3^{1/2}$ and by Eq.~\eqtag{eq-ip-with-r'} we have 
$\ip{r'}{v}=h^{-1}(a+B)$, so conclusion \rom2 applies.

In each of the remaining cases ($\ring=\Hurwitz$ and $h=1$;
$\ring=\Eisenstein$ and 
$h=1$ or $\theta$), we observe that
$\im(h\ring)=\im\ring$ and recall that $\im\ring$ has covering
radius $3^{1/2}\negspace/2$. That is, by making an appropriate choice of
$z$ we may suppose $|B|^2\leq3/4$. 
We have assumed that
$D^2\leq1/|h|^2$, so $a'\leq1/2$. Therefore
$|\height(v')|^2=(a')^2+|B|^2$
is less than $1=|\height(v)|^2$ (so that conclusion \rom1
applies) unless $a'=1/2$ and $|B|^2=3/4$.
In this case we see that
$D^2=1/|h|^2$ and that
$B$ is a deep hole of $\im\ring$. Since all deep holes of
$\im\ring$ are equivalent by translations of $\im\ring$, by
choice of $z$ we may take $B=\theta/2$. Then by 
Eq.~\eqtag{eq-ip-with-r'} we have 
$\ip{r'}{v}=h^{-1}(a+B)=h^{-1}(\w+1)$, so conclusion \rom3
applies. 
%
\endproof

\beginproclaim Theorem
\Tag{thm-what-short-roots-do}.
Let $L=\Lambda\oplus\II1,1$ for some  positive-definite
$\ring$-lattice $\Lambda$. Let $h$ be one of
$$
\eqalign{
\hbox{$1$, $\theta$, $2$, or $2\theta$}\qquad\qquad
&\hbox{if $\ring=\eisen$, or}\cr
\hbox{\rlap{$1$, $1+\i $, or $2$}%
\phantom{\hbox{$1$, $\theta$, $2$, or $2\theta$}}}\qquad\qquad
&\hbox{if $\ring=\Hurwitz$ or $\Gaussian$.}\cr}
$$
Let $r$ be a
short root of $L$ of height $h$ lying over $\lambda h^{-1}$, with
$\lambda\in \Lambda$. 
Let $v\in L \tensor\R$ be isotropic, have height one and
lie over $\ell\in \Lambda\tensor\R$. Set $D^2=(\ell-\lambda
h^{-1})^2$ and suppose  $D^2\leq1/|h|^2$. Then there exists a
short root $r'$ of $L$ also of height $h$ and lying over
$\lambda h^{-1}$ such 
that either
%
\item{\rom1} 
some reflection in $r'$ carries $v$ to a vector of
height smaller than that of $v$,
\par\noindent
or else $D^2=1/|h|^2$ and
(exactly) one of the following holds:
\item{\rom2}  $\ip{r'}{v}=0$. 
\item{\rom3}  $\ring=\gauss$, $h=1+\i $ or $2$,
and $\ip{r'}{v}=h^{-1}\i $.
\item{\rom4} $\ring=\eisen$, $h=2$ or $2\theta$,
and $\ip{r'}{v}=h^{-1}\theta$.
\item{\rom5} $\ring=\Hurwitz$, $h=1+\i $ and 
$\ip{r'}{v}=h^{-1}\i =(\i +1)/2$.
\item{\rom6} $\ring=\Hurwitz$, $h=2$ and 
$\ip{r'}{v}=h^{-1}(b\i +c\j +d\k )$ for
some $b,c,d\in\{0,1\}$, not all zero.
\par\endproclaim

\remark{Remarks:}
For each $h$ treated, the
elements of $\ring$ with absolute value $|h|$ are unit scalar multiples of
each other and generate a 2-sided ideal in $\ring$.  It appears
that the values of $h$ 
treated here are the only ones for which conclusions
similar to these can be obtained.
The only place in this paper that cases \rom3, \rom4 and \rom6
are used is in the proof of
Theorem~\tag{thm-well-covered-implies-reflective}. Furthermore,
none of 
these cases are required for the examples treated with that
theorem. Thus, the reader may ignore these cases if
desired. Omitting their treatment here would not significantly
shorten or simplify the proof.
%If one were
%interested in lattices of the form $\Lambda\oplus\II1,1$, with $\Lambda$
%indefinite then one could redefine $D^2$ as $|(\ell-\lambda
%h^{-1})^2|$ and then make
%several minor
%changes to the  proof to obtain a similar theorem.


\beginproof{Proof:}
Following the first part of the proof of
Theorem~\tag{thm-what-long-roots-do}, we have 
$$ 
v=(\ell;1,w-\ell^2/2)
$$
for some $w\in\im\K$, and
$$ 
r=\(\lambda;h,z_0+{1-\lambda^2\over2|h|^2}h\)
\eqno\eqTag{eq-coordinates-of-r-2} 
$$
for some $z_0\in\K$ such that $\re(\bar hz_0)=0$. Continuing to
follow that argument, the other short roots $r'$ of $L$ lying over
$\lambda h^{-1}$ all have the form $r+(0;0,z)$ for $z\in\ring$
such that $\re(\bar hz)=0$. Modifying the derivation of Eq.~\eqtag{eq-ip-with-r'}
slightly we find
$$\eqalignno{
\ip{r'}{v}
=&\ip{\lambda}{\ell}+\bar h\(w-{\ell^2\over2}\)
	+\overline{\(z_0+z+{(1-\lambda^2)h\over2|h|^2}\)}\cr
=&h^{-1}\[\({1\over2}-{|h|^2\over2}D^2\) 
	+\(|h|^2\im\ip{\lambda h^{-1}}{\ell}+|h|^2w
	+h{\bar z}_0+h\bar z\)\]\cr
=&h^{-1}[a+B]&\eqTag{eq-ip-with-r'-2}
\cr}$$
where $a=(1-|h|^2D^2)/2$ is the real part of the term in
brackets and $B$ is the imaginary part. The slight difference
between the terms $a$ in Eqs.~\eqtag{eq-ip-with-r'} and
\eqtag{eq-ip-with-r'-2} is due to the 
replacement of  $(2-\lambda^2)$ in Eq.~\eqtag{eq-coordinates-of-r} by
$(1-\lambda^2)$ in Eq.~\eqtag{eq-coordinates-of-r-2}, which 
is due to the fact that $r$ is now a short root.

We take $v'$ to be the image of $v$ under $\xi$-reflection in
$r'$ (we will choose $\xi$ later). Since
$v'=v-r'(1-\xi)\ip{r'}{v}$, we have
$$\eqalignno{
%\height v'=
\ip{\rho}{v'}
&=\ip{\rho}{v}-\ip{\rho}{r'}(1-\xi)\ip{r'}{v}\cr
&=1+{h(\xi-1)\bar h\over|h|^2}[a+B]\;.&\eqTag{eq-height-of-v'-2}
\cr}$$
By hypothesis, $D^2\leq1/|h|^2$, so $a\in[0,1/2]$.
As before, by changing our choice of $z$ we may change $B$ by
any element of $\im(h\ring)$. Now we will treat
$\Gaussian$, $\Eisenstein$ and $\Hurwitz$ separately; the
analysis is just like that of the proof of
Theorem~\tag{thm-what-long-roots-do} except that our
freedom to choose $\xi$ creates opportunities 
which require more complicated computations to exploit.

Suppose $\ring=\gauss$. If  $h=1$ (resp. $1+\i $ or $2$) then
$\im(h\gauss)$ is the set of integral
multiples of $\i $ (resp. $2\i $). Writing $B=b\i $ with
$b\in\R$, we may take $b\in[0,1)$ (resp. $b\in(-1,1]$).
Taking $\xi=\pm \i $ we find by Eq.~\eqtag{eq-height-of-v'-2} that 
$\ip{\rho}{v'}=1+(\pm \i -1)[a+b\i ]$. Computing the norm of this,
or drawing pictures in $\C$, shows that if
$b\in[0,1)$  then upon taking $\xi=+\i $
we find $\height(v')<1=\height(v)$ (so conclusion
{\rom1} applies) unless $a=b=0$, in which case
Eq.~\eqtag{eq-ip-with-r'-2} shows that conclusion {\rom2}
applies. If $b\in(-1,0]$ then taking $\xi=-\i $ we obtain the same
result. Finally, if $b=1$, which can only happen if $h=1+\i $ or
$2$, then taking $\xi=+\i $ yields
$\height(v')<\height(v)$ unless $a=0$. If $a=0$ then
$D^2=1/|h|^2$ and by
Eq.~\eqtag{eq-ip-with-r'-2} we find that
$$
\ip{r'}{v}=h^{-1}[a+b\i ]=h^{-1}(0+\i )\;,
$$
so conclusion {\rom3} applies.

Now suppose $\ring=\eisen$. 
If  $h=1$ (resp. $\theta$, $2$, $2\theta$) 
then $\im(h\eisen)$ is the set of
integer multiples of $\theta$ (resp. $\theta$, $2\theta$,
$2\theta$). Writing $B=b\i $ with $b\in\R$ we may take
$b\in[0,3^{1/2})$
(resp. $b\in[0,3^{1/2})$, $b\in(-3^{1/2},3^{1/2}]$,
$b\in(-3^{1/2},3^{1/2}]$). Using Eq.~\eqtag{eq-height-of-v'-2} and drawing pictures
in $\C$, we see that if 
$b\in[0,3^{1/2})$  then upon taking
$\xi=-\bar{\w}$  we find $|\height(v')|<|\height(v)|$
(so that conclusion {\rom1} applies) unless $a=b=0$, in
which case {\rom2} applies. If $b\in(-3^{1/2},0]$ then taking
$\xi=-\w$ we obtain the same result. Finally, if $b=3^{1/2}$,
which can only happen if $h=2$ or $2\theta$, then
taking $\xi=-\bar{\w}$ yields  $|\height(v')|<|\height(v)|$ unless
$a=0$. If $a=0$ then $D^2=1/|h|^2$ and by
Eq.~\eqtag{eq-ip-with-r'-2} we have
$$
\ip{r'}{v}=h^{-1}[0+3^{1/2}\i ]=h^{-1}\theta\;,
$$
so conclusion {\rom4} applies.

Now suppose $\ring=\Hurwitz$. We  write $B$ as $b\i +c\j +d\k $
with $b,c,d\in\R$. 
We first carry out a computation that will allow us
to use the $24$ units of $\Hurwitz$ effectively: we claim that
there is a unit $\xi'$ of $\Hurwitz$ with $\re\xi'=-1/2$ such
that 
$$
|1+\xi'(a+B)|^2=(a-1/2)^2+(|b|-1/2)^2+(|c|-1/2)^2+(|d|-1/2)^2\;.
\eqno\eqTag{eq-foo3}
$$
For any unit $\xi'$, the left side is just
the square of the distance between $a+B$ and $-\bar{\xi}'$
(proof: left-multiply by $1=|-\bar{\xi}'|^2$). Setting
$-\bar{\xi}'=(1\pm \i \pm \j \pm \k )/2$, with each of its $\i $, 
$\j $ and $\k $ components having the same sign as the corresponding
component of $B$ (or a random sign if that component of $B$
vanishes), the right hand side becomes another expression for
this squared distance, proving the claim.

If $h=1$ (resp. $h=2$)
then $\im(h\Hurwitz)$ is the integral
span of $\i ,\j $ and $\k $ (resp. of $2\i $, $2\j $ and $2\k $), so we may
take each of $b$, $c$ and $d$ to lie in $[0,1)$ (resp. in
$(-1,1]$). 
Now suppose $h=1+\i $. It is easy to check that
$$ 
\im((1+\i )\Hurwitz)=\set{b\i +c\j +d\k }{b,c,d\in\Z,\ 
b+c+d\congruent0\pmod2}\;.  
$$
That
is, $\im(h\Hurwitz)$ is spanned by $\j +\k $, $\k -\i $ and $\i +\k $,
so by choice of $z$ we may take $b\in(-1,1]$ and $c,d\in[0,1)$.

Let $\xi'$ be a unit of $\Hurwitz$ with $\re\xi'=-1/2$ satisfying
Eq.~\eqtag{eq-foo3}, and suppose for a moment that there is a unit
$\xi$ of $\Hurwitz$ such that 
$$
\xi'=h(\xi-1)\bar h/|h|^2\;.
\eqno\eqTag{eq-relation-between-xi's}
$$ 
Then by
Eqs.~\eqtag{eq-height-of-v'-2}, \eqtag{eq-relation-between-xi's} and
\eqtag{eq-foo3}, 
$$\eqalignno{
|\height v'|^2
%&=|\ip{\rho}{v'}|^2\cr
&=|1+\xi'(a+B)|^2\cr
&=(a-1/2)^2+(|b|-1/2)^2+(|c|-1/2)^2+(|d|-1/2)^2\;.
&\eqTag{eq-foo4}
\cr}$$
By hypothesis $D^2\leq1/|h|^2$, so
$a\in[0,1/2]$. By this and our constraints on $b$, $c$ and $d$
obtained above, we see that the right hand side of Eq.~\eqtag{eq-foo4} 
is less than $1=|\height v|^2$ (so that conclusion \rom1
applies) unless
$a=0$ and  $b,c,d\in\{0,1\}$. In each of these cases,
$a=0$ implies that $D^2=1/|h|^2$, and $\ip{r'}{v}$ can be read from 
Eq.~\eqtag{eq-ip-with-r'-2}. We obtain the following possibilities:
$$
\vbox{
\halign{\hfil#\hfil&\qquad\hfil#\hfil&\qquad\hfil#\hfil&\qquad\hfil#\hfil\cr
\bf possibility&\bf value of $h$&%
$\bf\langle$\relax$r'$\relax$\bf|$\relax$v$\relax$\bf\rangle$%
&\bf conclusion\cr
\noalign{\smallskip}
$b=c=d=0$&$1$, $1+\i $ or $2$&0&\rom2\cr
$b=1$, $c=d=0$&$1+\i $ or $2$&$h^{-1}\i $&\rom5 or \rom6\cr
any other&$2$&$2^{-1}(b\i +c\j +d\k )$&\rom6.\cr}}
$$
(The last column refers to the various cases listed in  the
statement of the theorem.)

It remains only to show that given a unit $\xi'$ of $\Hurwitz$ with
$\re\xi'=-1/2$, there is another unit $\xi$ of $\Hurwitz$ satisfying
Eq.~\eqtag{eq-relation-between-xi's}. If $h=1$ or $2$ then this is
trivial: take $\xi=\xi'+1$. If $h=1+\i $ then we solve
Eq.~\eqtag{eq-relation-between-xi's} for $\xi$:
$$
\xi=|h|^2\cdot h^{-1}\xi'\bar{h}^{-1}+1 = 
{(1-\i )\xi'(1+\i )\over\sqrt2\phantom{\i (\xi')q}\sqrt2}+1\;.
\eqno\eqTag{eq-relation-between-xi's-2}
$$
The most straightforward way to show that $\xi$ is a unit of
$\Hurwitz$ 
is to simply evaluate the right hand side of
Eq.~\eqtag{eq-relation-between-xi's-2} for each of
the eight possibilities $\xi'=(-1\pm \i \pm \j \pm \k )/2$.
(What is really going on here is that the units of $\Hurwitz$
together with $2^{-1/2}(1\pm \i )$ generate the binary octahedral
group, which normalizes the binary tetrahedral group consisting
of the units of $\Hurwitz$.)
%
\endproof

\section{\Tag{sec-reflection-groups}. The reflection groups}

\def\wc{well-covered}%
This section is the heart of the paper: we will apply
the results of Section~\tag{sec-reflections} to 
find Lorentzian lattices that are reflective.
The most basic of our results is

\beginproclaim Theorem
\Tag{thm-cov-radius-implies-reflective}.
Suppose $\Lambda$ is a positive-definite $\ring$-lattice which is
spanned by its roots and has covering radius $\leq1$. Then 
$L=\Lambda\oplus\II1,1$ is reflective.
\QED
\endproclaim 

\noindent
(This follows immediately from
Theorem~\tag{thm-well-covered-implies-reflective} below.) 
The proof of such a theorem has two parts: first that
$\reflec L$ acts with only finitely many orbits on the primitive
null vectors  in $L$ and second that the stabilizer in $\reflec
L$ of one such
vector, namely $\rho$, has finite index in its stabilizer in
$\aut L$. 
The second part has already been proven, in Theorem~\tag{thm-rho-stabilizer-finite-index}.
The first and more interesting part of the argument
proceeds by supposing 
$v$ to be a primitive null vector in $L$ and repeatedly applying
reflections to decrease the height of
$v$. Theorems~\tag{thm-what-long-roots-do} and 
\tag{thm-what-short-roots-do} assure us that we can find
height-decreasing reflections if $v$ 
lies over a point of $\Lambda\tensor\K$ sufficiently close to an element of
$\Lambda$ or to any of various  other points of
$\Lambda\tensor\K$.  In particular, if 
the covering radius of $\Lambda$ is small enough then we may
always find suitable reflections and this  allows us to
reduce the height of $v$ indefinitely. Since the
details of Theorems~\tag{thm-what-long-roots-do} and
\tag{thm-what-short-roots-do} 
are quite complicated, conditions on $\Lambda$ that allow us to
bring to bear the full force of these results are  bound to also
be
complicated. The condition defined below, that $\Lambda$ be
well-covered, is just a generalization of the requirement
of Theorem~\tag{thm-cov-radius-implies-reflective} that
$\Lambda$ have covering radius $1$. 

There are two ``tracks'' through this section. The
first deals with \wc\ lattices and the general theory, with details
worked out for a few low-dimensional examples. This track yields
reflective lattices in $\C^{n,1}$ for $n\leq5$ and in $\H^{n,1}$
for $n\leq3$, and a detailed study of  the reflection
groups of $\eI n,1$ for $n\leq4$ 
and $\qI n,1$ for $n\leq3$. The second
track treats in detail two high-dimensional examples, $\eI7,1$ and
$\qI5,1$. This track begins with
Lemma~\tag{thm-linear-independence-descends} and 
requires none of the earlier material of the section except for
Theorem~\tag{thm-rho-stabilizer-finite-index-2}. Furthermore, it
does not require the use of 
Theorem~\tag{thm-what-long-roots-do}. The reader who wishes to
know why these two 
lattices are reflective but is not interested in further details
need read only Lemmas~\tag{thm-coxeter-todd-well-covered} and
\tag{thm-barnes-wall-well-covered}, which assert that $\cct$ 
and $\qbwl$ are \wc, and then apply
Theorem~\tag{thm-well-covered-implies-reflective}. 

Suppose $\Lambda$ is a positive-definite
$\ring$-lattice. We define a family $\calc$ (for ``covering'')
of closed balls centered at various points of
$\Lambda\tensor\R$. If the union of these balls is all of
$\Lambda\tensor\R$ then we say that $\Lambda$ is \wc. The
definition of $\calc$ depends on $\ring$, and is given in 
Table~\tag{tab-table-of-balls}.

\topinsert
\centerline{Table~\tag{tab-table-of-balls}. The balls used to
define the notion of a lattice being \wc}
$$
\vcenter{
%
%
%
\halign{\hfil$#$\hfil&\qquad\hfil$#$\hfil&\qquad%
$#$\hfil&\qquad\hfil$#$\hfil&\qquad\hfil #\hfil\cr
\hbox{\bf The ring $\ring$}&(\hbox{\bf Radius})^2&\hbox{\bf Center}&%
\hbox{\bf Condition}&\bf Root length(s)\cr
\noalign{\smallskip}
\Gaussian&\sqrt3&\lambda&\lambda^2\congruent 0 (2)&% 
	long\cr
&1&\lambda &\lambda^2\congruent 1 (2)&%
	short\cr
&1/2&\lambda(1+\i )^{-1}&\lambda^2\congruent 1(2)&%
	short\cr
&1/4&\lambda /2&\lambda^2\congruent 1(4)&%
	short\cr
\noalign{\smallskip}
\Eisenstein&1&\lambda&\hbox{none}&long, short\cr
&1/3&\lambda\theta^{-1}&\lambda^2\congruent2(3)&%
	long\cr
&1/3&\lambda\theta^{-1}&\lambda^2\congruent1(3)&%
	short\cr
&1/4&\lambda /2&\lambda^2\congruent1(2)&%
	short\cr
&1/12&\lambda \theta^{-1}/2&\lambda^2\congruent1(6)&%
	short\cr
\noalign{\smallskip}
\Hurwitz&1&\lambda&\hbox{none}&long, short\cr
&1/2&\lambda(1+\i )^{-1}&\lambda^2\congruent1(2)&%
	short\cr
&1/4&\lambda /2&\lambda^2\congruent1(2)&short\cr}
%
}$$
\endinsert

The table should be interpreted as follows. A closed ball is a member
of $\calc$ just if for one of the rows listed under $\ring$, it
has the given radius and center, where $\lambda\in \Lambda$
satisfies the condition in the `condition' column. If $\Lambda$ is
\wc\ then a point of $\Lambda\tensor\R$ is called a
$\calc$-hole if it lies in the interior of none of the balls of
$\calc$; the $\calc$-holes of a \wc\ lattice obviously form a
discrete set. We say that $\Lambda$ is strictly \wc\ if it has
no $\calc$-holes; that is, if the interiors of the balls cover
$\Lambda\tensor\R$. 

The last column of Table~\tag{tab-table-of-balls} indicates
whether short or long 
roots of $L=\Lambda\oplus\II1,1$ (or roots of both lengths) lie
over the center of the 
ball. The verifications that there are such roots is straightforward. For
example, if $\ring=\Eisenstein$ and $\lambda\in \Lambda$ we can
look for short roots of the form $r=(\lambda;\theta,\nu)$, which
would
lie over $\lambda\theta^{-1}$. Our search will succeed if we
can choose $\nu\in\Eisenstein$ so that $r^2=1$. We may restate the
condition $r^2=1$ as
$\re(\theta\bar{\nu})=(1-\lambda^2)/2$. Since the set of values taken
by the left hand side is ${3\over2}\Z$, as $\nu$ varies over
$\Eisenstein$, we can solve for $\nu\in\Eisenstein$ exactly when
$1-\lambda^2\in3\Z$, that is, when $\lambda^2\congruent 1(3)$. This sort of
computation is the source of the entries in the `condition' column.

\beginproclaim Theorem
\Tag{thm-well-covered-implies-reflective}.
Suppose $L=\Lambda\oplus\II1,1$ for some \wc\ positive-definite
$\ring$-lattice  $\Lambda$.
%
\item{\rom1} If no vector of $\Lambda$ is fixed
by every reflection of $\Lambda$ then 
$L$ is reflective. 
\item{\rom2} If $\Lambda$ is strictly
\wc\ then any two primitive null sublattices of $L$ are
equivalent under $\reflec L$.\par\endproclaim 

\beginproof{Proof:}
Suppose $v$ is a primitive null vector of $L$ that has
minimal height in its 
orbit under $\reflec L$,  is not a
multiple of $\rho$, and lies over $\ell\in \Lambda \tensor\R$. If
$\ell$ lies in the interior of any ball of $\calc$ then the
image $v'$ of $v$ under some reflection of $L$ has
$\height(v')<\height(v)$, contradicting our hypothesis on
$v$. Here's why: if a ball of $\calc$ whose interior contains
$\ell$ is centered at $\lambda h^{-1}$ for $\lambda\in \Lambda$
then 
there is a root $r$ of $L$ lying over
$\lambda h^{-1}$. 
By applying Theorem~\tag{thm-what-short-roots-do} in the case of a
short root, or Theorem~\tag{thm-what-long-roots-do} in the
case of a long root, we see 
that there is another root $r'$ of $L$ and a reflection in $r'$
which carries $v$ to a vector 
of smaller height. If $\Lambda$ is strictly \wc\ then this
shows that  $v$ cannot exist, so every primitive isotropic
vector of $L$ is equivalent to one in the span of $\rho$. This
proves \rom2.

If $\Lambda$ is \wc\ but not strictly \wc\ then we
conclude from the argument above that $\ell$ is a $\calc$-hole
of $\Lambda$, and from the conclusions of
Theorems~\tag{thm-what-long-roots-do} and 
\tag{thm-what-short-roots-do} that for one of finitely many
pairs $h,k\in\K$ there is a 
root $r'$ of $L$ of height $h$ such that $\ip{r'}{v}=k$. 
Under the hypothesis of \rom1,
Theorem~\tag{thm-rho-stabilizer-finite-index}\rom4 shows that
the 
stabilizer of $\rho$ in $\reflec L$ has finite index in the
stabilizer in $\aut L$. Since the translations of $L$ act with
only finitely many orbits on the vectors of any given norm and
(nonzero) height, we see that after applying an element of
$\reflec L$ we may take $r'$ to be one of some fixed finite
set of roots. If $r'$ is one of these roots then the conditions
$\ip{r'}{v}=k$ and $v^2=0$ together with the fact that $v$ lies
over a $\calc$-hole of $\Lambda$ determine $v$ to within
finitely many possibilities. Therefore $\reflec L$ acts with
only finitely many orbits on the  primitive null vectors
of $L$. This fact, together with the finite index of the
stabilizer in $\reflec L$ of one such vector, $\rho$, in its
stabilizer in $\aut L$, proves \rom1.
%
\endproof

\remark{Remark:}
If we were to restrict the definition of the covering $\calc$ to
only involve balls with centers over which lie short roots of $L$
(i.e., those entries in Table~\eqtag{tab-table-of-balls} with `short' in the last
column), then we would obtain  new definitions of `\wc'
and `strictly \wc'. The proof of \rom2  shows that if
$\Lambda$ is strictly \wc\ in this sense then $\shr L$ 
acts
transitively on the primitive null lattices in $L$. 


\beginproclaim Corollary
\Tag{thm-13-reflection-groups}.
Let $\Lambda$ be any one of the $\ring$-lattices
$$
\vbox{
\halign{#\hfil&\qquad#\hfil\cr
$\Gaussian$, $2^{1/2}\Gaussian$, $D_4^\Gaussian$, 
$D_6^\Gaussian$, or $\geeight$& if $\ring=\Gaussian$,\cr
%
$\Eisenstein$, $\Eisenstein^2$, $\Eisenstein^3$, or
$D_3(\sqrt{-3})$&if $\ring=\Eisenstein$, or\cr
%
$\Hurwitz$, $2^{1/2}\Hurwitz$, $\Hurwitz^2$, or
$\qeeight$&if $\ring=\Hurwitz$.\cr}}
$$ 
Then $L=\Lambda\oplus\rII1,1$ is reflective. Unless $\Lambda$ is
$\Eisenstein^3$, $D_3(\sqrt{-3})$, $D_4^\Hurwitz$, $\Hurwitz^2$ or
$\qeeight$, all primitive null sublattices of $L$ are equivalent
under $\reflec L$.\endproclaim 

\remark{Remark:}
The lattices appearing
here are all described in Section~\tag{sec-lattice-reference}.
Theorems~\tag{thm-eI2,1-eI3,1-eI4,1} and
\tag{thm-qI2,1-qI3,1} give much more precise information about 
$\reflec L$ for $\Lambda=\Eisenstein$, $\Eisenstein^2$,
$\Eisenstein^3$, $\Hurwitz$ or $\Hurwitz^2$.


\beginproof{Proof:}
The covering radius of $\Gaussian$ is $2^{-1/2}$, so $\Gaussian$
is strictly \wc. The remaining Gaussian lattices are all even and have
covering radii $1$, $1$, $(3/2)^{1/2}$ and $1$, respectively, 
so they are also strictly \wc. The covering radii of
the Eisenstein lattices are $(1/3)^{1/2}$,
$(2/3)^{1/2}$,  $1$ and $1$,
respectively, so these lattices are \wc\ and the first two
strictly so. The Hurwitz lattices have covering radii
$2^{-1/2}$, $1$, $1$ and $1$, respectively, so all are \wc\ and
$\Hurwitz$ strictly so. In each case the roots of $\Lambda$ span
$\Lambda\tensor\R$, so no element  of $\Lambda$ is fixed by all
reflections of $\Lambda$. Our conclusions follow from
Theorem~\tag{thm-well-covered-implies-reflective}. 
%
\endproof

We can also apply the theorem when $\Lambda=\{0\}$, to deduce that
$\reflec\rII1,1$ acts on $\kh1$ with finite covolume and 
that there is only one orbit 
of primitive isotropic lattices  in $\rII1,1$. A little
picture-drawing  reveals that
$\reflec\eII1,1$ acts on  the right half-plane (a copy of $\ch1$)
as the triangle group $\trianglegroup{2}{6}{\infty}$. 
%$\aut\gI1,1$ acts on $\ch1$ as $\trianglegroup{2}{4}{\infty}$,
%and its reflection subgroup acts as 
%$\trianglegroup{4}{4}{\infty}$. 
One can also show that $\aut\gII1,1$ acts on $\ch1$
as $\trianglegroup{2}{3}{\infty}$ and its subgroup
of index 2 consisting of elements with determinant $+1$ is conjugate in
$\GL_2(\gauss)$ to $\SL_2\Z$. 
The group  $\reflec\gII1,1$ 
%has index 6 in $\aut\gII1,1$ and
is generated by 3 biflections, which act by rotations by $\pi$
around the three finite corners of a quadrilateral in $\ch1$
with corner angles $\pi/2$, $\pi/2$, $\pi/2$ and
$\pi/\infty$. 
See \cite{cox:gens-and-rlns} for descriptions of the
groups $\trianglegroup{p}{q}{r}$ and other information.
By adapting the argument of \ecite{allcock:oh2}{Thm.~5.3\rom1}, one can
also show that $\reflec\qII1,1$ acts on $\qh1\isomorphism\rh4$
as the rotation subgroup of the real hyperbolic reflection group
with the Coxeter diagram
below.
%
Note that the 6 outer nodes generate an affine reflection
group, so this strange-looking graph is just a special case of
the usual procedure of ``hyperbolizing'' an affine reflection
group by adjoining an extra node.
$$
%
%  a Coxeter Diagram
%
\beginpicture
\def\magfactor{2}
\newdimen\vstep
\vstep=17.13 pt
\newdimen\hstep
\hstep=20 pt
\multiply\vstep by\magfactor
\multiply\hstep by\magfactor
\setcoordinatesystem units <\hstep,\vstep>
\put {$\bullet$} at  .5 -1 
\put {$\bullet$} at  -.5 -1 
\put {$\bullet$} at  1 0
\put {$\bullet$} at  0 0  
\put {$\bullet$} at  -1 0 
\put {$\bullet$} at  .5 1  
\put {$\bullet$} at  -.5 1   
\plot 0 0 .5 1 -.5 1 0 0 -1 0 -.5 -1 0 0 .5 -1 1 0 0 0 /
\put {$\infty$} [b]  <0pt,2pt> at 0 1 
\put {$\infty$} [tr] <0pt,0pt> at -.75 -.5
\put {$\infty$} [tl] <0pt,0pt> at  .75 -.5
\endpicture
$$
%
%
%
%
%
%
%

We will now study in more detail the reflection groups of several Lorentzian
lattices over $\Eisenstein$ and $\Hurwitz$. The reader may
follow two paths. One requires \rom1--\rom3 of
Theorem~\tag{thm-rho-stabilizer-finite-index-2} 
below but then nothing else until
Lemma~\tag{thm-linear-independence-descends}. This path leads 
to our highest-dimensional examples, $\reflec\eI7,1$ acting on
$\ch7$ and $\reflec\qI5,1$ acting on $\qh5$. The other path
requires \romp1--\romp3 of
Theorem~\tag{thm-rho-stabilizer-finite-index-2} and leads to
detailed 
analyses of several low-dimensional examples, namely $\eI n,1$
for $n=2,3,4$ and $\qI n,1$ for
$n=2,3$. Theorem~\tag{thm-rho-stabilizer-finite-index-2}
is 
a more specialized version of Theorem~\tag{thm-rho-stabilizer-finite-index}.
As usual, the different rings behave slightly differently.

\beginproclaim Theorem
\Tag{thm-rho-stabilizer-finite-index-2}.
Suppose $\ring=\Eisenstein$ or $\Hurwitz$, that $\Lambda$ is
a positive-definite selfdual $\ring$-lattice of positive dimension
that is spanned by its roots, and that $\aut \Lambda$ is generated by
reflections. Let $L=\Lambda\oplus\rII1,1$. Then
%
\item{\rom1}
If $\ring=\Eisenstein$ (resp. $\Hurwitz$) then $\reflec L$
contains all (resp. at least a quarter) of the translations of
$L$. (If $\ring=\Hurwitz$ then $\reflec L$ contains
$T_{0,-\theta}$, and coset representatives for the translations
of $\reflec L$ in those of $\aut L$ may be
taken from $\{T_{0,0},T_{0,\i },T_{0,\j },T_{0,\k }\}$.)
\item{\rom2}
$\reflec L$ contains a
transformation acting trivially on $\Lambda$ and on $\rII1,1$ by left scalar
multiplication by any given unit of $\ring$.
\item{\rom3}
If $\ring=\Eisenstein$ (resp. $\Hurwitz$) then the stabilizer of
$\spanof{\rho}$ in $\reflec L$ coincides with (resp. has index
at most four in) the stabilizer in $\aut L$.
\par\noindent
Furthermore, if $\Lambda$ is spanned by its short roots then
\item{\romp1}
The conclusion of \rom1 holds with $\shr L$ in place of $\reflec
L$.
\item{\romp2}
If $\ring=\Hurwitz$ then the conclusion of \rom2 holds with
$\shr L$ in place of $\reflec 
L$. If $\ring=\Eisenstein$ then $\shr L$ contains a
transformation acting trivially on $\Lambda$ and on $\eII1,1$ by
the scalar $\w$.
\item{\romp3}
If $\ring=\Hurwitz$ then the conclusion of \rom3 holds with
$\shr L$ in place of $\reflec 
L$. If $\ring=\Eisenstein$ then the stabilizer of
$\spanof{\rho}$ in $\aut L$ is generated by the stabilizer in
$\shr L$ together with the central involution of $\eII1,1$ (or
alternately that of $L$). \par\endproclaim 

\beginproof{Proof:}
\rom1,\romp1
The selfduality of $\Lambda$ together with
Theorem~\tag{thm-rho-stabilizer-finite-index}\rom2 
show that for each root $r$ of $\Lambda$ there exists
$z\in\im\K$ such that $T_{r,z}\in\reflec L$. Since $\Lambda$ is
spanned by its roots, Eq.~\eqtag{eq-product-of-translations} shows that for all
$\lambda\in\Lambda$ there exists $z\in\im\K$ such that
$T_{\lambda,z}\in\reflec L$. Taking commutators of these
translations and using the selfduality of $\Lambda$ we find by
Theorem~\tag{thm-rho-stabilizer-finite-index}\rom3 that for each
$z\in\ring$, the central 
translation $T_{0,2\im z}$ lies in $\reflec L$. (This is where
we use the hypothesis  $\dimension \Lambda>0$.) If
$\ring=\Eisenstein$ then this yields all  the central
translations. If $\ring=\Hurwitz$ then this yields the central
translations $T_{0,b\i +c\j +d\k }$ with $b,c,d\in\Z$ and $b\congruent
c\congruent d\ (2)$, a subgroup of index $4$ in the group of
all central translations. (Coset representatives are $T_{0,0}$,
$T_{0,\i }$, $T_{0,\j }$ and $T_{0,\k }$.) This proves \rom1.
If the short roots of $\Lambda$ span $\Lambda$ then the entire
argument goes through with $\shr L$ in place of $\reflec L$,
because for a short root $r$ of $\Lambda$, the proof of
Theorem~\tag{thm-rho-stabilizer-finite-index}\rom2 shows that
there exists $z\in\im\K$ such 
that $T_{r,z}\in\shr L$. 

\smallskip
In the rest of the proof, unless otherwise specified, the term
`scalar matrix' will have a nonstandard meaning: an isometry of
$L$ fixing $\Lambda$ 
pointwise and acting on $\rII1,1$ by left-multiplication by some
unit of $\ring$. (Note that if $\ring=\Hurwitz$ then there may
be no concept of ``left-multiplication by scalars'' on $L$.)

\romp2
We have $T_{0,-\theta}\in\shr L$
by \romp1. Let $F$ be the transformation composed of
$T_{0,-\theta}$ followed by $(-\w)$-reflection in the short root
$(0;1,-\w)$. It is obvious that $F$ acts trivially on $\Lambda$
and computation reveals that it acts on $\rII1,1$ by left
multiplication by the matrix
$$ 
\pmatrix{0&\wbar\cr\wbar&0\cr}\;.
$$
The square of this matrix is the scalar matrix $\w$, which
proves the claim in the case $\ring=\Eisenstein$. If
$\ring=\Hurwitz$ then $\shr L$ contains the 8  matrices
that are the images of $F^2$  under $\aut\Hurwitz$. (This is
because $\aut\Hurwitz$, acting on $\rII1,1$, normalizes $\reflec
L$ even though it doesn't act by $\Hurwitz$-linear
transformations.) These matrices generate the group of 
scalar matrices, proving the claim in the case
$\ring=\Hurwitz$. 

\rom2
If we no longer assume that $\Lambda$ is spanned by its short roots
then all of the above goes through with $\reflec L$ in place of
$\shr L$ and the reference to \romp1 replaced by a reference to
\rom1. Therefore it remains only to show that if
$\ring=\Eisenstein$ then $\reflec L$ contains all of the scalar
matrices. This 
follows because the biflection $B$ in $b=(0;1,1)$ fixes
$\Lambda$ pointwise and acts on $\eII1,1$ 
by the matrix
$$ 
\pmatrix{0&-1\cr-1&0}\;.
$$
Observe that $FB$ is the scalar matrix $-\wbar$, which generates the
group of all scalar matrices.

\rom3
By \rom2, $\reflec L$ acts transitively on the primitive vectors
in $\spanof{\rho}$. By \rom1, $\reflec L$ contains all the
translations of $L$ (or at least a quarter, if
$\ring=\Hurwitz$), and hence the stabilizer of $\rho$ in
$\reflec L$ acts with only one orbit (or at most 4 orbits, if
$\ring=\Hurwitz$) on the null vectors of $L$ of height 1. The
simultaneous stabilizer in $\aut L$ of $\rho$ and one such,
namely $(0;1,0)$, is just $\aut \Lambda$, which is generated by reflections by
hypothesis. This proves \rom3.

\romp3
Let $G$ be the group generated by $\shr L$ and the central
involution of $L$; it is obvious that $G$ is normal in $\aut
L$. Since $\shr L$ contains the central involution of $\Lambda$,
we see that $G$ contains the central involution $J$ of
$\rII1,1$. By the proof of \rom2, $J=WFB$ where  
$B$ is the biflection in
$b=(0,\ldots,0;1,1)$ and $W$ denotes the scalar matrix $\w$. Since
$WF\in\shr L$, we see that $G$ is generated by $\shr L$ and
$B$, so $G\sset\reflec L$. 
We claim that $G$ contains
$\aut \Lambda$. Since $\Lambda$ is spanned by its short roots,
$\Lambda\isomorphism \ring^n$ for some $n$, and we may introduce
an orthonormal basis for $\Lambda$. If $n=1$ then $\aut \Lambda$
is generated by reflections in its short roots, proving the
claim. If $n>1$ then it suffices to prove that $G$ contains the
coordinate permutations with respect to the chosen basis of
$\Lambda$. That is, we must show that $G$ contains the
biflections in vectors like $x=(1,-1,0,\ldots,0;0,0)$. 
Since $G$ is normal in $\aut L$ and contains $B$, it suffices to
show that $x$ and $b$ are equivalent under $\aut L$. To see this,
observe
that $T_{(1,0,\ldots,0),\theta/2}$ followed by $F$, followed by
the  scalar matrix $-\w$, followed by 
$T_{(-1,1,0,\ldots,0),0}$,
carries $x$ to $b$. 

Repeating
the argument of \rom3, replacing $\reflec L$ by $G\sset\reflec L$ and
the references to \rom1 and \rom2 by references to \romp1 and
\romp2, proves that the 
stabilizers of $\spanof{\rho}$ in $G$ and $\aut L$ coincide.
If $\ring=\Hurwitz$ then by \romp2 we have $J\in\shr L$, so $\shr L=G$,
proving \romp3. If $\ring=\Eisenstein$ then \romp3 follows from
the several descriptions of $G$ given above.
%
\endproof

\remark{Remarks:}
The 
condition $\dimension \Lambda>0$ is necessary; one can show
that $\reflec \eII1,1$ contains no scalars except the
identity. It is interesting to note 
that when $\ring=\Hurwitz$ and $\Lambda$ (of dimension $>0$) is
spanned by its short 
roots then $\shr L$ contains biflections such as $B$ in long
roots of $L$. This follows from the fact that as in the proof of
\rom2, $B$ may be written as a product of $F$ and 
a scalar matrix. I suspect that in some
cases, such as those studied in Theorem~\tag{thm-qI2,1-qI3,1} below, 
$\shr L$ contains all the reflections of $L$.


\beginproclaim Lemma
\Tag{thm-braiding-reflections}.
If $r$ and $r'$ are short roots in a lattice  over
$\ring=\Eisenstein$ or $\Hurwitz$ and $\ip{r}{r'}=1$ then $r$
and $r'$ are equivalent under the group generated by the
reflections in them.\endproclaim

\beginproof{Proof:}
One checks that the $(-\w)$-reflections $R$ and $R'$ in $r$ and
$r'$ satisfy the braid relation $RR'R=R'RR'$. (Because the
Hermitian form is degenerate on the span of $r$ and $r'$, one
must check that this relation holds by using
Eq.~\eqtag{eq-def-of-reflection}, not by just multiplying matrices for
the actions of $R$ 
and $R'$ on the span of $r$ and $r'$.) Rewriting this as
$R'^{-1}RR'=RR'R^{-1}$ we see that $R$ and $R'$ are conjugate in
the group they generate, which implies the lemma.
%
\endproof

\remark{Remark:}
The proof suggests connections between the braid groups and
complex reflection groups. This connection was first observed in
\cite{coxeter:quotients-of-braid-groups}, and the braid groups
play a central role in the work of  Deligne and Mostow \cite{deligne:monodromy-of-hypergeometric-functions},
Mostow \cite{mostow:picard-lattices-from-half-inegral-conditions} and Thurston \cite{thurston:shapes-of-polyhedra}. They are also important in
work of the author, J. Carlson and D. Toledo \cite{allcock:cubic-surf} on moduli
of cubic surfaces.


A useful
tool for studying Eisenstein lattices
$L=\Lambda\oplus\eII1,1$ with $\Lambda$ selfdual
is the reduction
of $L$ modulo $\theta$. We write $q$ for the natural map
$q:\Eisenstein\to\Eisenstein/\theta\Eisenstein\isomorphism{\F_3}$,
and also for the natural map $q:L\to L/L\theta$; we write $V$ for
the $\F_3$-vector space $L/L\theta$. The Hermitian inner product
on $L$ gives rise to a symmetric bilinear form on $V$, given by
$\ip{q(v)}{q(w)}=q(\ip{v}{w})$. Since $L$ is selfdual this
pairing is a nondegenerate bilinear form on $V$ and yields a
natural homomorphism from $\aut L$ to $H=\isom V$, an orthogonal
group over $\F_3$. There is a homomorphism from $H$ to the set
$\{\pm1\}$ of nonzero square classes of ${\F_3}$, called the
spinor norm map and characterized by the following property: if
$v\in V$ has nonzero 
norm then the spinor norm of the reflection in $v$ is the square
class of the norm of $v$. 
%See \oldcite{dieudonne?} for a proof that this
%defines a homomorphism of $H$. 
We define the spinor norm of an
element of $\aut L$ to be the spinor norm of its image in
$H$. If $v$ is a long (resp. short) root of $L$ then the
biflection (resp. either $6$-fold reflection) in $r$ acts on $V$
as the reflection 
in the image of $r$, and thus has spinor norm $-1$ (resp. $+1$).
By choosing an orthogonal basis for $L$ we obtain an orthogonal  basis
for $V$ and then it is clear that the central involution has
spinor norm $-1$.  (There is an orthogonal basis for $L$ by
Theorem~\tag{thm-indefinite-selfdual-classification}.)  

\beginproclaim Theorem
\Tag{thm-eI2,1-eI3,1-eI4,1}.
Let $\ring=\Eisenstein$, $\Lambda=\Eisenstein^n$ and
$L=\Lambda\oplus\eII1,1$. 
%
\item{\rom1} 
If $n=1$ then $\shr L$ acts with exactly $2$ orbits
of primitive null vectors, represented by $\pm\rho$. If $n=2$ or
$3$ then $\shr L$ acts transitively on the primitive null vectors of $L$.
\item{\rom2} 
If $n=1$, $2$ or $3$ then 
$\aut L=\reflec L=\shr L\times\{\pm I\}$.\endproclaim 

\beginproof{Proof:}
For $n=1$ or $2$, $\Eisenstein^n$ is strictly \wc\ in the sense
of the remark following
Theorem~\tag{thm-well-covered-implies-reflective} and so $\shr
L$ acts 
transitively on the primitive null lattices in $L$. Now suppose
$n=3$.  
Suppose $v\in L$ is a
primitive null vector not proportional to $\rho$ and of smallest
height in its orbit under $\shr L$. Since the covering radius of
$\Eisenstein^3$ is $1$,
Theorem~\tag{thm-what-short-roots-do}\rom2 implies that $v$ is 
orthogonal to a  short root of height $1$. By applying a
translation we may suppose that this root is
$r_1=(0,0,0;1,-\w)$. Taking $r_2=(0,0,1;0,1)$ and
$r_3=(0,0,1;0,0)$ we see that
$\ip{r_1}{r_2}=\ip{r_2}{r_3}=1$, so by
Lemma~\tag{thm-braiding-reflections}, $r_1$ 
is equivalent to 
$r_3$ under $\shr L$. Thus $v$ is equivalent to an element of
$r_3^\perp$, which is a copy of
$\Eisenstein^2\oplus\eII1,1$. Applying the $n=2$ case, we see
that for $n=3$, $\shr L$ acts transitively on the 
primitive null sublattices of $L$.

By considering the spinor norm, we see that $-I\notin\shr L$. 
The transitivity above together with the equality
(Theorem~\tag{thm-rho-stabilizer-finite-index-2}\romp3) of the
stabilizers of $\spanof{\rho}$ in 
$\aut L$ and $\shr L\times\{\pm I\}$ proves that $\aut L=\shr
L\times\{\pm I\}$ and  that $\shr L$ is the kernel of
the spinor norm map. By considerations of the spinor norm,  no
biflection  in a long root of $L$ lies in $\shr 
L$. This proves \rom2 

From the above, we conclude that any primitive null vector of
$L$ is equivalent under $\shr L$ to one of $\pm\rho$. If $n>1$
then $\pm\rho$ are equivalent, because the central involution
followed by biflection in a long root of $\Lambda$ exchanges
them and has spinor
norm 1. If $n=1$ then $\pm\rho$ are not
equivalent, or else the fact that the stabilizers of $\rho$ in
$\shr L$ and $\aut L$ coincide would prove $\shr L=\aut L$. 
%
\endproof

\remark{Remark:}
The quotient by  $\shr \eI4,1$ of $\ch4$ minus the union of the hyperplanes
orthogonal to short roots of $\eI4,1$
may be identified with the
moduli space of smooth cubic surfaces over $\C$. The analogous
construction with $\eI3,1$ in place of $\eI4,1$ yields the
moduli space of smooth genus 2 curves over $\C$.
One can also construct the (fine) moduli space of marked smooth cubic
surfaces by taking the quotient of $\ch4$ (minus the same
hyperplanes as above) by the congruence
subgroup of $\shr \eI4,1$ associated to the prime
$\theta\in\Eisenstein$. The quotient of
$\shr \eI4,1$ 
by this normal subgroup is the $E_6$ Weyl group, also known as ``the group of
the 27 lines on a cubic surface''.
See \cite{allcock:cubic-surf} for details.


\beginproclaim Theorem
\Tag{thm-qI2,1-qI3,1}.
Let $\ring=\Hurwitz$, $\Lambda=\Hurwitz^n$ for  $n=1$ or $2$, and
$L=\Lambda\oplus\qII1,1$. Then $\shr L$ acts transitively on the
primitive null vectors of $L$ and has index at most $4$ in
$\aut L$.\endproclaim 

\beginproof{Proof:}
We observe that
$\shr L$ acts transitively on primitive null lattices in
$L$. For $n=1$ this follows from
the
fact 
that $\Lambda=\Hurwitz$ is strictly \wc\ in the sense of the remark
following the proof of 
Theorem~\tag{thm-well-covered-implies-reflective}. 
For $n=2$ it follows from an
argument similar to the $n=3$ case of
Theorem~\tag{thm-eI2,1-eI3,1-eI4,1}.
That is, the covering radius of
$\Lambda=\Hurwitz^2$ is $1$, so if $v\in L$ is primitive, isotropic and
of smallest height in its orbit under $\shr L$ then by
Theorem~\tag{thm-what-short-roots-do}\rom2 we see that $v$
is either proportional to $\rho$ or orthogonal to a short
root $r_1$ of height $1$. In the latter case, after applying a
translation of $\shr 
L$, courtesy of Theorem~\tag{thm-rho-stabilizer-finite-index-2}\romp1, 
we may take $r=(0,0;1,x-\w)$ for $x=0$, $\i $, $\j $ or $\k $. In
any of these cases, upon taking $r_2=(0,1;0,1)$ and $r_3=(0,1;0,0)$ we have
$\ip{r_1}{r_2}=\ip{r_2}{r_3}=1$. By Lemma~\tag{thm-braiding-reflections},
$v$ is equivalent under
$\shr L$ to an element of $r_3^\perp$. Since $r_3^\perp$ is a
copy of
$\Hurwitz^1\oplus\qII1,1$, the transitivity follows from the case $n=1$.

Using Theorem~\tag{thm-rho-stabilizer-finite-index-2}\romp2,
the transitivity on primitive null vectors follows. The bound
on the index of $\shr L$ in 
$\aut L$ follows from Theorem~\tag{thm-rho-stabilizer-finite-index-2}\romp3.
%
\endproof

Now we move on to higher-dimensional examples\emdash we will 
construct a group acting on $\ch7$ and 
another acting on $\qh5$. These arise  from our basic
construction by taking $\Lambda=\cct$ or
$\qbwl$. 

\beginproclaim Lemma
\Tag{thm-linear-independence-descends}.
Suppose $v,r_1,\ldots,r_m\in\K^n\oplus\K^{1,1}$ lie
over $\ell,\lambda_1,\ldots,\lambda_m\in\K^n$ respectively. Suppose
$v^2=0$, that $\ip{r_i}{v}=0$ for all $i$, and that the vectors
$\lambda_i-\ell$ are linearly independent in $\K^n$. Then the
images of the $r_i$ in $v^\perp/\spanof{v}$ are linearly
independent. \endproclaim

\beginproof{Proof:}
We may obviously replace $v$ and the $r_i$ by any scalar
multiples of themselves and so suppose that they have height
$1$. Thus $v=(\ell;1,?)$ and $r_i=(\lambda_i;1,?)$ where the
question marks denote irrelevant and possibly distinct elements
of $\K$. Let $T$ be the translation carrying $v$ to $(0;1,0)$,
so $T(r_i)=(\lambda_i-\ell;1,0)$. (The last coordinate vanishes
because $\ip{T(r_i)}{Tv}=0$.) Since the image of $T(r_i)$ in
$(Tv)^\perp/\spanof{Tv}$ may be identified with its first
coordinate, namely $\lambda_i-\ell$, the images of the $T(r_i)$ in
$(Tv)^\perp/\spanof{Tv}$ are linearly independent. The theorem
follows immediately.
%
\endproof

\beginproclaim Lemma
\Tag{thm-coxeter-todd-well-covered}.
$\cct\tensor\R$ is covered by the closed balls of radius $1$ centered
at points of $\cct$, together with those of radius
$3^{-1/2}$ centered at points $\lambda\theta^{-1}$ with
$\lambda\in\cct$ and $\lambda^2\congruent1(3)$. In particular,
$\cct$ is \wc.\endproclaim

\beginproof{Proof:}
Section~7 of \cite{jhc:Coxeter-Todd} defines a linear ``gluing map'' 
$g:{\textstyle{1\over\theta}}\cct/\cct\to{\textstyle{1\over\theta}}\cct/\cct$
with the property that the Leech lattice $\leech$, scaled
down by $2^{1/2}$, is the real form of the lattice of  vectors 
$(x_1,x_2)\in\({1\over\theta}\cct\)^2$ satisfying 
$g(x_1+\cct)=x_2+\cct$. Identifying $\cct$ with the set of such
$(x_1,x_2)$ with 
$x_2=0$, we see that the only points of $2^{-1/2}\leech$ 
at distance $<1$ from $\cct\tensor\R$ are those in $\cct$ and
those of the form $(x_1\theta^{-1},x_2\theta^{-1})$ with $x_2$ a
minimal vector of $\cct$ (a long root). 
The definition of $g$ (see \cite{jhc:Coxeter-Todd}) shows that
$x_1^2\congruent1(3)$ if and only if there is a long root $x_2$
of $\cct$ such that 
$(x_1\theta^{-1},x_2\theta^{-1})\in2^{-1/2}\leech$.
By \cite{jhc:leech-radius}, the covering radius of
$2^{-1/2}\leech$ is $1$, so the intersections of the balls
of radius $1$ centered at the points of $\cct$ and at the points
$(x_1\theta^{-1},x_2\theta^{-1})$ with $x_1^2\congruent1(3)$ and
$x_2^2=2$ cover 
$\cct\tensor\R$. Computing the radius of the intersection of a
ball of the second family with $\cct\tensor\R$, we obtain the lemma.
%
\endproof

\beginproclaim Theorem
\Tag{thm-eI7,1}.
Let $\Lambda=\cct$ and $L=\Lambda\oplus\eII1,1\isomorphism\eI7,1$. 
%
\item{\rom1} If $v\in L$ is primitive, isotropic and not
equivalent to a multiple of $\rho$ under $\reflec_0 L$, then
$v^\perp/\spanof{v}\isomorphism\Eisenstein^6$.
\item{\rom2}  $\aut L=\reflec L$. In particular, $L$ is
reflective. \endproclaim 

\beginproof{Proof:}
\rom1 
Suppose that $v$ is a primitive isotropic vector of smallest
height in its orbit under $\reflec_0 L$. Suppose $v$ is not a
multiple of $\rho$, so that it lies over some $\ell\in
\Lambda\tensor \R$. By the minimality of its height and
Theorem~\tag{thm-what-short-roots-do}, $\ell$ lies at 
distance $\geq1$ from each lattice point $\lambda\in \Lambda$
and at distance $\geq 3^{-1/2}$ from each $\lambda\theta^{-1}$ with
$\lambda\in \Lambda$ and $\lambda^2\congruent1(3)$. By
Lemma~\tag{thm-coxeter-todd-well-covered} the set $S$ of such
points in $\Lambda\tensor\R$ is 
discrete. Let $\mu_1,\ldots,\mu_n$ be the elements of
$\Lambda$ with $(\ell-\mu_i)^2=1$ and let $\nu_1,\ldots,\nu_m$
be those vectors  of the form $\lambda\theta^{-1}$ with
$\lambda\in \Lambda$ and $\lambda^2\congruent1(3)$ such that
$(\ell-\nu_i)^2=1/3$. Over each $\mu_i$ (resp. $\nu_i$) there is
a short root of $L$, say $r_i$ (resp. $s_i$), of height $1$
(resp. $\theta$). By
Theorem~\tag{thm-what-short-roots-do}\rom2, we may suppose that
the $r_i$ and $s_i$ are orthogonal to $v$. Because $S$ is discrete the vectors
$\mu_i-\ell$ and $\nu_i-\ell$ span $\Lambda\tensor\R$. By
Lemma~\tag{thm-linear-independence-descends} this implies that among
the images of the vectors $r_i$ 
and $s_i$ in $v^\perp/\spanof{v}$ are  $6$  short
roots that are linearly independent over $\Eisenstein$. Since
$v^\perp/\spanof{v}$ is positive-definite, it 
follows that $v^\perp/\spanof{v}\isomorphism 
\Eisenstein^6$. 

\rom2
Follows from the transitivity on primitive null sublattices that
are orthogonal to no short roots (such as $\spanof{\rho}$) and
Theorem~\tag{thm-rho-stabilizer-finite-index-2}\rom3. 
%
\endproof

We now study the quaternionic lattice $\qI5,1$. The analysis is
surprisingly similar to our study of $\eI7,1$.
In particular,  Lemma~\tag{thm-barnes-wall-well-covered} below
is very similar to 
Lemma~\tag{thm-coxeter-todd-well-covered}.  
Section~5 of \cite{jhc:laminated-lattices} describes an
embedding 
of the real form $\rbwl$ of $2^{1/2}\qbwl$ into
the Leech lattice $\leech$. 
(Up to isometry of $\leech$, there is only one embedding.)
When we refer to concepts involving
$\leech$ while 
discussing $\qbwl$, we implicitly refer to this embedding.

\beginproclaim Lemma
\Tag{thm-barnes-wall-well-covered}.
$\qbwl\tensor\R$ is covered by the closed balls of radius $1$ centered
at points of $\qbwl$ together with those of radius
$2^{-1/2}$ centered at points $\lambda(1+\i )^{-1}$ with
$\lambda\in\qbwl$ of odd norm. In particular, $\qbwl$ is \wc.
Any point of
$\qbwl\tensor\R$ not in the interior of one of these balls is a
deep hole of $2^{-1/2}\leech$.\endproclaim

\beginproof{Proof:}
The orthogonal complement of $\qbwl$ in $2^{-1/2}\leech$ is
a copy of the $E_8$ lattice. Properties of the embedding are
described in \cite{jhc:laminated-lattices} and include the following.
If 
$(x,y)\in(\qbwl\tensor\R)\times(E_8\tensor\R)$ lies in
$2^{-1/2}\leech$, then $y\in{1\over2}E_8$ and hence has
norm $n/2$ for some nonnegative integer $n$. 
We write $B(x,y)$ for the ball of radius 1 with center
$(x,y)\in2^{-1/2}\leech$. Only if the norm of $y$ is $0$ or
$1/2$ does the interior of $B(x,y)$ meet $\qbwl\tensor\R$. Those
$x$ for which $(x,y)\in2^{-1/2}\leech$ for some $y$ of norm
$1/2$ are exactly the deep
holes of $\qbwl$. By Theorem~\tag{thm-barnes-wall-properties},
the set of such $x$ 
coincides exactly with 
$\set{\lambda(1+\i )^{-1}}{\lambda\in\qbwl,\lambda^2\congruent1(2)}$.
For such $(x,y)$, the ball $B(x,y)$ meets $\qbwl\tensor\R$ in a
ball of radius $2^{-1/2}$. The theorem follows from the fact
\cite{jhc:leech-radius} that the covering radius of
$2^{-1/2}\leech$ is 1. 
%
\endproof

The deep holes of $2^{-1/2}\leech$ played a role in our
study of $L^\Eisenstein=\cct\oplus\eII1,1$ in Theorem~\tag{thm-eI7,1}. We
showed there that if $v$ is a primitive null vector of
$L^\Eisenstein$ lying over such a hole in $\cct\tensor\R$, then
there are  short
roots of $v^\perp/\spanof{v}$ associated to certain elements of
$\cct$ and $\cct\theta^{-1}$ near the hole.
Given the similarity between
Lemmas~\tag{thm-coxeter-todd-well-covered} and
\tag{thm-barnes-wall-well-covered}, one should expect something
similar to 
happen in the study of $L^\Hurwitz=\qbwl\oplus\qII1,1$. Indeed
it does, but because the conclusions of
Theorem~\tag{thm-what-short-roots-do} for short 
roots of height $1+\i $ in Hurwitz lattices  are slightly weaker than
those for short roots of heights $1$ and $\theta$ in Eisenstein
lattices, the arguments are more
complicated. Lemma~\tag{thm-leech-holes-in-barnes-wall} 
provides the necessary technical information about the deep
holes of $\leech$. The language of affine Coxeter-Dynkin
diagrams is the natural way to discuss these deep holes; see
\cite{jhc:leech-radius} for background. Since the real form $\rbwl$ of
$2^{1/2}\,\qbwl$ is the  sublattice of $\leech$ fixed by an
involution, we will study the actions of involutions on affine
diagrams. 

\def\D{\Delta}
If $\D$ is an affine Coxeter-Dynkin diagram each of whose
components has type $A_n$, $D_n$ or $E_n$, then an affine simple
root system of type $\D$ is a set of vectors $v_i$ of norm $2$ 
in real Euclidean
space, one for each node of $\D$, satisfying $\ip{v_i}{v_j}=0$,
$-1$ or $-2$ according as the corresponding nodes of $\D$ are
disjoined, joined or doubly joined. (The last case occurs only
for the affine diagram $A_1$, where the doubly joined vectors
are each others negatives.) We usually identify the nodes of
$\D$ with the vectors $v_i$, so we may speak of vectors being
disjoined or of nodes being linearly independent. Each component
$X_n$ has $n+1$ nodes and spans an $n$-dimensional space; any
$n$ of its vectors are linearly independent. The spaces spanned 
by distinct components are orthogonal.

\beginproclaim Lemma
\Tag{thm-affine-diagram-automorphisms}.
Suppose $\D$ is an affine simple root system spanning a Euclidean
space $E$ and that $\phi$ is an isometry of $E$ permuting the vectors
of $\D$. Let $A$ (resp. $B$) be the number of orbits of $\phi$
on the vertices (resp. components) of $\D$. Then $A-B$ is the dimension
of the space $F$ of points fixed by $\phi$. Furthermore, if
$\phi$ has prime order $p$ then
$$ 
p\dimension F-\dimension E =(P_V-P_C)(p-1)\;,
\eqno\eqTag{eq-dimension-of-fixed-space}
$$
where $P_V$ (resp. $P_C$) is the number of vertices
(resp. components) of $\D$ preserved by $\phi$.\endproclaim

\beginproof{Proof:}
Since the vectors $v_i$ of $\D$
span $E$, $\phi$ is determined by its action on $\D$. 
Let $M$ be the order of $\phi$ and 
let $\pi$
denote the projection $x\mapsto{1\over M}\sum_{j=1}^M\phi^j(x)$
of $E$ to $F$. Naturally, $F$ is spanned by the vectors
$\pi(v_i)$; there are $A$ distinct such vectors. Let the minimal
affine subdiagrams of $\D$ preserved by $\phi$ be denoted
$\D_1,\ldots,\D_B$. (Such a diagram is just a union of
components of $\D$ permuted cyclically by $\phi$.) Let $F_j$
(for $1\leq j \leq B$) be the intersection of  the span of
$\D_j$ with $F$. If $j\neq k$ then $\D_j$ is orthogonal to
$\D_k$ and thus $F_j$ and $F_k$ are also orthogonal. Since each
vector $\pi(v_i)$ lies in some $F_j$, we see that
$F=\oplus_{j=1}^BF_j$.  Therefore the lemma holds if it holds
with $\D$ replaced by each $\D_j$ in turn.

It now suffices to prove the lemma under the hypothesis that
$\phi$ acts transitively on the components of $\D$. 
Each of the 
vectors $\pi(v_i)$ has the form ${1\over |J_i|}\sum_{j\in J_i}v_i$ for
some $\phi$-orbit $J_i$ of nodes of $\D$ that meets each component of
$\D$. 
If $v_i$ and $v_j$ are not
$\phi$-equivalent  then the sets $J_i$ and $J_j$ are disjoint. 
In order for some subset of the $\pi(v_i)$ to be linearly independent,
the union of the corresponding $J_i$ must contain an affine
diagram. Since such a diagram would be a component of $\D$ and
$\phi$ permutes the components transitively, the union of the
$J_i$ would have to be all of $\D$. Therefore there is at most
one linear dependence among the $\pi(v_i)$. There is at least
one linear dependence because the vectors of any given component
of $\D$ are dependent and therefore their images under
$\pi$ are also. (Indeed, some of the $\pi(v_i)$ might vanish, as
happens when $\D$ is an $A_1$ diagram and $\phi$ is the
nontrivial automorphism.) Therefore $F$ has dimension
$A-1=A-B$. 

Now suppose $\phi$ has prime order $p$. If $\D$ has $C$
components then it has $C+\dimension E$ vertices. Then
$B=P_C+(C-P_C)/p$ and $A=P_V+(C+\dimension E-P_V)/p$. Arithmetic
proves Eq.~\eqtag{eq-dimension-of-fixed-space}.
%
\endproof

\beginproclaim Lemma
\Tag{thm-leech-holes-in-barnes-wall}.
Let $\ell$ be a deep hole of $2^{-1/2} \leech$ that lies
in $\qbwl\tensor\R$. Then there are at least $5$ vertices $v_i$ of
$\ell$ that lie in $\qbwl$ such that the vectors $v_i-\ell$ are
linearly independent over $\R$.\endproclaim

\beginproof{Proof:}
In this proof we'll scale everything up by $2^{1/2}$, so $\ell$
is a deep hole of $\leech$ that lies in
$\rbwl\tensor\R$. By~\cite{jhc:leech-radius}, if we take
coordinates centered at $\ell$ then 
the vertices in $\leech$ of the hole form an affine simple root system
spanning $\R^{24}$. Let $\D$ be the associated Coxeter-Dynkin
diagram, which is a union of affine diagrams of types $A_n$,
$D_n$ and $E_n$.
The space $\rbwl\tensor\R$ is the fixed space of an
involution $\phi$ of $\leech$. Since $\phi$ fixes $\ell$, it
acts on $\D$. 
Let $F$, $P_V$ and $P_C$ be as in
Lemma~\tag{thm-affine-diagram-automorphisms}. Then  we have
$2\cdot16=24+P_V-P_C$, 
so $P_V-P_C=8$. This shows $P_V\geq8$; since $P_V>0$
we have $P_C>0$ and so we actually have $P_V>8$.

Therefore there are $P_V>8$ vertices of $\D$ fixed by $\phi$,
which is to say, lying in $\rbwl$. The number of independent
linear
conditions satisfied by a set $S$ of vertices of $\D$ equals the
number of components of $\D$ all of whose vertices lie in
$S$. Since each affine diagram has at least $2$ vertices, there
are at most $P_V/2$ conditions, so among the
vertices of $\ell$ lying in $\rbwl$ there are at least $5$ that
are linearly independent over $\R$. The lemma follows because we
took $\ell$ as the origin.
%
\endproof

\remark{Remark:}
The proof shows that any deep hole of $\leech$ that lies in in
$\rbwl\tensor\R$ has at least $9$ vertices in $\rbwl$. This is the
best possible result, because there are holes of type $E_8^3$ in
$\rbwl$; the involution $\phi$ fixes one $E_8$ diagram and
exchanges the other two.

\beginproclaim Theorem
\Tag{thm-qI5,1}.
Let $\Lambda=\qbwl$ and $L=\Lambda\oplus\qII1,1\isomorphism\qI5,1$. 
%
\item{\rom1} 
If $v\in L$ is a primitive isotropic vector not
equivalent under  $\shr L$ to a multiple of $\rho$ then $v^\perp/\spanof{v}$
contains at least two linearly independent short roots.
\item{\rom2} 
The index of  $\reflec L$ in $\aut L$ is at most $4$.
In particular, $L$ is reflective.\endproclaim 


\beginproof{Proof:}
\rom1
This is very similar to the proof of
Theorem~\tag{thm-eI7,1}\rom1. Suppose 
that $v$ is a primitive isotropic vector of $L$ of smallest
height in its orbit under $\reflec L$. Suppose $v$ is not a multiple
of $\rho$, so that $v$ lies over some $\ell\in
\Lambda\tensor\R$. By
Theorem~\tag{thm-what-short-roots-do}, $\ell$ lies at 
distance $\geq1$ from each lattice point $\lambda\in \Lambda$
and at distance $\geq 2^{-1/2}$ from each point
$\lambda(1+\i )^{-1}$ with $\lambda\in \Lambda$ and
$\lambda^2\congruent1(2)$. By Lemma~\tag{thm-barnes-wall-well-covered},
$\ell$ must be a deep 
hole of $2^{-1/2}\leech$. By
Lemma~\tag{thm-leech-holes-in-barnes-wall} there are $5$ 
vertices $w_1,\ldots,w_5$ of the hole that lie in $\qbwl$ such
that the vectors $w_i-\ell$ are linearly independent over
$\R$. Since there are $5$ of them, we may suppose that  $w_1-\ell$
and $w_2-\ell$ are linearly independent over $\H$. Because
$w_1,w_2\in\qbwl$, There are
short roots $r_1$ and $r_2$ of $L$ of height $1$ lying over
them. Since their heights are $1$, 
Theorem~\tag{thm-what-short-roots-do}\rom2 assures us that
$r_1$ and $r_2$ may be chosen 
orthogonal to $v$. By Lemma~\tag{thm-linear-independence-descends}, the
images of $r_1$ and $r_2$ 
in $v^\perp/\spanof{v}$ are linearly independent, proving \rom1.

\rom2
Follows from
Theorem~\tag{thm-rho-stabilizer-finite-index-2}\rom3 and the
fact \rom1 that 
$\reflec L$ acts transitively on the primitive null lattices
(such as $\spanof{\rho}$) in
$L$ that are orthogonal to no short roots.
%
\endproof

\remark{Remark:}
In light of the fact that $\shr\qI n,1$ contains the biflections in
some long roots (see the remark following the proof of
Theorem~\tag{thm-rho-stabilizer-finite-index-2}), 
it is possible that $\reflec L=\shr L$.


%We now consider odd lattices over $\gauss$.
%!!Theorem  
%The automorphism group of the Gaussian Lorentzian
%lattice $E_8\oplus\I1,1$ contains a subgroup generated by
%reflections that acts on $\ch5$ with quotient having finite
%volume and exactly 2 cusps.
%!!

\section{\Tag{sec-selfdual-classification}. Enumeration of
selfdual lattices}

As we explain below, the orbits of primitive isotropic lattices
in the Lorentzian 
lattice $\rI n+1,1$ are in natural $1$-$1$ correspondence with
the equivalence classes of positive-definite selfdual lattices of
dimension $n$ over $\ring$. This means that one may classify
such lattices by studying $\aut\rI n+1,1$. Since we did just
that in Section~\tag{sec-reflection-groups}, for various $\ring$
and $n$, we can now 
provide geometric proofs of such classifications. We begin with
an analogue of a  result well-known for lattices over $\Z$.

\beginproclaim Theorem
\Tag{thm-indefinite-selfdual-classification}.
An indefinite selfdual lattice $L$ over $\ring=\Eisenstein$ or
$\Hurwitz$ is characterized up to isometry by its dimension and
signature. An indefinite selfdual lattice $L$ over
$\ring=\Gaussian$ is characterized up to isometry by its
dimension, signature, and whether it is even; if $L$ is even
then its signature is divisible by $4$.\endproclaim

\beginproof{Proof:}
First we
show that $L$ contains an isotropic vector. If
$\ring=\Hurwitz$, or if $\dimension L>2$, then the real
form of $L\tensor\Q$ 
is an indefinite  rational bilinear form of rank $>4$, so Meyer's theorem
\ecite{milnor:bilinear-forms}{Chap.~2} asserts the existence of an
isotropic vector. If $\dimension L=2$ and $\ring=\gauss$ or
$\eisen$, then we consider the $2\times2$ matrix of inner
products of the elements of a basis for $L$. This may be
diagonalized by row and column operations
over $\ring\tensor\Q$ to a diagonal matrix
$[a,-a^{-1}]$ with $a\in\Q$. (Each term is real because the
matrix is Hermitian, and each determines the other because the
determinant is $-1$.)  Then  the vector $(1,a)$ is
isotropic. Having obtained an isotropic vector in
$L\tensor \Q$, we may multiply by a scalar to obtain an
isotropic vector of $L$. 

If $L$ is odd then the proof of  Theorem~4.3 in
\ecite{milnor:bilinear-forms}{Chap.~2} applies, and $L\isomorphism
\rI n,m$ for some $n$ and $m$. This completes the proof of the
first claim, since any selfdual lattice over $\Eisenstein$ or
$\Hurwitz$ is odd: if $v,w\in L$ satisfy $\ip{v}{w}=\w$ then
$v^2$, $w^2$ and $(v+w)^2$ cannot all have the same
parity. Existence of such $v$ and $w$ is assured by the
selfduality of $L$. This also proves that an odd indefinite
selfdual Gaussian lattice is characterized by its dimension
and signature.

One may construct lattices $N$ from an odd Gaussian lattice $M$ by
considering the sublattice $M^e$ consisting of the elements  of
$M$ of even norm, and considering the 3 lattices $N$ such that
$M_e'\sset N\sset M_e$. When $M$ is $\gI1,1$, then
$N$ may be chosen to be $\gII1,1$.
Now consider an indefinite 
even selfdual $\gauss$-lattice $L$. We know that $L$ contains
an isotropic vector, and as in \cite{milnor:bilinear-forms} there
is a  decomposition $L=\Lambda\oplus\gII1,1$. We see that $L$
arises from applying the construction above to the odd selfdual lattice
$\Lambda\oplus\gI1,1$. Since  $\Lambda\oplus\gI1,1$
is isomorphic to $\gI n,m$, with $n$ and $m$ 
determined by the dimension and signature of $L$, is is clear
that $L$ can be constructed by applying our
construction to $\gI n,m$. No even lattices arise unless
$n-m\congruent0(4)$, when two isometric ones do.
%
\endproof

Special cases of
Theorem~\tag{thm-indefinite-selfdual-classification}  are
$\eI7,1\isomorphism\cct\oplus\eII1,1$ and
$\eI6,1\isomorphism\qbwl\oplus\qII1,1$, which are the lattices
studied in
Theorems~\tag{thm-eI7,1} and
\tag{thm-qI5,1}. 
Theorem~\tag{thm-indefinite-selfdual-classification} also provides
the correspondence mentioned above: 
if $V$ is a primitive isotropic lattice in $\rI n+1,1$ then  it
is easy to check that
$V^\perp/V$ is an $n$-dimensional positive-definite selfdual
lattice. Every such lattice $\Lambda$ arises this way, because we may
write $\rI n+1,1\isomorphism \Lambda\oplus\rI1,1$, and $\rI1,1$ contains
isotropic lattices. Furthermore, if $V$ and $V'$ are primitive isotropic
lattices in $\rI n+1,1$ and 
$V^\perp/V\isomorphism {V'}^\perp/V'$ then one may find an
isometry of $\rI n+1,1$ carrying $V$ to $V'$. This provides a
one-to-one correspondence between orbits of primitive isotropic lattices of 
$\rI n+1,1$ and selfdual positive-definite lattices in dimension
$n$. Similarly, the orbits of primitive isotropic lattices of
$\gII n+1,1$ 
correspond to even positive-definite  selfdual Gaussian lattices
of dimension 
$n$. 
%Finally, the cusps of $\khnp/\aut L$, where $L$
%is a Lorentzian lattice also correspond the the orbits of isotropic
%lattices in $L$.

\beginproclaim Theorem
\Tag{thm-definite-selfdual-classification}.
There are exactly two positive-definite 
selfdual $\eisen$-lattices in dimension
$6$,  namely $\cct$ and  $\eisen^6$. 
There are exactly two positive-definite selfdual
$\Hurwitz$-lattices of dimension $4$, namely $\qbwl$ and $\Hurwitz^4$.\endproclaim

\beginproof{Proof:}
In light of the correspondence between selfdual lattices over
$\eisen$  and orbits of
primitive isotropic
lattices in $\eI n,1$, to prove the first claim it
suffices to show that if $V$ is such a lattice then
$V^\perp/ V$ is isomorphic to one of $\cct$ and $\eisen^6$.
This follows from Theorem~\tag{thm-eI7,1}\rom1.

Before proving the second claim, we show that the only positive-%
definite selfdual 
$\Hurwitz$-lattice of dimension $2$ is $\Hurwitz^2$. This
follows from the correspondence between such lattices and the
primitive null sublattices of $\qI3,1$ and the fact
(Theorem~\tag{thm-qI2,1-qI3,1}) that all such sublattices
are equivalent. 
Now suppose that $\Lambda$ is a positive-definite selfdual
$\Hurwitz$-lattice of dimension $4$. 
By the correspondence between such lattices and primitive null
sublattices of $\qI5,1$ we know that $\Lambda\isomorphism
V^\perp/V$ for some primitive null lattice 
$V$ in $\qI5,1\isomorphism \qbwl\oplus\qII1,1$. By
Theorem~\tag{thm-qI5,1}\rom1, either
$\Lambda\isomorphism\qbwl$ or else 
$\Lambda$ contains two linearly independent short roots $r_1$
and $r_2$. In the latter case  we observe that in a
positive-definite  integral
lattice any two short roots are either 
proportional or orthogonal, so $r_1\perp r_2$. Thus their span $S$ in
$\Lambda$ is a copy of $\Hurwitz^2$, and their orthogonal
complement  $S^\perp$  is a selfdual
$\Hurwitz$-lattice of dimension $2$. By the above,
$S^\perp\isomorphism\Hurwitz^2$ so $\Lambda=S\oplus
S^\perp\isomorphism\Hurwitz^4$. 
%
\endproof

The theorem implies that the only positive-definite 
selfdual $\eisen$-lattices of
dimension $n\leq6$ are $\eisen^n$ and $\cct$ and that the only
such $\Hurwitz$-lattices of dimension $n\leq4$ are $\qbwl$
and $\Hurwitz^n$.
Similarly, 
Theorem~\tag{thm-13-reflection-groups} shows that
$\gII5,1\isomorphism E_8^\gauss\oplus\gII1,1$ contains only one
orbit of primitive isotropic lattices,  so
$E_8^\gauss$ is the only
$4$-dimensional even positive-definite selfdual $\gauss$-lattice.
%the cusps of the 6-dimensional Gaussian
%lattice $E_8\oplus\I1,1$ correspond to the 2 selfdual
%lattices in dimension 4 over $\gauss$. 

These results have been obtained before but only by  computational
means. Feit \cite{feit:unimodular-Eisenstein} found examples of
many positive-definite 
selfdual $\Eisenstein$-lattices. He derived and used a
version of the
mass formula to verify that his list was complete for dimensions
$n\leq12$. Conway and Sloane \ecite{jhc:Coxeter-Todd}{Thm.~3}
provide a nice proof of 
this classification in dimensions $n\leq6$ based on theta series
and modular forms. (Their proof does not apply for $6<n<12$: in
the second-to-last sentence of the proof, ``12'' 
should be replaced by ``7''.)
Although selfdual $\Gaussian$-lattices have not been tabulated,
it would be easy (and boring) to enumerate them through
dimension $12$  by using the fact that the real form of a
selfdual $\Gaussian$-lattice is selfdual over $\Z$. Selfdual
lattices over $\Z$ have been extensively tabulated;
see \ecite{splag}{Chap.~16} and \cite{reb:thesis}.
An enumeration of positive-definite
selfdual $\Hurwitz$-lattices for dimensions 
$n\leq7$ has recently been completed by Bachoc
\cite{bachoc:selfdual-Hurwitz-lattices} and for $n=8$ by Bachoc
and Nebe \cite{bachoc:8-dimensional-Hurwitz-lattices}. 
These enumerations are based on a generalization of Kneser's notion
of ``neighboring'' lattices, together with a suitable version of
the mass formula.

\section{\Tag{sec-comparison}. Comparison with the lists of Mostow and
Thurston}

In this section we justify the word ``new'' in our title, by
showing that our ``largest'' three reflection groups do not
appear on the lists of Mostow
\cite{mostow:monodromy-groups-on-the-complex-n-ball} and
Thurston 
\cite{thurston:shapes-of-polyhedra}. Deligne and Mostow
\cite{deligne:monodromy-of-hypergeometric-functions} and Mostow
\cite{mostow:picard-lattices-from-half-inegral-conditions} 
constructed 94 reflection groups acting on $\chn$ for various
$n=2,\ldots,9$ by considering the monodromy of hypergeometric
functions. Thurston \cite{thurston:shapes-of-polyhedra}
constructed the same set of groups 
in terms of moduli of flat metrics (with specified
sorts of singularities) on the sphere $S^2$. We will sometimes refer to
these groups as ``the monodromy groups''. We show here that none
of the groups $\reflec\eI n,1$ ($n\geq4$) or $\reflec\gII
4n+1,1$ ($n\geq1$) appear on their lists. In particular, our
groups $\reflec\eI7,1$, $\reflec\eI4,1$ and $\reflec\gII5,1$ are
new. According to Conjecture~\tag{thm-conjecture} and the commentary
following it, the lattices $\eI n,1$ ($n\leq13$) and $\gII
4n+1,1$ ($n\leq2$) are reflective; if this is correct then the
reflection groups of these lattices are also new.  We will also identify 
$\reflec\eI3,1$ with one of the monodromy groups. We leave open
the question of whether our other groups appear on their
lists and also the question of the commensurability of our
groups and theirs.

We write $G$ for $\reflec L$ where $L$ is one of the lattices
$\eI n,1$ ($n\geq4$) or 
$\gII4n+1,1$ ($n\geq1$). We will show
that each monodromy group that acts on $\chn$ for $n\geq4$ is
either cocompact or contains a primitive reflection of order~3
or 4. We will also show that $G$ contains no such reflections,
so $G$ cannot be a monodromy group. (A reflection is
primitive if it is not a power of a reflection of larger order.)
It is easy to identify all the reflections of $L$
(Lemma~\tag{thm-classify-honest-reflections}), but to compare
$G$ with the monodromy groups we
must also consider elements $R$ of $G$ that are not reflections of
$L$ but still act on $\chn$ as reflections\emdash such an $R$ differs
from a reflection of $L$ by a complex scalar of norm one. To see that
this is a nontrivial issue, consider $\aut\gII1,1$. The subgroup
of elements of determinant one is conjugate to $\SL_2\Z$ and
hence contains an element acting on $\ch1$ as a
triflection. This happens despite the fact that (by
Lemma~\tag{thm-classify-honest-reflections}) the only 
reflections of $\gII1,1$ are biflections. We deal with this
issue by calling a reflection of $\C^n$ an honest reflection
and a reflection of $\cpnm$ a projective reflection. This
distinction requires a more careful definition of a primitive
reflection. If $H$ is a group acting on $\C^n$
(resp. $\cpnm$) and $R\in H$ is an honest (resp. projective)
reflection, then we say that $R$ is primitive in $H$ if it is
not a power of an 
honest (resp. projective) reflection of $H$ of larger order. Our
first lemma assures that the behavior of $\gII1,1$ discussed
above is a low-dimensional phenomenon, and gives a condition for
a projective reflection in $G$ to ``come from'' an honest
reflection.

\beginproclaim Lemma
\Tag{thm-projective-reflections-are-honest}.
Suppose $M$ is an $n$-dimensional lattice over
$\ring=\Eisenstein$ or $\Gaussian$ and that $R$ is an element of
$\aut M$ whose action on $\cpnm$ is a
projective reflection of order $m<n$. Then $R$ differs by a unit
of $\ring$ from an honest reflection of $M$.\endproclaim

\beginproof{Proof:}
Since $R$ acts on $\cpnm$ as a projective reflection, it has two distinct
eigenvalues $\lambda$ and $\lambda'$, with one (say $\lambda$)
having multiplicity $n-1$. Furthermore, since $R^m$ preserves
$M$ and acts trivially on $\cpnm$, we see that there is a unit
$\a$ of $\ring$ such that $\lambda^m=\lambda'^m=\a$. The
characteristic polynomial of $R$ is
$(x-\lambda)^{n-1}(x-\lambda')$; since $R\in \GL_n\ring$, the
coefficient of each power of $x$ must lie in
$\ring$. Considering the coefficients of $x^{n-1}$ and
$x^{n-m-1}$ we find that 
$$\eqalign{
{n-1\choose 1}\lambda^{\phantom{m+1}} + 
	{n-1\choose 0} \lambda'\phantom{\lambda^m}&=
	y\rlap{\qquad and}\cr
{n-1\choose m+1}\lambda^{m+1} + 
	{n-1\choose m}\lambda'\lambda^{m} &= z\cr 
\cr}$$
for some $y,z\in\ring$. Because $\lambda^m=\a\in\ring$, the
second equation reduces to a linear equation in $\lambda$ and
$\lambda'$. For $n>m$ this is a nonsingular system of equations,
so $\lambda,\lambda'\in\ring\tensor\Q$. Since $\lambda,\lambda'$
are roots of unity they must actually lie in $\ring$. Then
$\lambda^{-1}R\in\aut M$ has eigenvalues $1$ (with multiplicity
$n-1$) and $\lambda^{-1}\lambda'$, completing the proof.
%
\endproof

Now we show that the only honest reflections of a selfdual
lattice are the obvious ones; the result is well-known for
lattices over $\Z$.

\beginproclaim Lemma
\Tag{thm-classify-honest-reflections}.
Any honest reflection $R$ of a selfdual lattice $M$ over
$\ring=\Eisenstein$ or $\Gaussian$ is either a reflection in a
lattice vector of norm $\pm1$ or a biflection in a lattice
vector of norm $\pm2$.\endproclaim

\beginproof{Proof:}
By considering the determinant of $R$ we discover that its only
nontrivial eigenvalue is a unit of $\ring$, so $M$ contains an
element of the corresponding eigenspace, so $R$ is a
reflection in some lattice vector $v$. Taking $v$ to be
primitive, every vector in the complex span of $v$ 
lies in the $\ring$-span of $v$. (This uses the fact that
$\ring$ is a principal ideal domain.) Furthermore, by
the selfduality of $M$, there exists $w\in M$ satisfying
$\ip{v}{w}=1$. Then $R(w)=w-v(1-\a)/v^2$ and so 
$w-R(w)=v(1-\a)/v^2$ lies in $M$. Therefore
$(1-\a)/v^2\in\ring$. Unless $\a=-1$ this requires $v^2=\pm1$
and if $\a=-1$ then it requires that $v^2$ divide $2$.
%
\endproof

\beginproclaim Theorem
\Tag{thm-primitive-reflections-in-my-groups}.
The image of $\aut L$ in the isometry group of complex
hyperbolic space contains no primitive projective reflection of
order $3$ or $4$.\endproclaim 

\beginproof{Proof:}
If $R\in\aut L$ acted on hyperbolic space as a
projective reflection of order $m=3$ or $4$, then because
$\dimension L>m$ we see by 
Lemmas~\tag{thm-projective-reflections-are-honest} and
\tag{thm-classify-honest-reflections} 
that $R$ differs by a unit from a reflection in a root $r$ of
$L$. Since $R$ acts as a reflection on hyperbolic space, $r^2$
must be positive.
Since $m\neq2$, $r^2$ must be $1$. If $L=\gII4n+1,1$
then this is impossible because even lattices contain no short
roots. If $L=\eI n,1$ then we must have $m=3$, and then there is
a $6$-fold reflection in $r$ whose square acts on hyperbolic
space as $R$, showing that $R$ is not primitive as a projective
reflection. 
%
\endproof

Now we will return to the monodromy groups and find primitive
reflections in them. We will discuss them in Thurston's terms;
here is a sketch of his construction. Let $n\geq4$ and let
$\a=(\a_1,\ldots,\a_n)$ be an $n$-tuple of  numbers in the
interval $(0,2\pi)$ that sum to $4\pi$. Let $P(\a)$ be the
moduli space of pairs $(p,g)$ where $p$ is an injective map from
$\{1,\ldots,n\}$ to an oriented sphere $S^2$
and $g$ is a singular Riemannian metric on 
$S^2$ which is flat  except on the image of $p$, with 
$p(i)$ being a ``cone point'' of curvature $\a_i$. (We denote
$p(i)$ by $p_i$.) Two such
pairs are considered equivalent if they differ by an
orientation-preserving similarity that identifies the
corresponding points $p_i$ with each other. This moduli space is a manifold of
real dimension 
$2(n-3)$ and admits a metric which is
locally isometric to $\ch{n-3}$. Let $H$ be the group of
elements $\sigma$ of the symmetric group $S_n$ satisfying
$\a_{\sigma(i)}=\a_i$ for all $i=1,\ldots,n$. Then $H$ acts by
isometries of $P(\a)$, by permuting the points $p_i$. We denote
the quotient orbifold by $C(\a)$ and its metric completion
by $\Cbar(\a)$. The fundamental group of $P(\a)$ is the pure
(spherical) braid
group on $n$ strands, and the orbifold fundamental group of
$C(\a)$ is the subgroup of the full (spherical) braid group the maps to $H$
under the usual map from the braid group to the symmetric
group. 

If the $\a_i$ satisfy certain combinatorial identities then
$\Cbar(\a)$ turns out to be the quotient of $\ch{n-3}$ by a
reflection group $\G(\a)$. There are only 94 choices for $\a$
(with $n\geq5$) satisfying these conditions, and the corresponding
$\G(\a)$ are the monodromy groups. The points of
$\Cbar(\a)\setminus C(\a)$
are the images of the mirrors of
certain reflections of $\G(\a)$. One can figure out the orders
of the primitive reflections associated to these mirrors by
finding the ``cone angle'' at each generic point of
$\Cbar(\a)\setminus C(\a)$: if the cone angle is $2\pi/m$ then the
corresponding (primitive) projective reflections have order
$m$. (This cone angle should not be confused with the cone angles
at the points $p_i\in S^2$.) The
generic points of $\Cbar(\a)\setminus C(\a)$ are associated to
``collisions'' between pairs of points $p_i$ and $p_j$ on
$S^2$ for which $\a_i+\a_j<2\pi$. We quote Thurston's
Proposition~3.5, which provides a way to compute the cone angles
at these points of $\Cbar(\a)\setminus C(\a)$.

\beginproclaim Proposition
\Tag{thm-cone-angles}. 
Let $S$ be the stratum of $\Cbar(\a_1,\ldots,\a_n)$ where two
cone points of curvature $\a_i$ and $\a_j$ collide. If $\a_i=\a_j$
then the cone angle around $S$ is $\pi-\a_i$; otherwise it is
$2\pi-\a_i-\a_j$. 
\QED
\endproclaim 

For example, take $\a$ to be the 10-tuple
$({2\pi\over3},{2\pi\over3},{\pi\over3},{\pi\over3},{\pi\over3},{\pi\over3},{\pi\over3},{\pi\over3},{\pi\over3},{\pi\over3})$,
which is number~13 on Thurston's list and number~4 on Mostow's.
Then at the singular strata of $\Cbar(\a)$ where two cone points of
curvature ${2\pi\over3}$ (resp. two of curvature $\pi\over3$,
resp. one of each curvature) collide, the cone angle is
$\pi-{2\pi\over3}={\pi\over3}$
(resp. $\pi-{\pi\over3}={2\pi\over3}$, resp.
$2\pi-{2\pi\over3}-{\pi\over3}=\pi$). We deduce that
$\G(\a)$ contains primitive projective reflections of orders~6,
3 and 2. 

\beginproclaim Theorem
\Tag{thm-primitive-reflections-in-their-groups}.
Each of the monodromy groups that acts on $\ch{n-3}$ for $n\geq7$ is
either cocompact or contains a primitive projective reflection
of order $3$ or $4$.\endproclaim 

\beginproof{Proof:}
Using Proposition~\tag{thm-cone-angles} and the list of $n$-tuples
$\a$ provided in
\cite{mostow:monodromy-groups-on-the-complex-n-ball} or
\cite{thurston:shapes-of-polyhedra}, it is easy 
to compute 
the cone angles at all the generic points of $\Cbar(\a)\setminus
C(\a)$. (The author wrote a short computer program to do this,
and also performed the computation by hand.) The only one with
$n\geq7$ for which none of the cone angles are $2\pi/4$ or
$2\pi/3$ is number~50 on Thurston's list (number~21 on
Mostow's). According to Thurston's table, $\G(\a)$ is cocompact
for this choice of $\a$.
%
\endproof

From Theorems~\tag{thm-primitive-reflections-in-my-groups} and
\tag{thm-primitive-reflections-in-their-groups} we immediately
deduce 

\beginproclaim Theorem
\Tag{thm-my-groups-are-new}.
If $L$ is $\eI n,1$ ($n\geq4$) or
$\gII4n+1,1$ ($n\geq1$) then $\reflec L$ does not appear among
the $94$ monodromy groups.
\QED
\endproclaim 

We close this section by sketching a proof that $\reflec\eI3,1$
is one of the monodromy groups\emdash it is the group $\G(\a)$ 
with  
$\a=({2\pi\over3},{2\pi\over3},{2\pi\over3},{2\pi\over3},{2\pi\over3},{2\pi\over3})$,
which is number~1 on Thurston's list and number~23 on Mostow's.
Because all the $\a_i$ are equal, the orbifold fundamental group
of $C(\a)$ is the spherical braid group $B_6$ on six strands. A standard
generator for $B_6$, braiding two points $p_i$ and $p_{i+1}$,
corresponds to a loop in $C(\a)$ encircling the singular stratum
$S$ of $\Cbar(\a)$ associated to a collision between $p_i$ and
$p_{i+1}$. Since the cone angle at $S$ is $\pi/3$ we find that
the standard generators map to 6-fold reflections. This fact,
together with the braid relations and the fact that the image of
$B_6$ is
not finite, specifies the representation uniquely up to
complex conjugation. The five standard generators may be taken to
map to $(-\w)$-reflections in short roots of $\eI3,1$, two of
these being orthogonal if the corresponding braid generators 
commute and having inner product $+1$ otherwise. One may then
use the techniques of Sections~\tag{sec-reflections} and
\tag{sec-reflection-groups} to show that 
the image of $B_6$ is all of $\shr\eI3,1$, which has the
same projective action as $\reflec\eI3,1$. The
arguments we have sketched here concerning the braid group
representation are carried out in detail in
\cite{allcock:cubic-surf}. 

\section{\Tag{sec-conclusion}. Comments}
 
The Leech lattice plays an interesting role in hyperbolic
geometry. Conway \cite{jhc:26dim} showed that
its special properties allow a remarkably simple description of
the symmetry group of the real Lorentzian lattice $\zII25,1$. In
Lemmas~\tag{thm-coxeter-todd-well-covered} and
\tag{thm-barnes-wall-well-covered}, we showed that it also 
``explains'' our  best examples, through 
embeddings of the Coxeter-Todd  and Barnes-Wall
lattices. We note that Borcherds   \ecite{reb:like-leech}{Sect.~3} 
used the same embeddings to produce interesting
{\it real} reflection groups acting on $\rh{13}$ and
$\rh{17}$. 

%The  lattices $\eI7,1$ and $\qI5,1$ are much like
%the lattice $\II25,1$, but there are some differences. 
%As we saw in the proof of
%Theorems~\tag{thm-eI7,1} and
%\tag{thm-qI5,1}, 
%there is a  correspondence between the Leech holes 
%of $\rct$ and isotropic vectors of 
%$\eI7,1$ whose heights cannot be decreased by reflection in
%the  roots of heights $1$ and
%$\theta$. These
%isotropic vectors 
%fall into several orbits under the stabilizer of $\rho$, but are
%all equivalent under $\aut\eI7,1$. 
%(For example, Leech holes of  types $A_1^{24}$ and $A_2^{12}$
%both occur.)
%This contrasts with the
%case of $\II25,1$, where two points in the boundary of the
%fundamental domain for the reflection group are equivalent  under
%$\aut\II25,1$ if and only if they are equivalent under the
%stabilizer of $\rho$. This touches on the difficulty of
%constructing fundamental domains for groups acting on $\chn$.
%%\annote{holy constructions of $\cct$?}

There are a number of interesting Eisenstein lattices besides
$\Eisenstein^n$ and $\cct$, but unfortunately
most seem  unsuitable for our purposes. 
The $D_4$, $E_6$, and $E_8$ root lattices are
$\eisen$-lattices, but at the smallest scale at which they are
integral, they have minimal norm $3$ and covering radii
$(3/2)^{1/2}$, $2^{1/2}$, and $(3/2)^{1/2}$, 
so Theorem~\tag{thm-cov-radius-implies-reflective} 
doesn't apply. 
There are also  complex and quaternionic forms of the Leech lattice (see
\cite{raw:quaternionic-leech} and \cite{raw:complex-leech}), but
they also have large 
covering radii. 
I do not know whether any 
of these lattices are \wc.
(Note also that the groups of the complex and quaternionic Leech
lattices contains no reflections.)

It would be nice to understand the groups $G$ generated by
reflections in the short roots of $L=\eI7,1$ or $\qI5,1$. The
transitivity of $G$ on primitive null lattices in $L$ that are
orthogonal to no short roots of $L$ proves that $(\aut L)/G$ is
the image of the stabilizer of $\spanof{\rho}$, where $\rho$ is
a primitive null vector corresponding to $\cct$ or $\qbwl$. One
can show that the central translations  (relative to the
decomposition $\cct\oplus\eII1,1$ or $\qbwl\oplus\qII1,1$ of
$L$) of $G$ have finite index in those of $\aut L$. This implies
that $(\aut L)/G$ is virtually free abelian (possibly
finite). We note the similar 
behavior of the integer lattices $\zI9,1$ and $\zII25,1$. That
is, $\shr \zI9,1$ acts transitively on the primitive null
sublattices orthogonal to no short roots, and
$$ 
\aut\zI9,1/(\shr\zI9,1\times\{\pm1\})\isomorphism E_8{:}\aut
E_8\;,  
$$
where the right hand side indicates a semidirect product of the
additive group of $E_8$ by $\aut E_8$. Similarly,
$\reflec\zII25,1$ acts transitively on the primitive null
sublattices  orthogonal to no roots at all, and
$$ 
\aut\zII25,1/(\reflec\zII25,1\times\{\pm1\})\isomorphism 
\leech{:}\aut\leech\;. 
$$
It is natural to wonder whether $\eI7,1$ and $\qI5,1$ behave
similarly. 

%The quaternionic lattice $\qbwl$ is surprisingly nice. Our
%results show that it shares a number of special properties of
%other good lattices, like $E_8$, $\cct$, and $\leech$. In
%particular, its automorphism group acts transitively on vectors
%of each given small norm, its classes modulo some scalar (in
%this case $1+\i$) fall into few orbits, each represented by some
%number of vectors of (uniform) small norm, and it admits a congruence
%base (in this case, the coordinate system provided by a frame). 
%It is also worth noting that $U_4(2)$, the nonabelian composition
%factor of $\aut\qbwl$, is 
%the main component of three(!) other reflection
%groups: the  Weyl group $E_6$ and two different
%complex reflection groups. See \oldcite{ATLAS} for  constructions
%of these groups.

We close with a conjecture that there are arithmetic complex and
quaternionic hyperbolic reflection groups in dimensions considerably
higher than we have so far considered:

\beginproclaim Conjecture
\Tag{thm-conjecture}.
The lattice $\rI n+1,1$ is reflective if and only if  each
positive-definite selfdual $\ring$-lattice $\Lambda$ of
dimension $n$ has finite index in the span
of its roots.\endproclaim

Inspiration for the conjecture comes from the real case, which
has been exhaustively studied (see for example
\cite{vin:groups-of-quad-forms}, \cite{vin:I18.1-and-I19.1},
\cite{jhc:vinberg-gps}, \cite{reb:lzl}). Namely, the groups of
the Lorentzian lattices $\zI n+1,1$ contain reflection
groups of finite index for $n\leq18$, and the failure of this in
higher dimension is associated with the existence of selfdual
$\Z$-lattices of dimension $\geq19$ that are not virtually
spanned by their roots.

If the conjecture is true, then it  provides reflection groups
acting with finite covolume on $\ch n$ for all $n\leq13$ and on
$\qh n$ for all $n\leq9$. The condition  on the lattices $\Lambda$
appearing in the conjecture has been verified in the Eisenstein
case for dimensions $\leq12$ by Feit
\cite{feit:unimodular-Eisenstein} and in the Hurwitz case
for dimensions $\leq 8$ by Bachoc
\cite{bachoc:selfdual-Hurwitz-lattices} and Bachoc and 
Nebe \cite{bachoc:8-dimensional-Hurwitz-lattices}. Furthermore,
since enumerations of selfdual 
lattices over $\Eisenstein$ and $\Hurwitz$ have not been
accomplished in higher dimensions, it might be that $\eI n+1,1$
(resp. $\qI n+1,1$) is reflective for $n$ even larger than $12$
(resp. $8$). The lowest-dimensional selfdual lattice over
$\Eisenstein$ or $\Hurwitz$ of which the author is aware
and whose roots fail to virtually span it is
$19$-dimensional, the tensor product of $\Eisenstein$
or $\Hurwitz$ with a $19$-dimensional $\Z$-lattice having
this property.

A similar conjecture could be made for the even Gaussian
lattices $\gII 4n+1,1$. If it were true then it would imply that
this lattice is reflective exactly for $n=0,1$ or $2$. The
failure to be reflective in higher dimensions would follow from
the fact that the Leech lattice may be regarded as a
$12$-dimensional selfdual $\Gaussian$-lattice with
no roots.

\section{References}
%

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