The accompanying file, between "BODY" and "ENDBODY" (which you may receive in several separate email messages), is a plain TeX file. When TeX-ed, it automatically unfolds 6 postscript files containing figures. Then when the dvi file is printed, these postscript files are "included". In other words one can print this file in the manner usual for a plain TeX file. If you have difficulty with this file please contact either author (radin@math.utexas.edu or sadun@math.utexas.edu) to obtain separately the plain TeX text and postscript figure files. BODY % This file was generated from % text12.tex Fig1.ps Fig2.ps Fig3.ps Fig4.ps Fig5.ps Fig6.ps by /public/bin/texpsinclude. % This file contains included PostScript. % TeX writes out the included PostScript to files. % Here are the macros for writing out the included PostScript: % Macros for dumping included Postscript to files. % Requires Plain TeX. Maybe other flavors will work too? % Jamie Stephens, jamies@math.utexas.edu, 28 Nov 94 % If you're in the UT Math Department, see % /usr/local/doc/tex/texpsinclude.text and /public/bin/texpsinclude. % Parts of these files are attached at the end of this file. % Adapted from Knuth's \answer macro in the TeXbook. \def\endofps{EndOfTheIncludedPostscriptMagicCookie} \chardef\other=12 \newwrite\psdumphandle \outer\def\psdump#1{\par\medbreak \immediate\openout\psdumphandle=#1 \copytoblankline} \def\copytoblankline{\begingroup\setupcopy\copypsline} \def\setupcopy{\def\do##1{\catcode`##1=\other}\dospecials \catcode`\\=\other \obeylines} {\obeylines \gdef\copypsline#1 {\def\next{#1}% \ifx\next\endofps\let\next=\endgroup % \else\immediate\write\psdumphandle{\next} \let\next=\copypsline\fi\next}} \outer\def\closepsdump{ \immediate\closeout\psdumphandle} %% %% Here's a shell script for automating the use of psdump.tex %% % #!/bin/bash % % PSDUMP=/usr/local/tex/macros/local/psdump.tex % TEXPSINCLUDEHELP=/usr/local/doc/tex/texpsinclude.help % % if [ $# -eq 0 ]; then % echo "Usage: $0 file.tex {files.ps} > output.tex" >&2 % echo "See $TEXPSINCLUDEHELP for more information." >&2 % exit 1 % fi % % echo "% This file was generated from" % echo "% $@ by $0." % echo "% This file contains included PostScript." % echo "% TeX writes out the included PostScript to files." % echo % echo "% Here are the macros for writing out the included PostScript:" % cat $PSDUMP % echo % % for argument in $@; do % if [ $argument != $1 ]; then % echo "% Here's the Postscript for $argument:" % echo "\message{Writing file $argument}" % echo -n "\psdump{$argument}" % echo "Including $argument." >&2 % cat $argument % echo "EndOfTheIncludedPostscriptMagicCookie" % echo % echo "\closepsdump" % fi % done % % echo % echo "% Finally, here is $1:" % % cat $1 %% %% Here's some documentation: %% % This file documents how to use the texpsinclude command in /public/bin. % Here's the problem that texpsinclude solves: % You have a TeX file called foo.tex that you want to distribute as a % single TeX file. The problem is that foo.tex needs two Postscript % files, bar1.ps and bar2.ps, for embedded figures. You'd like a single % TeX file which somehow includes bar1.ps and bar2.ps. When TeX % processes foo.tex, TeX should extract bar1.ps and bar2.ps from % foo.tex. % Here's how to do what you want to do: % texpsinclude foo.tex. bar1.ps bar2.ps > bigfoo.tex % (In general: "texpsinclude > ".) If you % enter this command, the result is a new file called bigfoo.tex. The % file bigfoo.tex contains bar1.ps and bar2.ps. If you give bigfoo.tex % to a friend, she can make your document with: % tex bigfoo.tex % This command writes out bar1.ps and bar2.ps, and the command also % TeX's foo.tex. % Note: If *you* run tex on bigfoo.tex in the same directory, then TeX % will write over your .ps files. Be careful when testing your % bigfoo.tex. %% %% Here's another way to use psdump.tex %% % EXAMPLE (remove the leading % signs to make it work): % %\psdump{example.ps}These three lines %are going be dumped "as is" %to the file example.ps %EndOfTheIncludedPostscriptMagicCookie %\closepsdump % Here's the Postscript for Fig1.ps: \message{Writing file Fig1.ps} \psdump{Fig1.ps}%!PS-Adobe-3.0 EPSF-3.0 %%Creator: Adobe Illustrator(TM) 3.0.1. %%For: () () %%Title: (/d1/grad/strauss/Penrose6.ai) %%CreationDate: (6/15/94) (16:04) %%BoundingBox: -46 175 389 646 %%DocumentProcessColors: Black %%DocumentSuppliedResources: procset Adobe_packedarray 2.0 0 %%+ procset Adobe_cmykcolor 1.1 0 %%+ procset Adobe_cshow 1.1 0 %%+ procset Adobe_customcolor 1.0 0 %%+ procset Adobe_IllustratorA_AI3 1.0 1 %AI3_ColorUsage: Black&White %AI3_TemplateBox: 306 396 306 396 %AI3_TileBox: 30 31 582 761 %AI3_DocumentPreview: Header %%Template: %%PageOrigin:30 31 %%AI3_PaperRect:0 792 612 0 %%AI3_Margin:30 31 30 31 %%EndComments %%BeginProlog %%BeginResource: procset Adobe_packedarray 2.0 0 %%Title: (Packed Array Operators) %%Version: 2.0 %%CreationDate: (8/2/90) () %%Copyright: ((C) 1987-1990 Adobe Systems Incorporated All Rights Reserved) userdict /Adobe_packedarray 5 dict dup begin put /initialize % - initialize - { /packedarray where { pop } { Adobe_packedarray begin Adobe_packedarray { dup xcheck { bind } if userdict 3 1 roll put } forall end } ifelse } def /terminate % - terminate - { } def /packedarray % arguments count packedarray array { array astore readonly } def /setpacking % boolean setpacking - { pop } def /currentpacking % - setpacking boolean { false } def currentdict readonly pop end %%EndResource Adobe_packedarray /initialize get exec %%BeginResource: procset Adobe_cmykcolor 1.1 0 %%Title: (CMYK Color Operators) %%Version: 1.1 %%CreationDate: (1/23/89) () %%Copyright: ((C) 1987-1990 Adobe Systems Incorporated All Rights Reserved) currentpacking true setpacking userdict /Adobe_cmykcolor 4 dict dup begin put /initialize % - initialize - { /setcmykcolor where { pop } { userdict /Adobe_cmykcolor_vars 2 dict dup begin put /_setrgbcolor /setrgbcolor load def /_currentrgbcolor /currentrgbcolor load def Adobe_cmykcolor begin Adobe_cmykcolor { dup xcheck { bind } if pop pop } forall end end Adobe_cmykcolor begin } ifelse } def /terminate % - terminate - { currentdict Adobe_cmykcolor eq { end } if } def /setcmykcolor % cyan magenta yellow black setcmykcolor - { 1 sub 4 1 roll 3 { 3 index add neg dup 0 lt { pop 0 } if 3 1 roll } repeat Adobe_cmykcolor_vars /_setrgbcolor get exec pop } def /currentcmykcolor % - currentcmykcolor cyan magenta yellow black { Adobe_cmykcolor_vars /_currentrgbcolor get exec 3 { 1 sub neg 3 1 roll } repeat 0 } def currentdict readonly pop end setpacking %%EndResource %%BeginResource: procset Adobe_cshow 1.1 0 %%Title: (cshow Operator) %%Version: 1.1 %%CreationDate: (1/23/89) () %%Copyright: ((C) 1987-1990 Adobe Systems Incorporated All Rights Reserved) currentpacking true setpacking userdict /Adobe_cshow 3 dict dup begin put /initialize % - initialize - { /cshow where { pop } { userdict /Adobe_cshow_vars 1 dict dup begin put /_cshow % - _cshow proc {} def Adobe_cshow begin Adobe_cshow { dup xcheck { bind } if userdict 3 1 roll put } forall end end } ifelse } def /terminate % - terminate - { } def /cshow % proc string cshow - { exch Adobe_cshow_vars exch /_cshow exch put { 0 0 Adobe_cshow_vars /_cshow get exec } forall } def currentdict readonly pop end setpacking %%EndResource %%BeginResource: procset Adobe_customcolor 1.0 0 %%Title: (Custom Color Operators) %%Version: 1.0 %%CreationDate: (5/9/88) () %%Copyright: ((C) 1987-1990 Adobe Systems Incorporated All Rights Reserved) currentpacking true setpacking userdict /Adobe_customcolor 5 dict dup begin put /initialize % - initialize - { /setcustomcolor where { pop } { Adobe_customcolor begin Adobe_customcolor { dup xcheck { bind } if pop pop } forall end Adobe_customcolor begin } ifelse } def /terminate % - terminate - { currentdict Adobe_customcolor eq { end } if } def /findcmykcustomcolor % cyan magenta yellow black name findcmykcustomcolor object { 5 packedarray } def /setcustomcolor % object tint setcustomcolor - { exch aload pop pop 4 { 4 index mul 4 1 roll } repeat 5 -1 roll pop setcmykcolor } def /setoverprint % boolean setoverprint - { pop } def currentdict readonly pop end setpacking %%EndResource %%BeginResource: procset Adobe_IllustratorA_AI3 1.0 1 %%Title: (Adobe Illustrator (R) Version 3.0 Abbreviated Prolog) %%Version: 1.0 %%CreationDate: (7/22/89) () %%Copyright: ((C) 1987-1990 Adobe Systems Incorporated All Rights Reserved) currentpacking true setpacking userdict /Adobe_IllustratorA_AI3 61 dict dup begin put % initialization /initialize % - initialize - { % 47 vars, but leave slack of 10 entries for custom Postscript fragments userdict /Adobe_IllustratorA_AI3_vars 57 dict dup begin put % paint operands /_lp /none def /_pf {} def /_ps {} def /_psf {} def /_pss {} def /_pjsf {} def /_pjss {} def /_pola 0 def /_doClip 0 def % paint operators /cf currentflat def % - cf flatness % typography operands /_tm matrix def /_renderStart [/e0 /r0 /a0 /o0 /e1 /r1 /a1 /i0] def /_renderEnd [null null null null /i1 /i1 /i1 /i1] def /_render -1 def /_rise 0 def /_ax 0 def % x character spacing (_ax, _ay, _cx, _cy follows awidthshow naming convention) /_ay 0 def % y character spacing /_cx 0 def % x word spacing /_cy 0 def % y word spacing /_leading [0 0] def /_ctm matrix def /_mtx matrix def /_sp 16#020 def /_hyphen (-) def /_fScl 0 def /_cnt 0 def /_hs 1 def /_nativeEncoding 0 def /_useNativeEncoding 0 def /_tempEncode 0 def /_pntr 0 def /_tDict 2 dict def % typography operators /Tx {} def /Tj {} def % compound path operators /CRender {} def % printing /_AI3_savepage {} def % color operands /_gf null def /_cf 4 array def /_if null def /_of false def /_fc {} def /_gs null def /_cs 4 array def /_is null def /_os false def /_sc {} def /_i null def Adobe_IllustratorA_AI3 begin Adobe_IllustratorA_AI3 { dup xcheck { bind } if pop pop } forall end end Adobe_IllustratorA_AI3 begin Adobe_IllustratorA_AI3_vars begin newpath } def /terminate % - terminate - { end end } def % definition operators /_ % - _ null null def /ddef % key value ddef - { Adobe_IllustratorA_AI3_vars 3 1 roll put } def /xput % key value literal xput - { dup load dup length exch maxlength eq { dup dup load dup length 2 mul dict copy def } if load begin def end } def /npop % integer npop - { { pop } repeat } def % marking operators /sw % ax ay string sw x y { dup length exch stringwidth exch 5 -1 roll 3 index 1 sub mul add 4 1 roll 3 1 roll 1 sub mul add } def /swj % cx cy fillchar ax ay string swj x y { dup 4 1 roll dup length exch stringwidth exch 5 -1 roll 3 index 1 sub mul add 4 1 roll 3 1 roll 1 sub mul add 6 2 roll /_cnt 0 ddef {1 index eq {/_cnt _cnt 1 add ddef} if} forall pop exch _cnt mul exch _cnt mul 2 index add 4 1 roll 2 index add 4 1 roll pop pop } def /ss % ax ay string matrix ss - { 4 1 roll { % matrix ax ay char 0 0 {proc} - 2 npop (0) exch 2 copy 0 exch put pop gsave false charpath currentpoint 4 index setmatrix stroke grestore moveto 2 copy rmoveto } exch cshow 3 npop } def /jss % cx cy fillchar ax ay string matrix jss - { 4 1 roll { % cx cy fillchar matrix ax ay char 0 0 {proc} - 2 npop (0) exch 2 copy 0 exch put gsave _sp eq { exch 6 index 6 index 6 index 5 -1 roll widthshow currentpoint } { false charpath currentpoint 4 index setmatrix stroke }ifelse grestore moveto 2 copy rmoveto } exch cshow 6 npop } def % path operators /sp % ax ay string sp - { { 2 npop (0) exch 2 copy 0 exch put pop false charpath 2 copy rmoveto } exch cshow 2 npop } def /jsp % cx cy fillchar ax ay string jsp - { { % cx cy fillchar ax ay char 0 0 {proc} - 2 npop (0) exch 2 copy 0 exch put _sp eq { exch 5 index 5 index 5 index 5 -1 roll widthshow } { false charpath }ifelse 2 copy rmoveto } exch cshow 5 npop } def % path construction operators /pl % x y pl x y { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add exch itransform } def /setstrokeadjust where { pop true setstrokeadjust /c % x1 y1 x2 y2 x3 y3 c - { curveto } def /C /c load def /v % x2 y2 x3 y3 v - { currentpoint 6 2 roll curveto } def /V /v load def /y % x1 y1 x2 y2 y - { 2 copy curveto } def /Y /y load def /l % x y l - { lineto } def /L /l load def /m % x y m - { moveto } def } {%else /c { pl curveto } def /C /c load def /v { currentpoint 6 2 roll pl curveto } def /V /v load def /y { pl 2 copy curveto } def /Y /y load def /l { pl lineto } def /L /l load def /m { pl moveto } def }ifelse % graphic state operators /d % array phase d - { setdash } def /cf {} def % - cf flatness /i % flatness i - { dup 0 eq { pop cf } if setflat } def /j % linejoin j - { setlinejoin } def /J % linecap J - { setlinecap } def /M % miterlimit M - { setmiterlimit } def /w % linewidth w - { setlinewidth } def % path painting operators /H % - H - {} def /h % - h - { closepath } def /N % - N - { _pola 0 eq { _doClip 1 eq {clip /_doClip 0 ddef} if newpath } { /CRender {N} ddef }ifelse } def /n % - n - {N} def /F % - F - { _pola 0 eq { _doClip 1 eq { gsave _pf grestore clip newpath /_lp /none ddef _fc /_doClip 0 ddef } { _pf }ifelse } { /CRender {F} ddef }ifelse } def /f % - f - { closepath F } def /S % - S - { _pola 0 eq { _doClip 1 eq { gsave _ps grestore clip newpath /_lp /none ddef _sc /_doClip 0 ddef } { _ps }ifelse } { /CRender {S} ddef }ifelse } def /s % - s - { closepath S } def /B % - B - { _pola 0 eq { _doClip 1 eq % F clears _doClip gsave F grestore { gsave S grestore clip newpath /_lp /none ddef _sc /_doClip 0 ddef } { S }ifelse } { /CRender {B} ddef }ifelse } def /b % - b - { closepath B } def /W % - W - { /_doClip 1 ddef } def /* % - [string] * - { count 0 ne { dup type (stringtype) eq {pop} if } if _pola 0 eq {newpath} if } def % group operators /u % - u - {} def /U % - U - {} def /q % - q - { _pola 0 eq {gsave} if } def /Q % - Q - { _pola 0 eq {grestore} if } def /*u % - *u - { _pola 1 add /_pola exch ddef } def /*U % - *U - { _pola 1 sub /_pola exch ddef _pola 0 eq {CRender} if } def /D % polarized D - {pop} def /*w % - *w - {} def /*W % - *W - {} def % place operators /` % matrix llx lly urx ury string ` - { /_i save ddef 6 1 roll 4 npop concat userdict begin /showpage {} def false setoverprint pop } def /~ % - ~ - { end _i restore } def % color operators /O % flag O - { 0 ne /_of exch ddef /_lp /none ddef } def /R % flag R - { 0 ne /_os exch ddef /_lp /none ddef } def /g % gray g - { /_gf exch ddef /_fc { _lp /fill ne { _of setoverprint _gf setgray /_lp /fill ddef } if } ddef /_pf { _fc fill } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /G % gray G - { /_gs exch ddef /_sc { _lp /stroke ne { _os setoverprint _gs setgray /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /k % cyan magenta yellow black k - { _cf astore pop /_fc { _lp /fill ne { _of setoverprint _cf aload pop setcmykcolor /_lp /fill ddef } if } ddef /_pf { _fc fill } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /K % cyan magenta yellow black K - { _cs astore pop /_sc { _lp /stroke ne { _os setoverprint _cs aload pop setcmykcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /x % cyan magenta yellow black name gray x - { /_gf exch ddef findcmykcustomcolor /_if exch ddef /_fc { _lp /fill ne { _of setoverprint _if _gf 1 exch sub setcustomcolor /_lp /fill ddef } if } ddef /_pf { _fc fill } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /X % cyan magenta yellow black name gray X - { /_gs exch ddef findcmykcustomcolor /_is exch ddef /_sc { _lp /stroke ne { _os setoverprint _is _gs 1 exch sub setcustomcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def % locked object operator /A % value A - { pop } def currentdict readonly pop end setpacking % annotate page operator /annotatepage { } def %%EndResource %AI3-Grid.0 18 18 3 0 0 0 3 %%EndProlog %%BeginSetup Adobe_cmykcolor /initialize get exec Adobe_cshow /initialize get exec Adobe_customcolor /initialize get exec Adobe_IllustratorA_AI3 /initialize get exec %%EndSetup 0 A u u u u 0 R 0 G 0 i 0 J 0 j 1 w 4 M []0 d %AI3_Note: 0 D 103.8711 412.8662 m 134.8572 390.3529 L 103.8706 367.8406 L 111.1857 390.3537 L 103.8706 412.8666 L 103.8711 412.8662 L s U u 65.5686 367.8405 m 34.5839 390.3542 L 22.7471 353.9286 L 46.4184 353.9279 L 65.5689 367.8411 L 65.5686 367.8405 L s U u 84.7177 308.9012 m 115.7044 331.4135 L 84.7182 353.9268 L 77.4031 331.4137 L 84.7181 308.9008 L 84.7177 308.9012 L s U u 84.7197 353.9272 m 115.7044 331.4135 L 127.5412 367.8391 L 103.8699 367.8398 L 84.7194 353.9266 L 84.7197 353.9272 L s U u 46.4183 353.9272 m 84.7197 353.9272 L 72.8838 390.3536 L 65.5686 367.8405 L 46.4178 353.9271 L 46.4183 353.9272 L s U u 111.1856 390.3527 m 72.8849 390.3538 L 84.7197 353.9272 L 103.8706 367.8406 L 111.1855 390.3533 L 111.1856 390.3527 L s U u 84.7197 353.9272 m 46.4189 353.9283 L 58.2537 317.5017 L 77.4046 331.4151 L 84.7196 353.9278 L 84.7197 353.9272 L s U u 103.8685 294.9884 m 65.5678 294.9895 L 77.4046 331.4151 L 84.7189 308.9021 L 103.8691 294.9883 L 103.8685 294.9884 L s U U u s U U u u u 1 w 127.5397 367.8398 m 115.7026 331.4149 L 154.0028 331.4132 L 146.6889 353.926 L 127.5391 367.8399 L 127.5397 367.8398 L s U u 103.8682 367.8407 m 134.8548 390.3523 L 146.6889 353.926 L 127.5388 367.8401 L 103.8677 367.8404 L 103.8682 367.8407 L s U u 158.5257 390.3513 m 196.8266 390.3507 L 184.9913 426.7769 L 177.676 404.2642 L 158.5251 390.3512 L 158.5257 390.3513 L s U u 134.8548 390.3523 m 146.6889 353.926 L 177.6755 376.4375 L 158.5253 390.3517 L 134.8543 390.352 L 134.8548 390.3523 L s U u 177.6759 404.2634 m 146.692 426.7772 L 134.8548 390.3523 L 158.5257 390.3513 L 177.6762 404.264 L 177.6759 404.2634 L s U U u s U U u u u 1 w 77.4019 258.5626 m 65.5666 294.989 L 103.8672 294.9884 L 84.7166 281.0748 L 77.402 258.562 L 77.4019 258.5626 L s U u 134.8538 272.4756 m 146.6877 236.0498 L 177.6739 258.561 L 158.524 272.475 L 134.8533 272.4753 L 134.8538 272.4756 L s U u 154.0052 331.4132 m 115.7046 331.4138 L 127.5398 294.9875 L 146.6905 308.9011 L 154.0051 331.4139 L 154.0052 331.4132 L s U u 127.5384 294.9881 m 115.7046 331.4138 L 84.7184 308.9027 L 103.8683 294.9887 L 127.539 294.9883 L 127.5384 294.9881 L s U u 158.5245 272.4755 m 127.5384 294.9881 L 115.7032 258.5619 L 134.8538 272.4756 L 158.525 272.4752 L 158.5245 272.4755 L s U u 84.7173 281.0756 m 115.7022 258.5624 L 127.5384 294.9881 L 103.8672 294.9884 L 84.717 281.0751 L 84.7173 281.0756 L s U u 127.5384 294.9881 m 158.5234 272.4749 L 170.3596 308.9005 L 146.6884 308.9009 L 127.5382 294.9875 L 127.5384 294.9881 L s U u 146.6896 353.9252 m 177.6745 331.4121 L 146.6884 308.9009 L 154.0037 331.4133 L 146.6892 353.9256 L 146.6896 353.9252 L s U U u s U U u u u 1 w 77.4017 213.5386 m 84.7163 191.0258 L S U u 108.388 236.0515 m 146.6877 236.0498 L 134.8538 272.4755 L 115.7032 258.5628 L 108.3881 236.0509 L 108.388 236.0515 L s U u 58.2536 272.4783 m 46.4173 236.0527 L 84.7179 236.0521 L 77.4033 258.565 L 58.2529 272.4785 L 58.2536 272.4783 L s U u 84.7169 236.051 m 46.4173 236.0527 L 58.2512 199.6269 L 77.4017 213.5397 L 84.7168 236.0516 L 84.7169 236.051 L s U u 115.7028 258.5634 m 84.7169 236.051 L 115.7027 213.5386 L 108.388 236.0515 L 115.7033 258.5638 L 115.7028 258.5634 L s U u 84.7157 191.0267 m 115.7019 213.5378 L 84.7169 236.051 L 77.4017 213.5386 L 84.7162 191.0263 L 84.7157 191.0267 L s U u 84.7169 236.051 m 115.7031 258.5622 L 84.7181 281.0753 L 77.4029 258.5629 L 84.7174 236.0506 L 84.7169 236.051 L s U u 34.5829 272.4776 m 65.569 294.9887 L 77.4029 258.5629 L 58.253 272.4769 L 34.5823 272.4772 L 34.5829 272.4776 L s U U u S U U u u u 1 w 34.5837 317.4952 m 65.5698 294.9819 L 34.5832 272.4696 L 41.8983 294.9827 L 34.5832 317.4956 L 34.5837 317.4952 L s U u 15.4303 213.5302 m 46.417 236.0425 L 15.4308 258.5558 L 8.1157 236.0427 L 15.4307 213.5298 L 15.4303 213.5302 L s U u 15.4323 258.5561 m 46.417 236.0425 L 58.2538 272.468 L 34.5825 272.4687 L 15.432 258.5556 L 15.4323 258.5561 L s U u -22.8691 258.5561 m 15.4323 258.5561 L 3.5963 294.9826 L -3.7188 272.4695 L -22.8697 258.5561 L -22.8691 258.5561 L s U u 41.8982 294.9817 m 3.5975 294.9828 L 15.4323 258.5561 L 34.5832 272.4696 L 41.8981 294.9822 L 41.8982 294.9817 L s U u 15.4323 258.5561 m -22.8685 258.5573 L -11.0337 222.1306 L 8.1172 236.0441 L 15.4322 258.5567 L 15.4323 258.5561 L s U u 34.5811 199.6174 m -3.7196 199.6185 L 8.1172 236.0441 L 15.4315 213.5311 L 34.5817 199.6173 L 34.5811 199.6174 L s U U u s U U u u u 1 w -3.7193 317.4988 m -34.7058 294.9879 L -3.7211 272.4744 L 3.5944 294.9867 L -3.7198 317.4992 L -3.7193 317.4988 L s U u -22.8695 331.4133 m 15.4312 331.4122 L 3.5944 294.9867 L -3.7199 317.4996 L -22.8701 331.4134 L -22.8695 331.4133 L s U u 34.5809 317.498 m 65.5667 294.985 L 77.4025 331.4111 L 58.2516 317.4977 L 34.5804 317.4983 L 34.5809 317.498 L s U u 15.4312 331.4122 m 3.5944 294.9867 L 41.8952 294.9856 L 34.5808 317.4986 L 15.4306 331.4123 L 15.4312 331.4122 L s U u 58.2511 317.4971 m 46.4177 353.9231 L 15.4312 331.4122 L 34.5809 317.498 L 58.2517 317.4974 L 58.2511 317.4971 L s U U u s U U u u u 1 w 3.5987 367.8332 m 15.4341 331.4065 L -22.867 331.4071 L -3.7161 345.3209 L 3.5985 367.8338 L 3.5987 367.8332 L s U U u S U U u u u 1 w 58.2529 199.6267 m 46.4195 236.0527 L 15.433 213.5419 L 34.5827 199.6276 L 58.2535 199.627 L 58.2529 199.6267 L s U u 34.5827 199.6276 m 58.2539 199.627 L 77.4044 213.5401 L 77.4042 213.5396 L S U u 34.5827 199.6276 m S U U u S U U U u u u u u 1 w 34.5812 463.2072 m 65.5678 485.7195 L 77.4026 449.2929 L 58.252 463.207 L 34.5807 463.2069 L 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603.5845 L 289.7826 603.5862 L 297.0965 581.0735 L 316.2462 567.1597 L 316.2456 567.1598 L s U U u S U U U U *u 0 O 1 g 1 w 18.2525 226.0637 m 18.2525 585.0015 L 335.4146 585.0015 L 335.4146 226.0637 L 18.2525 226.0637 L f 1 D 387.5 175.5 m 387.5 641.5 L -44.5 641.5 L -44.5 175.5 L 387.5 175.5 L f *U %%PageTrailer %%Trailer Adobe_IllustratorA_AI3 /terminate get exec Adobe_customcolor /terminate get exec Adobe_cshow /terminate get exec Adobe_cmykcolor /terminate get exec Adobe_packedarray /terminate get exec %%EOF EndOfTheIncludedPostscriptMagicCookie \closepsdump % Here's the Postscript for Fig2.ps: \message{Writing file Fig2.ps} \psdump{Fig2.ps}%!PS-Adobe-2.0 EPSF-2.0 %%Creator: gnuplot %%DocumentFonts: Courier %%BoundingBox: 50 50 554 554 %%EndComments /gnudict 40 dict def gnudict begin /Color false def /gnulinewidth 5.000 def /vshift -46 def /dl {10 mul} def /hpt 31.5 def /vpt 31.5 def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke moveto 0 vshift rmoveto show } def /Rshow { currentpoint stroke moveto dup stringwidth pop neg vshift rmoveto show } def /Cshow { currentpoint stroke moveto dup stringwidth pop -2 div vshift rmoveto show } def /DL { Color {setrgbcolor [] 0 setdash pop} {pop pop pop 0 setdash} ifelse } def /BL { stroke gnulinewidth 2 mul setlinewidth } def /AL { stroke gnulinewidth 2 div setlinewidth } def /PL { stroke gnulinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 0 1 0 DL } def /LT1 { PL [4 dl 2 dl] 0 0 1 DL } def /LT2 { PL [2 dl 3 dl] 1 0 0 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /M {moveto} def /L {lineto} def /P { stroke [] 0 setdash currentlinewidth 2 div sub moveto 0 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radin@zariski.ma.utexas.edu (Charles Radin,department,faculty,0000) %%BoundingBox: 98 251 514 541 %%Pages: 1 %%EndComments /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /l {lineto} bind def /m {moveto} bind def /s {stroke} bind def /n {newpath} bind def /gs {gsave} bind def /gr {grestore} bind def /clp {closepath} bind def /graycol {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul setrgbcolor} bind def /col-1 {} def /col0 {0 0 0 setrgbcolor} bind def /col1 {0 0 1 setrgbcolor} bind def /col2 {0 1 0 setrgbcolor} bind def /col3 {0 1 1 setrgbcolor} bind def /col4 {1 0 0 setrgbcolor} bind def /col5 {1 0 1 setrgbcolor} bind def /col6 {1 1 0 setrgbcolor} bind def /col7 {1 1 1 setrgbcolor} bind def end /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def %%EndProlog $F2psBegin 0 setlinecap 0 setlinejoin -1.0 645.0 translate 0.720 -0.720 scale 0.500 setlinewidth % Polyline n 569 229 m 610 164 l gs col0 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51 -26 V 6087 211 L -251 0 R 125 0 V -25 52 V 5836 211 L 201 104 R 5961 211 L -25 52 V 101 52 V 5961 211 M 76 104 V 25 -52 V 5961 211 L 126 0 V -25 52 V 5961 211 L 126 0 R 0 130 V -50 -26 V 50 -104 V 0 130 R -126 0 V 26 -52 V 100 52 V 5886 237 M 75 104 V 26 -52 V 5886 237 L 75 104 R 5886 237 L -25 52 V 100 52 V -125 0 V 25 -52 V 100 52 V -125 0 R 0 -130 V 50 26 V -50 104 V 251 0 R -126 0 V 26 52 V 100 -52 V 5886 445 M 75 -104 V 26 52 V -101 52 V 75 -104 R -75 104 V -25 -52 V 100 -52 V -125 0 V 25 52 V 100 -52 V -125 0 R 0 130 V 50 -26 V 5836 341 L 0 130 R 0 -130 V -51 26 V 51 104 V 5735 263 M 101 78 V -51 26 V 5735 263 L 101 78 R 5735 263 L 50 -26 V 51 104 V 0 -130 V -51 26 V 51 104 V 0 -130 R -126 0 V 25 52 V 101 -52 V 5332 730 M 75 -104 V 26 52 V -101 52 V 5533 626 M -126 0 V 26 52 V 100 -52 V -126 0 R 126 0 V -25 -51 V -101 51 V 76 -103 V 25 52 V -101 51 V 76 -103 R 101 78 V -51 25 V 5483 523 L 5634 315 M -76 104 V 56 10 V 20 -114 V -40 228 R 5558 419 L 56 10 V -20 114 V 5558 419 M 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104 R 5256 237 L -25 52 V 101 52 V -126 0 V 25 -52 V 101 52 V -126 0 R 0 -130 V 50 26 V -50 104 V 252 0 R -126 0 V 25 52 V 101 -52 V 5256 445 M 76 -104 V 25 52 V -101 52 V 76 -104 R -76 104 V -25 -52 V 101 -52 V -126 0 V 25 52 V 101 -52 V -126 0 R 0 130 V 50 -26 V 5206 341 L 0 130 R 0 -130 V -51 26 V 51 104 V 5105 263 M 101 78 V -51 26 V 5105 263 L 101 78 R 5105 263 L 50 -26 V 51 104 V 0 -130 V -51 26 V 51 104 V 0 -130 R -126 0 V 25 52 V 101 -52 V -126 0 R 0 130 V 50 -26 V 5080 211 L 101 208 R 5080 341 L 50 -26 V 51 104 V 5080 341 M 101 78 V -51 26 V 5080 341 L 0 130 V 50 -26 V 5080 341 L 0 130 R 126 0 V -25 -52 V -101 52 V 0 259 R 0 -129 V 50 25 V -50 104 V 5181 523 M -101 78 V 50 25 V currentpoint stroke M 51 -103 V -101 78 R 101 -78 V -51 -26 V -50 104 V 0 -130 V 50 26 V -50 104 V 0 -130 R 126 0 V -25 52 V 5080 471 L 126 0 R 0 130 V -51 -26 V 51 -104 V 5105 678 M 101 -77 V -51 -26 V -50 103 V 101 -77 R -101 77 V 50 26 V 51 -103 V 0 129 V -51 -26 V 51 -103 V 0 129 R -126 0 V 25 -52 V 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grestore}}ifelse B /QV{gsave newpath transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail {dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail}B /c{-4 M} B /d{-3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{3 M}B /k{ 4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w}B /q{ p 1 w}B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{3 2 roll p a}B /bos{/SS save N}B /eos{SS restore}B end %%EndProcSet %%BeginProcSet: special.pro TeXDict begin /SDict 200 dict N SDict begin /@SpecialDefaults{/hs 612 N /vs 792 N /ho 0 N /vo 0 N /hsc 1 N /vsc 1 N /ang 0 N /CLIP 0 N /rwiSeen false N /rhiSeen false N /letter{}N /note{}N /a4{}N /legal{}N}B /@scaleunit 100 N /@hscale{@scaleunit div /hsc X}B /@vscale{@scaleunit div /vsc X}B /@hsize{/hs X /CLIP 1 N}B /@vsize{/vs X /CLIP 1 N}B /@clip{ /CLIP 2 N}B /@hoffset{/ho X}B /@voffset{/vo X}B /@angle{/ang X}B /@rwi{ 10 div /rwi X 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rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop 90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr 2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4 -1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S TR}if}N /cp {pop pop showpage pm restore}N end}if}if}N /normalscale{Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale}if 0 setgray} N /psfts{S 65781.76 div N}N /startTexFig{/psf$SavedState save N userdict maxlength dict begin /magscale false def normalscale currentpoint TR /psf$ury psfts /psf$urx psfts /psf$lly psfts /psf$llx psfts /psf$y psfts /psf$x psfts currentpoint /psf$cy X /psf$cx X /psf$sx psf$x psf$urx psf$llx sub div N /psf$sy psf$y psf$ury psf$lly sub div N psf$sx psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub TR /showpage{}N /erasepage{}N /copypage{}N /p 3 def @MacSetUp}N /doclip{ psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2 roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath moveto}N /endTexFig{end psf$SavedState restore}N /@beginspecial{SDict begin /SpecialSave save N gsave normalscale currentpoint TR @SpecialDefaults count /ocount X /dcount countdictstack N}N /@setspecial {CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR }{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury lineto closepath clip}if /showpage{}N /erasepage{}N /copypage{}N newpath }N /@endspecial{count ocount sub{pop}repeat countdictstack dcount sub{ end}repeat grestore SpecialSave restore end}N /@defspecial{SDict begin} N /@fedspecial{end}B /li{lineto}B /rl{rlineto}B /rc{rcurveto}B /np{ /SaveX currentpoint /SaveY X N 1 setlinecap newpath}N /st{stroke SaveX SaveY moveto}N /fil{fill SaveX SaveY moveto}N /ellipse{/endangle X /startangle X /yrad X /xrad X /savematrix matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix}N end %%EndProcSet TeXDict begin 40258431 52099146 1200 400 400 (/tmp_mnt/d6/faculty/radin/graphics/pinwheel3.dvi) @start end %%EndProlog %%BeginSetup %%Feature: *Resolution 400dpi TeXDict begin %%EndSetup %%Page: 1 1 %1 0 bop eop %%Page: 2 2 2 1 bop 448 3204 a @beginspecial -3 @llx -4 @lly 621 @urx 1227 @ury 2436 @rwi @setspecial %%BeginDocument: pinwheel.eps %AI3_ColorUsage: Black&White %AI3_TemplateBox: 306 396 306 396 %AI3_TileBox: 30 31 582 761 %AI3_DocumentPreview: Header userdict /Adobe_packedarray 5 dict dup begin put /initialize % - initialize - { /packedarray where { pop } { Adobe_packedarray begin Adobe_packedarray { dup xcheck { bind } if userdict 3 1 roll put } forall end } ifelse } def /terminate % - terminate - { } def /packedarray % arguments count packedarray array { array astore readonly } def /setpacking % boolean setpacking - { pop } def /currentpacking % - setpacking boolean { false } def currentdict readonly pop end Adobe_packedarray /initialize get exec currentpacking true setpacking userdict /Adobe_cmykcolor 4 dict dup begin put /initialize % - initialize - { /setcmykcolor where { pop } { userdict /Adobe_cmykcolor_vars 2 dict dup begin put /_setrgbcolor /setrgbcolor load def /_currentrgbcolor /currentrgbcolor load def Adobe_cmykcolor begin Adobe_cmykcolor { dup xcheck { bind } if pop pop } forall end end Adobe_cmykcolor begin } ifelse } def /terminate % - terminate - { currentdict Adobe_cmykcolor eq { end } if } def /setcmykcolor % cyan magenta yellow black setcmykcolor - { 1 sub 4 1 roll 3 { 3 index add neg dup 0 lt { pop 0 } if 3 1 roll } repeat Adobe_cmykcolor_vars /_setrgbcolor get exec pop } def /currentcmykcolor % - currentcmykcolor cyan magenta yellow black { Adobe_cmykcolor_vars /_currentrgbcolor get exec 3 { 1 sub neg 3 1 roll } repeat 0 } def currentdict readonly pop end setpacking currentpacking true setpacking userdict /Adobe_cshow 3 dict dup begin put /initialize % - initialize - { /cshow where { pop } { userdict /Adobe_cshow_vars 1 dict dup begin put /_cshow % - _cshow proc {} def Adobe_cshow begin Adobe_cshow { dup xcheck { bind } if userdict 3 1 roll put } forall end end } ifelse } def /terminate % - terminate - { } def /cshow % proc string cshow - { exch Adobe_cshow_vars exch /_cshow exch put { 0 0 Adobe_cshow_vars /_cshow get exec } forall } def currentdict readonly pop end setpacking currentpacking true setpacking userdict /Adobe_customcolor 5 dict dup begin put /initialize % - initialize - { /setcustomcolor where { pop } { Adobe_customcolor begin Adobe_customcolor { dup xcheck { bind } if pop pop } forall end Adobe_customcolor begin } ifelse } def /terminate % - terminate - { currentdict Adobe_customcolor eq { end } if } def /findcmykcustomcolor % cyan magenta yellow black name findcmykcustomcolor object { 5 packedarray } def /setcustomcolor % object tint setcustomcolor - { exch aload pop pop 4 { 4 index mul 4 1 roll } repeat 5 -1 roll pop setcmykcolor } def /setoverprint % boolean setoverprint - { pop } def currentdict readonly pop end setpacking currentpacking true setpacking userdict /Adobe_IllustratorA_AI3 61 dict dup begin put % initialization /initialize % - initialize - { % 47 vars, but leave slack of 10 entries for custom Postscript fragments userdict /Adobe_IllustratorA_AI3_vars 57 dict dup begin put % paint operands /_lp /none def /_pf {} def /_ps {} def /_psf {} def /_pss {} def /_pjsf {} def /_pjss {} def /_pola 0 def /_doClip 0 def % paint operators /cf currentflat def % - cf flatness % typography operands /_tm matrix def /_renderStart [/e0 /r0 /a0 /o0 /e1 /r1 /a1 /i0] def /_renderEnd [null null null null /i1 /i1 /i1 /i1] def /_render -1 def /_rise 0 def /_ax 0 def % x character spacing (_ax, _ay, _cx, _cy follows awidthshow naming convention) /_ay 0 def % y character spacing /_cx 0 def % x word spacing /_cy 0 def % y word spacing /_leading [0 0] def /_ctm matrix def /_mtx matrix def /_sp 16#020 def /_hyphen (-) def /_fScl 0 def /_cnt 0 def /_hs 1 def /_nativeEncoding 0 def /_useNativeEncoding 0 def /_tempEncode 0 def /_pntr 0 def /_tDict 2 dict def % typography operators /Tx {} def /Tj {} def % compound path operators /CRender {} def % printing /_AI3_savepage {} def % color operands /_gf null def /_cf 4 array def /_if null def /_of false def /_fc {} def /_gs null def /_cs 4 array def /_is null def /_os false def /_sc {} def /_i null def Adobe_IllustratorA_AI3 begin Adobe_IllustratorA_AI3 { dup xcheck { bind } if pop pop } forall end end Adobe_IllustratorA_AI3 begin Adobe_IllustratorA_AI3_vars begin newpath } def /terminate % - terminate - { end end } def % definition operators /_ % - _ null null def /ddef % key value ddef - { Adobe_IllustratorA_AI3_vars 3 1 roll put } def /xput % key value literal xput - { dup load dup length exch maxlength eq { dup dup load dup length 2 mul dict copy def } if load begin def end } def /npop % integer npop - { { pop } repeat } def % marking operators /sw % ax ay string sw x y { dup length exch stringwidth exch 5 -1 roll 3 index 1 sub mul add 4 1 roll 3 1 roll 1 sub mul add } def /swj % cx cy fillchar ax ay string swj x y { dup 4 1 roll dup length exch stringwidth exch 5 -1 roll 3 index 1 sub mul add 4 1 roll 3 1 roll 1 sub mul add 6 2 roll /_cnt 0 ddef {1 index eq {/_cnt _cnt 1 add ddef} if} forall pop exch _cnt mul exch _cnt mul 2 index add 4 1 roll 2 index add 4 1 roll pop pop } def /ss % ax ay string matrix ss - { 4 1 roll { % matrix ax ay char 0 0 {proc} - 2 npop (0) exch 2 copy 0 exch put pop gsave false charpath currentpoint 4 index setmatrix stroke grestore moveto 2 copy rmoveto } exch cshow 3 npop } def /jss % cx cy fillchar ax ay string matrix jss - { 4 1 roll { % cx cy fillchar matrix ax ay char 0 0 {proc} - 2 npop (0) exch 2 copy 0 exch put gsave _sp eq { exch 6 index 6 index 6 index 5 -1 roll widthshow currentpoint } { false charpath currentpoint 4 index setmatrix stroke }ifelse grestore moveto 2 copy rmoveto } exch cshow 6 npop } def % path operators /sp % ax ay string sp - { { 2 npop (0) exch 2 copy 0 exch put pop false charpath 2 copy rmoveto } exch cshow 2 npop } def /jsp % cx cy fillchar ax ay string jsp - { { % cx cy fillchar ax ay char 0 0 {proc} - 2 npop (0) exch 2 copy 0 exch put _sp eq { exch 5 index 5 index 5 index 5 -1 roll widthshow } { false charpath }ifelse 2 copy rmoveto } exch cshow 5 npop } def % path construction operators /pl % x y pl x y { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add exch itransform } def /setstrokeadjust where { pop true setstrokeadjust /c % x1 y1 x2 y2 x3 y3 c - { curveto } def /C /c load def /v % x2 y2 x3 y3 v - { currentpoint 6 2 roll curveto } def /V /v load def /y % x1 y1 x2 y2 y - { 2 copy curveto } def /Y /y load def /l % x y l - { lineto } def /L /l load def /m % x y m - { moveto } def } {%else /c { pl curveto } def /C /c load def /v { currentpoint 6 2 roll pl curveto } def /V /v load def /y { pl 2 copy curveto } def /Y /y load def /l { pl lineto } def /L /l load def /m { pl moveto } def }ifelse % graphic state operators /d % array phase d - { setdash } def /cf {} def % - cf flatness /i % flatness i - { dup 0 eq { pop cf } if setflat } def /j % linejoin j - { setlinejoin } def /J % linecap J - { setlinecap } def /M % miterlimit M - { setmiterlimit } def /w % linewidth w - { setlinewidth } def % path painting operators /H % - H - {} def /h % - h - { closepath } def /N % - N - { _pola 0 eq { _doClip 1 eq {clip /_doClip 0 ddef} if newpath } { /CRender {N} ddef }ifelse } def /n % - n - {N} def /F % - F - { _pola 0 eq { _doClip 1 eq { gsave _pf grestore clip newpath /_lp /none ddef _fc /_doClip 0 ddef } { _pf }ifelse } { /CRender {F} ddef }ifelse } def /f % - f - { closepath F } def /S % - S - { _pola 0 eq { _doClip 1 eq { gsave _ps grestore clip newpath /_lp /none ddef _sc /_doClip 0 ddef } { _ps }ifelse } { /CRender {S} ddef }ifelse } def /s % - s - { closepath S } def /B % - B - { _pola 0 eq { _doClip 1 eq % F clears _doClip gsave F grestore { gsave S grestore clip newpath /_lp /none ddef _sc /_doClip 0 ddef } { S }ifelse } { /CRender {B} ddef }ifelse } def /b % - b - { closepath B } def /W % - W - { /_doClip 1 ddef } def /* % - [string] * - { count 0 ne { dup type (stringtype) eq {pop} if } if _pola 0 eq {newpath} if } def % group operators /u % - u - {} def /U % - U - {} def /q % - q - { _pola 0 eq {gsave} if } def /Q % - Q - { _pola 0 eq {grestore} if } def /*u % - *u - { _pola 1 add /_pola exch ddef } def /*U % - *U - { _pola 1 sub /_pola exch ddef _pola 0 eq {CRender} if } def /D % polarized D - {pop} def /*w % - *w - {} def /*W % - *W - {} def % place operators /` % matrix llx lly urx ury string ` - { /_i save ddef 6 1 roll 4 npop concat userdict begin /showpage {} def false setoverprint pop } def /~ % - ~ - { end _i restore } def % color operators /O % flag O - { 0 ne /_of exch ddef /_lp /none ddef } def /R % flag R - { 0 ne /_os exch ddef /_lp /none ddef } def /g % gray g - { /_gf exch ddef /_fc { _lp /fill ne { _of setoverprint _gf setgray /_lp /fill ddef } if } ddef /_pf { _fc fill } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /G % gray G - { /_gs exch ddef /_sc { _lp /stroke ne { _os setoverprint _gs setgray /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /k % cyan magenta yellow black k - { _cf astore pop /_fc { _lp /fill ne { _of setoverprint _cf aload pop setcmykcolor /_lp /fill ddef } if } ddef /_pf { _fc fill } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /K % cyan magenta yellow black K - { _cs astore pop /_sc { _lp /stroke ne { _os setoverprint _cs aload pop setcmykcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /x % cyan magenta yellow black name gray x - { /_gf exch ddef findcmykcustomcolor /_if exch ddef /_fc { _lp /fill ne { _of setoverprint _if _gf 1 exch sub setcustomcolor /_lp /fill ddef } if } ddef /_pf { _fc fill } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /X % cyan magenta yellow black name gray X - { /_gs exch ddef findcmykcustomcolor /_is exch ddef /_sc { _lp /stroke ne { _os setoverprint _is _gs 1 exch sub setcustomcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def % locked object operator /A % value A - { pop } def currentdict readonly pop end setpacking % annotate page operator /annotatepage { } def %AI3-Grid.0 18 18 3 0 0 0 3 Adobe_cmykcolor /initialize get exec Adobe_cshow /initialize get exec Adobe_customcolor /initialize get exec Adobe_IllustratorA_AI3 /initialize get exec 0 A u u u u 0 R 0 G 0 i 0 J 0 j 3 w 4 M []0 d %AI3_Note: 0 D 330.6263 275.4581 m 277.0221 285.2985 l 223.4178 295.1389 l 340.4667 329.0623 l 330.6263 275.4581 L s U u 1 w 270.2364 308.7083 m 246.8271 301.9236 l 223.4178 295.1389 l 277.0211 285.2991 l 270.2364 308.7083 L s U u 323.8396 298.8685 m 300.4303 292.0838 l 277.0211 285.2991 l 330.6243 275.4592 l 323.8396 298.8685 L s U u 323.8401 298.8678 m 300.4306 292.0834 l 277.0211 285.2991 l 317.0558 322.2773 l 323.8401 298.8678 L s U U u u 3 w 233.2582 348.7432 m 286.8625 338.9027 l 340.4667 329.0623 l 223.4178 295.1389 l 233.2582 348.7432 L s U u 1 w 293.6482 315.4929 m 317.0574 322.2776 l 340.4667 329.0623 l 286.8635 338.9022 l 293.6482 315.4929 L s U u 240.0449 325.3328 m 263.4542 332.1175 l 286.8635 338.9022 l 233.2602 348.742 l 240.0449 325.3328 L s U u 240.0444 325.3335 m 263.4539 332.1178 l 286.8635 338.9022 l 246.8288 301.9239 l 240.0444 325.3335 L s U U u u 3 w 233.2582 348.7432 m 179.654 358.5836 l 126.0498 368.424 l 223.4178 295.1389 l 233.2582 348.7432 L s U u 1 w 164.9961 339.1103 m 145.5229 353.7671 l 126.0498 368.424 l 179.6529 358.5834 l 164.9961 339.1103 L s U u 218.5992 329.2697 m 199.126 343.9265 l 179.6529 358.5834 l 233.256 348.7428 l 218.5992 329.2697 L s U u 218.5999 329.2702 m 199.1264 343.9268 l 179.6529 358.5834 l 203.9433 309.7967 l 218.5999 329.2702 L s U U u u 3 w 330.6263 275.4581 m 277.0221 285.2985 l 223.4178 295.1389 l 320.7859 221.8539 l 330.6263 275.4581 L s U u 1 w 262.3641 265.8252 m 242.8909 280.4821 l 223.4178 295.1389 l 277.0209 285.2983 l 262.3641 265.8252 L s U u 315.9672 255.9846 m 296.494 270.6415 l 277.0209 285.2983 l 330.624 275.4578 l 315.9672 255.9846 L s U u 315.9679 255.9851 m 296.4944 270.6417 l 277.0209 285.2983 l 301.3113 236.5116 l 315.9679 255.9851 L s U U u u 3 w 340.4667 329.0623 m 330.6263 275.4581 l 320.7859 221.8539 l 394.0709 319.2219 l 340.4667 329.0623 L s U u 1 w 350.0996 260.8001 m 335.4427 241.327 l 320.7859 221.8539 l 330.6265 275.4569 l 350.0996 260.8001 L s U u 359.9402 314.4032 m 345.2833 294.9301 l 330.6265 275.4569 l 340.4671 329.0601 l 359.9402 314.4032 L s U u 359.9397 314.4039 m 345.2831 294.9304 l 330.6265 275.4569 l 379.4132 299.7473 l 359.9397 314.4039 L s U U U U u u u u 3 w 189.4944 412.1878 m 243.0987 402.3474 l 296.7029 392.507 l 179.654 358.5836 l 189.4944 412.1878 L s U u 1 w 249.8844 378.9376 m 273.2936 385.7223 l 296.7029 392.507 l 243.0997 402.3468 l 249.8844 378.9376 L s U u 196.2811 388.7774 m 219.6904 395.5621 l 243.0997 402.3468 l 189.4964 412.1867 l 196.2811 388.7774 L s U u 196.2806 388.7781 m 219.6901 395.5625 l 243.0997 402.3468 l 203.065 365.3686 l 196.2806 388.7781 L s U U u u 3 w 286.8625 338.9027 m 233.2582 348.7432 l 179.654 358.5836 l 296.7029 392.507 l 286.8625 338.9027 L s U u 1 w 226.4725 372.153 m 203.0633 365.3683 l 179.654 358.5836 l 233.2573 348.7437 l 226.4725 372.153 L s U u 280.0758 362.3131 m 256.6665 355.5284 l 233.2573 348.7437 l 286.8605 338.9039 l 280.0758 362.3131 L s U u 280.0763 362.3124 m 256.6668 355.5281 l 233.2573 348.7437 l 273.292 385.7219 l 280.0763 362.3124 L s U U u u 3 w 286.8625 338.9027 m 340.4667 329.0623 l 394.0709 319.2219 l 296.7029 392.507 l 286.8625 338.9027 L s U u 1 w 355.1247 348.5356 m 374.5978 333.8787 l 394.0709 319.2219 l 340.4679 329.0625 l 355.1247 348.5356 L s U u 301.5216 358.3762 m 320.9947 343.7193 l 340.4679 329.0625 l 286.8647 338.9031 l 301.5216 358.3762 L s U u 301.5209 358.3757 m 320.9944 343.7191 l 340.4679 329.0625 l 316.1775 377.8492 l 301.5209 358.3757 L s U U u u 3 w 189.4944 412.1878 m 243.0987 402.3474 l 296.7029 392.507 l 199.3349 465.792 l 189.4944 412.1878 L s U u 1 w 257.7567 421.8207 m 277.2298 407.1638 l 296.7029 392.507 l 243.0998 402.3475 l 257.7567 421.8207 L s U u 204.1536 431.6613 m 223.6267 417.0044 l 243.0998 402.3475 l 189.4967 412.1882 l 204.1536 431.6613 L s U u 204.1528 431.6608 m 223.6263 417.0042 l 243.0998 402.3475 l 218.8094 451.1343 l 204.1528 431.6608 L s U U u u 3 w 179.654 358.5836 m 189.4944 412.1878 l 199.3349 465.792 l 126.0498 368.424 l 179.654 358.5836 L s U u 1 w 170.0212 426.8458 m 184.678 446.3189 l 199.3349 465.792 l 189.4943 412.1889 l 170.0212 426.8458 L s U u 160.1806 373.2427 m 174.8374 392.7158 l 189.4943 412.1889 l 179.6537 358.5858 l 160.1806 373.2427 L s U u 160.181 373.242 m 174.8376 392.7155 l 189.4943 412.1889 l 140.7076 387.8986 l 160.181 373.242 L s U U U U u u u u 3 w 271.9997 197.5631 m 247.7092 246.3509 l 223.4187 295.1388 l 223.2118 173.2725 l 271.9997 197.5631 L s U u 1 w 223.3356 246.3933 m 223.3771 270.766 l 223.4187 295.1388 l 247.7083 246.3517 l 223.3356 246.3933 L s U u 247.6252 197.6062 m 247.6668 221.9789 l 247.7083 246.3517 l 271.998 197.5647 l 247.6252 197.6062 L s U u 247.6261 197.6059 m 247.6672 221.9788 l 247.7083 246.3517 l 223.2532 197.647 l 247.6261 197.6059 L s U U u u 3 w 174.6308 270.8483 m 198.9213 222.0604 l 223.2118 173.2725 l 223.4187 295.1388 l 174.6308 270.8483 L s U u 1 w 223.2949 222.0181 m 223.2533 197.6453 l 223.2118 173.2725 l 198.9221 222.0596 l 223.2949 222.0181 L s U u 199.0052 270.8051 m 198.9636 246.4324 l 198.9221 222.0596 l 174.6324 270.8467 l 199.0052 270.8051 L s U u 199.0044 270.8054 m 198.9632 246.4325 l 198.9221 222.0596 l 223.3773 270.7643 l 199.0044 270.8054 L s U U u u 3 w 174.6308 270.8483 m 150.3403 319.6361 l 126.0498 368.424 l 223.4187 295.1388 l 174.6308 270.8483 L s U u 1 w 164.9968 339.1108 m 145.5233 353.7674 l 126.0498 368.424 l 150.3401 319.6373 l 164.9968 339.1108 L s U u 189.2871 290.324 m 169.8136 304.9807 l 150.3401 319.6373 l 174.6305 270.8505 l 189.2871 290.324 L s U u 189.2869 290.3232 m 169.8135 304.9802 l 150.3401 319.6373 l 203.9439 309.7965 l 189.2869 290.3232 L s U U u u 3 w 271.9997 197.5631 m 247.7092 246.3509 l 223.4187 295.1388 l 320.7875 221.8535 l 271.9997 197.5631 L s U u 1 w 262.3656 265.8255 m 242.8921 280.4821 l 223.4187 295.1388 l 247.709 246.352 l 262.3656 265.8255 L s U u 286.656 217.0388 m 267.1825 231.6954 l 247.709 246.352 l 271.9994 197.5653 l 286.656 217.0388 L s U u 286.6557 217.038 m 267.1823 231.695 l 247.709 246.352 l 301.3128 236.5113 l 286.6557 217.038 L s U U u u 3 w 223.2118 173.2725 m 271.9997 197.5631 l 320.7875 221.8535 l 247.5023 124.4847 l 223.2118 173.2725 L s U u 1 w 291.4743 182.9066 m 306.1309 202.38 l 320.7875 221.8535 l 272.0008 197.5632 l 291.4743 182.9066 L s U u 242.6875 158.6162 m 257.3441 178.0897 l 272.0008 197.5632 l 223.214 173.2729 l 242.6875 158.6162 L s U u 242.6867 158.6165 m 257.3437 178.0898 l 272.0008 197.5632 l 262.16 143.9594 l 242.6867 158.6165 L s U U U U u u u u 3 w 150.5472 441.5023 m 126.2567 490.2902 l 101.9662 539.078 l 101.7593 417.2118 l 150.5472 441.5023 L s U u 1 w 101.8831 490.3325 m 101.9246 514.7053 l 101.9662 539.078 l 126.2558 490.291 l 101.8831 490.3325 L s U u 126.1728 441.5455 m 126.2143 465.9182 l 126.2558 490.291 l 150.5455 441.5039 l 126.1728 441.5455 L s U u 126.1736 441.5452 m 126.2147 465.9181 l 126.2558 490.291 l 101.8007 441.5863 l 126.1736 441.5452 L s U U u u 3 w 53.1783 514.7876 m 77.4688 465.9997 l 101.7593 417.2118 l 101.9662 539.078 l 53.1783 514.7876 L s U u 1 w 101.8424 465.9574 m 101.8009 441.5846 l 101.7593 417.2118 l 77.4696 465.9989 l 101.8424 465.9574 L s U u 77.5527 514.7445 m 77.5111 490.3717 l 77.4696 465.9989 l 53.1799 514.786 l 77.5527 514.7445 L s U u 77.5519 514.7447 m 77.5107 490.3718 l 77.4696 465.9989 l 101.9248 514.7036 l 77.5519 514.7447 L s U U u u 3 w 53.1783 514.7876 m 28.8878 563.5754 l 4.5973 612.3633 l 101.9662 539.078 l 53.1783 514.7876 L s U u 1 w 43.5443 583.05 m 24.0708 597.7067 l 4.5973 612.3633 l 28.8877 563.5766 l 43.5443 583.05 L s U u 67.8346 534.2633 m 48.3611 548.9199 l 28.8877 563.5766 l 53.178 514.7898 l 67.8346 534.2633 L s U u 67.8344 534.2625 m 48.361 548.9195 l 28.8877 563.5766 l 82.4914 553.7358 l 67.8344 534.2625 L s U U u u 3 w 150.5472 441.5023 m 126.2567 490.2902 l 101.9662 539.078 l 199.335 465.7929 l 150.5472 441.5023 L s U u 1 w 140.9131 509.7648 m 121.4396 524.4215 l 101.9662 539.078 l 126.2565 490.2913 l 140.9131 509.7648 L s U u 165.2035 460.9781 m 145.73 475.6347 l 126.2565 490.2913 l 150.5469 441.5046 l 165.2035 460.9781 L s U u 165.2032 460.9773 m 145.7299 475.6343 l 126.2565 490.2913 l 179.8603 480.4506 l 165.2032 460.9773 L s U U u u 3 w 101.7593 417.2118 m 150.5472 441.5023 l 199.335 465.7929 l 126.0498 368.424 l 101.7593 417.2118 L s U u 1 w 170.0218 426.8459 m 184.6784 446.3194 l 199.335 465.7929 l 150.5483 441.5025 l 170.0218 426.8459 L s U u 121.235 402.5555 m 135.8917 422.029 l 150.5483 441.5025 l 101.7615 417.2121 l 121.235 402.5555 L s U u 121.2342 402.5558 m 135.8912 422.0291 l 150.5483 441.5025 l 140.7076 387.8987 l 121.2342 402.5558 L s U U U U u u u u 3 w 418.3614 270.434 m 369.5736 246.1435 l 320.7857 221.853 l 442.6519 221.6462 l 418.3614 270.434 L s U u 1 w 369.5312 221.77 m 345.1585 221.8115 l 320.7857 221.853 l 369.5728 246.1427 l 369.5312 221.77 L s U u 418.3183 246.0596 m 393.9455 246.1012 l 369.5728 246.1427 l 418.3598 270.4324 l 418.3183 246.0596 L s U u 418.3186 246.0605 m 393.9457 246.1016 l 369.5728 246.1427 l 418.2775 221.6876 l 418.3186 246.0605 L s U U u u 3 w 345.0762 173.0652 m 393.8641 197.3557 l 442.6519 221.6462 l 320.7857 221.853 l 345.0762 173.0652 L s U u 1 w 393.9064 221.7292 m 418.2792 221.6877 l 442.6519 221.6462 l 393.8649 197.3565 l 393.9064 221.7292 L s U u 345.1194 197.4396 m 369.4921 197.398 l 393.8649 197.3565 l 345.0778 173.0668 l 345.1194 197.4396 L s U u 345.1191 197.4387 m 369.492 197.3976 l 393.8649 197.3565 l 345.1602 221.8116 l 345.1191 197.4387 L s U U u u 3 w 345.0762 173.0652 m 296.2884 148.7747 l 247.5005 124.4842 l 320.7857 221.853 l 345.0762 173.0652 L s U u 1 w 276.8137 163.4311 m 262.1571 143.9577 l 247.5005 124.4842 l 296.2872 148.7745 l 276.8137 163.4311 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255.7781 l 418.3613 270.4352 l 471.965 260.5944 l 457.308 241.1211 L s U U U U u u u u 3 w 165.4108 582.8419 m 219.015 573.0015 l 272.6193 563.161 l 155.5704 529.2377 l 165.4108 582.8419 L s U u 1 w 225.8007 549.5916 m 249.21 556.3764 l 272.6193 563.161 l 219.016 573.0009 l 225.8007 549.5916 L s U u 172.1975 559.4315 m 195.6068 566.2162 l 219.016 573.0009 l 165.4128 582.8407 l 172.1975 559.4315 L s U u 172.197 559.4322 m 195.6065 566.2165 l 219.016 573.0009 l 178.9813 536.0227 l 172.197 559.4322 L s U U u u 3 w 262.7789 509.5568 m 209.1746 519.3973 l 155.5704 529.2377 l 272.6193 563.161 l 262.7789 509.5568 L s U u 1 w 202.3889 542.8071 m 178.9796 536.0224 l 155.5704 529.2377 l 209.1736 519.3978 l 202.3889 542.8071 L s U u 255.9922 532.9672 m 232.5829 526.1825 l 209.1736 519.3978 l 262.7769 509.558 l 255.9922 532.9672 L s U u 255.9927 532.9665 m 232.5832 526.1821 l 209.1736 519.3978 l 249.2083 556.376 l 255.9927 532.9665 L s U U u u 3 w 262.7789 509.5568 m 316.3831 499.7164 l 369.9873 489.876 l 272.6193 563.161 l 262.7789 509.5568 L s U u 1 w 331.041 519.1897 m 350.5142 504.5328 l 369.9873 489.876 l 316.3842 499.7166 l 331.041 519.1897 L s U u 277.438 529.0303 m 296.9111 514.3734 l 316.3842 499.7166 l 262.7811 509.5572 l 277.438 529.0303 L s U u 277.4372 529.0298 m 296.9107 514.3732 l 316.3842 499.7166 l 292.0938 548.5033 l 277.4372 529.0298 L s U U u u 3 w 165.4108 582.8419 m 219.015 573.0015 l 272.6193 563.161 l 175.2512 636.4461 l 165.4108 582.8419 L s U u 1 w 233.673 592.4748 m 253.1461 577.8179 l 272.6193 563.161 l 219.0162 573.0017 l 233.673 592.4748 L s U u 180.0699 602.3154 m 199.543 587.6585 l 219.0162 573.0017 l 165.4131 582.8422 l 180.0699 602.3154 L s U u 180.0692 602.3149 m 199.5427 587.6582 l 219.0162 573.0017 l 194.7258 621.7884 l 180.0692 602.3149 L s U U u u 3 w 155.5704 529.2377 m 165.4108 582.8419 l 175.2512 636.4461 l 101.9662 539.0781 l 155.5704 529.2377 L s U u 1 w 145.9375 597.4999 m 160.5944 616.973 l 175.2512 636.4461 l 165.4106 582.843 l 145.9375 597.4999 L s U u 136.0969 543.8968 m 150.7538 563.3699 l 165.4106 582.843 l 155.57 529.2399 l 136.0969 543.8968 L s U u 136.0974 543.8961 m 150.754 563.3695 l 165.4106 582.843 l 116.6239 558.5527 l 136.0974 543.8961 L s U U U U u u u u 3 w 306.5426 446.1122 m 252.9384 455.9526 l 199.3342 465.793 l 316.3831 499.7164 l 306.5426 446.1122 L s U u 1 w 246.1527 479.3624 m 222.7434 472.5777 l 199.3342 465.793 l 252.9374 455.9532 l 246.1527 479.3624 L s U u 299.7559 469.5226 m 276.3467 462.7379 l 252.9374 455.9532 l 306.5407 446.1133 l 299.7559 469.5226 L s U u 299.7565 469.5219 m 276.347 462.7375 l 252.9374 455.9532 l 292.9721 492.9314 l 299.7565 469.5219 L s U U u u 3 w 209.1746 519.3973 m 262.7789 509.5568 l 316.3831 499.7164 l 199.3342 465.793 l 209.1746 519.3973 L s U u 1 w 269.5645 486.147 m 292.9738 492.9317 l 316.3831 499.7164 l 262.7798 509.5563 l 269.5645 486.147 L s U u 215.9613 495.9869 m 239.3706 502.7716 l 262.7798 509.5563 l 209.1766 519.3961 l 215.9613 495.9869 L s U u 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m 306.5426 446.1122 l 296.7022 392.508 l 369.9873 489.876 l 316.3831 499.7164 L s U u 1 w 326.016 431.4542 m 311.3591 411.9811 l 296.7022 392.508 l 306.5428 446.1111 l 326.016 431.4542 L s U u 335.8565 485.0573 m 321.1997 465.5842 l 306.5428 446.1111 l 316.3834 499.7142 l 335.8565 485.0573 L s U u 335.8561 485.058 m 321.1994 465.5845 l 306.5428 446.1111 l 355.3295 470.4014 l 335.8561 485.058 L s U U U U u u u u 3 w 224.0374 660.7369 m 248.3279 611.9491 l 272.6184 563.1612 l 272.8253 685.0274 l 224.0374 660.7369 L s U u 1 w 272.7015 611.9067 m 272.66 587.534 l 272.6184 563.1612 l 248.3288 611.9483 l 272.7015 611.9067 L s U u 248.4118 660.6938 m 248.3703 636.321 l 248.3288 611.9483 l 224.0391 660.7353 l 248.4118 660.6938 L s U u 248.411 660.6941 m 248.3699 636.3212 l 248.3288 611.9483 l 272.7839 660.6529 l 248.411 660.6941 L s U U u u 3 w 321.4063 587.4517 m 297.1158 636.2396 l 272.8253 685.0274 l 272.6184 563.1612 l 321.4063 587.4517 L s U u 1 w 272.7422 636.2819 m 272.7838 660.6547 l 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247.2892 2.6191 l 223.0002 51.4068 l 247.3731 51.3649 L s U u 223.0841 100.1526 m 223.0422 75.7798 l 223.0002 51.4068 l 198.7112 100.1946 l 223.0841 100.1526 L s U u 223.0833 100.1529 m 223.0417 75.7799 l 223.0002 51.4068 l 247.4563 100.1114 l 223.0833 100.1529 L s U U U U u u u u 3 w 539.8106 26.4926 m 515.5207 75.2812 l 491.2309 124.0697 l 588.5991 50.7824 l 539.8106 26.4926 L s U u 1 w 530.1776 94.7557 m 510.7042 109.4127 l 491.2309 124.0697 l 515.5206 75.2823 l 530.1776 94.7557 L s U u 554.4673 45.9682 m 534.994 60.6253 l 515.5206 75.2823 l 539.8103 26.4949 l 554.4673 45.9682 L s U u 554.4671 45.9674 m 534.9938 60.6248 l 515.5206 75.2823 l 569.1245 65.4406 l 554.4671 45.9674 L s U U u u 3 w 540.0195 148.3595 m 564.3093 99.571 l 588.5991 50.7824 l 491.2309 124.0697 l 540.0195 148.3595 L s U u 1 w 549.6524 80.0965 m 569.1258 65.4394 l 588.5991 50.7824 l 564.3095 99.5699 l 549.6524 80.0965 L s U u 525.3628 128.8839 m 544.8361 114.2269 l 564.3095 99.5699 l 540.0198 148.3573 l 525.3628 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273.453 1050.6257 m 322.2409 1074.9161 l 371.0288 1099.2066 l 297.7436 1001.8378 l 273.453 1050.6257 L s U u 1 w 341.7156 1060.2597 m 356.3722 1079.7332 l 371.0288 1099.2066 l 322.242 1074.9163 l 341.7156 1060.2597 L s U u 292.9288 1035.9693 m 307.5854 1055.4428 l 322.242 1074.9163 l 273.4553 1050.626 l 292.9288 1035.9693 L s U u 292.928 1035.9696 m 307.585 1055.4429 l 322.242 1074.9163 l 312.4013 1021.3126 l 292.928 1035.9696 L s U U u u 3 w 200.1678 953.2568 m 248.9557 977.5473 l 297.7436 1001.8378 l 224.4583 904.4689 l 200.1678 953.2568 L s U u 1 w 268.4303 962.8908 m 283.0869 982.3643 l 297.7436 1001.8378 l 248.9568 977.5475 l 268.4303 962.8908 L s U u 219.6436 938.6005 m 234.3002 958.074 l 248.9568 977.5475 l 200.1701 953.2571 l 219.6436 938.6005 L s U u 219.6427 938.6007 m 234.2998 958.0741 l 248.9568 977.5475 l 239.1161 923.9437 l 219.6427 938.6007 L s U U u u 3 w 175.8774 1002.0447 m 200.1678 953.2568 l 224.4583 904.4689 l 127.0895 977.7542 l 175.8774 1002.0447 L s U u 1 w 185.5114 933.7822 m 204.9849 919.1256 l 224.4583 904.4689 l 200.168 953.2557 l 185.5114 933.7822 L s U u 161.221 982.5689 m 180.6945 967.9123 l 200.168 953.2557 l 175.8776 1002.0424 l 161.221 982.5689 L s U u 161.2212 982.5698 m 180.6946 967.9127 l 200.168 953.2557 l 146.5642 963.0964 l 161.2212 982.5698 L s U U U U u u u u 3 w 453.1185 640.8489 m 399.5142 650.6894 l 345.91 660.5298 l 462.9589 694.4532 l 453.1185 640.8489 L s U u 1 w 392.7286 674.0992 m 369.3193 667.3145 l 345.91 660.5298 l 399.5132 650.6899 l 392.7286 674.0992 L s U u 446.3318 664.2593 m 422.9225 657.4746 l 399.5132 650.6899 l 453.1165 640.8501 l 446.3318 664.2593 L s U u 446.3323 664.2586 m 422.9228 657.4743 l 399.5132 650.6899 l 439.5479 687.6681 l 446.3323 664.2586 L s U U u u 3 w 355.7504 714.134 m 409.3547 704.2936 l 462.9589 694.4532 l 345.91 660.5298 l 355.7504 714.134 L s U u 1 w 416.1404 680.8838 m 439.5496 687.6685 l 462.9589 694.4532 l 409.3556 704.293 l 416.1404 680.8838 L s U u 362.5371 690.7236 m 385.9464 697.5083 l 409.3556 704.293 l 355.7524 714.1329 l 362.5371 690.7236 L s U u 362.5366 690.7243 m 385.9461 697.5087 l 409.3556 704.293 l 369.3209 667.3148 l 362.5366 690.7243 L s U U u u 3 w 355.7504 714.134 m 302.1462 723.9744 l 248.542 733.8148 l 345.91 660.5298 l 355.7504 714.134 L s U u 1 w 287.4882 704.5011 m 268.0151 719.158 l 248.542 733.8148 l 302.1451 723.9743 l 287.4882 704.5011 L s U u 341.0913 694.6605 m 321.6182 709.3174 l 302.1451 723.9743 l 355.7482 714.1337 l 341.0913 694.6605 L s U u 341.0921 694.661 m 321.6186 709.3176 l 302.1451 723.9743 l 326.4354 675.1875 l 341.0921 694.661 L s U U u u 3 w 453.1185 640.8489 m 399.5142 650.6894 l 345.91 660.5298 l 443.278 587.2447 l 453.1185 640.8489 L s U u 1 w 384.8562 631.2161 m 365.3831 645.8729 l 345.91 660.5298 l 399.5131 650.6892 l 384.8562 631.2161 L s U u 438.4594 621.3755 m 418.9862 636.0323 l 399.5131 650.6892 l 453.1162 640.8486 l 438.4594 621.3755 L s U u 438.4601 621.3759 m 418.9866 636.0326 l 399.5131 650.6892 l 423.8035 601.9024 l 438.4601 621.3759 L s U U u u 3 w 462.9589 694.4532 m 453.1185 640.8489 l 443.278 587.2447 l 516.5631 684.6127 l 462.9589 694.4532 L s U u 1 w 472.5917 626.191 m 457.9349 606.7178 l 443.278 587.2447 l 453.1186 640.8478 l 472.5917 626.191 L s U u 482.4324 679.7941 m 467.7755 660.3209 l 453.1186 640.8478 l 462.9592 694.4509 l 482.4324 679.7941 L s U u 482.4319 679.7948 m 467.7753 660.3213 l 453.1186 640.8478 l 501.9054 665.1382 l 482.4319 679.7948 L s U U U U u u u u 3 w 311.9866 777.5787 m 365.5908 767.7382 l 419.1951 757.8978 l 302.1462 723.9744 l 311.9866 777.5787 L s U u 1 w 372.3766 744.3284 m 395.7858 751.1131 l 419.1951 757.8978 l 365.5918 767.7377 l 372.3766 744.3284 L s U u 318.7733 754.1683 m 342.1826 760.953 l 365.5918 767.7377 l 311.9886 777.5776 l 318.7733 754.1683 L s U u 318.7728 754.169 m 342.1823 760.9533 l 365.5918 767.7377 l 325.5571 730.7594 l 318.7728 754.169 L s U U u u 3 w 409.3547 704.2936 m 355.7504 714.134 l 302.1462 723.9744 l 419.1951 757.8978 l 409.3547 704.2936 L s U u 1 w 348.9647 737.5438 m 325.5555 730.7591 l 302.1462 723.9744 l 355.7494 714.1346 l 348.9647 737.5438 L s U u 402.568 727.704 m 379.1587 720.9193 l 355.7494 714.1346 l 409.3527 704.2947 l 402.568 727.704 L s U u 402.5685 727.7033 m 379.1589 720.9189 l 355.7494 714.1346 l 395.7841 751.1128 l 402.5685 727.7033 L s U U u u 3 w 409.3547 704.2936 m 462.9589 694.4532 l 516.5631 684.6127 l 419.1951 757.8978 l 409.3547 704.2936 L s U u 1 w 477.6168 713.9265 m 497.09 699.2696 l 516.5631 684.6127 l 462.96 694.4533 l 477.6168 713.9265 L s U u 424.0138 723.767 m 443.4869 709.1102 l 462.96 694.4533 l 409.3569 704.2939 l 424.0138 723.767 L s U u 424.013 723.7666 m 443.4865 709.11 l 462.96 694.4533 l 438.6696 743.2401 l 424.013 723.7666 L s U U u u 3 w 311.9866 777.5787 m 365.5908 767.7382 l 419.1951 757.8978 l 321.827 831.1829 l 311.9866 777.5787 L s U u 1 w 380.2488 787.2115 m 399.722 772.5547 l 419.1951 757.8978 l 365.592 767.7384 l 380.2488 787.2115 L s U u 326.6457 797.0521 m 346.1188 782.3953 l 365.592 767.7384 l 311.9889 777.579 l 326.6457 797.0521 L s U u 326.645 797.0516 m 346.1185 782.395 l 365.592 767.7384 l 341.3016 816.5251 l 326.645 797.0516 L s U U u u 3 w 302.1462 723.9744 m 311.9866 777.5787 l 321.827 831.1829 l 248.542 733.8148 l 302.1462 723.9744 L s U u 1 w 292.5133 792.2366 m 307.1702 811.7098 l 321.827 831.1829 l 311.9865 777.5798 l 292.5133 792.2366 L s U u 282.6727 738.6335 m 297.3296 758.1067 l 311.9865 777.5798 l 302.1459 723.9767 l 282.6727 738.6335 L s U u 282.6732 738.6328 m 297.3298 758.1063 l 311.9865 777.5798 l 263.1997 753.2894 l 282.6732 738.6328 L s U U U U u u u u 3 w 394.4918 562.9539 m 370.2014 611.7418 l 345.9108 660.5296 l 345.7039 538.6634 l 394.4918 562.9539 L s U u 1 w 345.8278 611.7841 m 345.8693 636.1569 l 345.9108 660.5296 l 370.2005 611.7425 l 345.8278 611.7841 L s U u 370.1174 562.997 m 370.159 587.3698 l 370.2005 611.7425 l 394.4902 562.9555 l 370.1174 562.997 L s U u 370.1183 562.9967 m 370.1594 587.3697 l 370.2005 611.7425 l 345.7454 563.0379 l 370.1183 562.9967 L s U U u u 3 w 297.123 636.2391 m 321.4135 587.4512 l 345.7039 538.6634 l 345.9108 660.5296 l 297.123 636.2391 L s U u 1 w 345.787 587.409 m 345.7455 563.0362 l 345.7039 538.6634 l 321.4143 587.4505 l 345.787 587.409 L s U u 321.4974 636.196 m 321.4558 611.8232 l 321.4143 587.4505 l 297.1246 636.2375 l 321.4974 636.196 L s U u 321.4965 636.1963 m 321.4554 611.8234 l 321.4143 587.4505 l 345.8694 636.1551 l 321.4965 636.1963 L s U U u u 3 w 297.123 636.2391 m 272.8325 685.027 l 248.542 733.8148 l 345.9108 660.5296 l 297.123 636.2391 L s U u 1 w 287.4889 704.5016 m 268.0154 719.1582 l 248.542 733.8148 l 272.8323 685.0281 l 287.4889 704.5016 L s U u 311.7793 655.7149 m 292.3058 670.3715 l 272.8323 685.0281 l 297.1227 636.2414 l 311.7793 655.7149 L s U u 311.779 655.714 m 292.3057 670.3711 l 272.8323 685.0281 l 326.4361 675.1874 l 311.779 655.714 L s U U u u 3 w 394.4918 562.9539 m 370.2014 611.7418 l 345.9108 660.5296 l 443.2797 587.2444 l 394.4918 562.9539 L s U u 1 w 384.8578 631.2164 m 365.3843 645.873 l 345.9108 660.5296 l 370.2012 611.7429 l 384.8578 631.2164 L s U u 409.1481 582.4297 m 389.6746 597.0863 l 370.2012 611.7429 l 394.4915 562.9562 l 409.1481 582.4297 L s U u 409.1479 582.4288 m 389.6745 597.0858 l 370.2012 611.7429 l 423.8049 601.9022 l 409.1479 582.4288 L s U U u u 3 w 345.7039 538.6634 m 394.4918 562.9539 l 443.2797 587.2444 l 369.9944 489.8756 l 345.7039 538.6634 L s U u 1 w 413.9664 548.2974 m 428.6231 567.7709 l 443.2797 587.2444 l 394.493 562.9541 l 413.9664 548.2974 L s U u 365.1797 524.0071 m 379.8363 543.4806 l 394.493 562.9541 l 345.7062 538.6637 l 365.1797 524.0071 L s U u 365.1789 524.0073 m 379.8359 543.4807 l 394.493 562.9541 l 384.6522 509.3503 l 365.1789 524.0073 L s U U U U u u u u 3 w 273.0393 806.8932 m 248.7488 855.6811 l 224.4583 904.4689 l 224.2515 782.6027 l 273.0393 806.8932 L s U u 1 w 224.3753 855.7234 m 224.4168 880.0962 l 224.4583 904.4689 l 248.748 855.6819 l 224.3753 855.7234 L s U u 248.6649 806.9363 m 248.7065 831.3091 l 248.748 855.6819 l 273.0377 806.8948 l 248.6649 806.9363 L s U u 248.6658 806.936 m 248.7069 831.309 l 248.748 855.6819 l 224.2929 806.9772 l 248.6658 806.936 L s U U u u 3 w 175.6705 880.1784 m 199.961 831.3906 l 224.2515 782.6027 l 224.4583 904.4689 l 175.6705 880.1784 L s U u 1 w 224.3346 831.3482 m 224.293 806.9755 l 224.2515 782.6027 l 199.9618 831.3898 l 224.3346 831.3482 L s U u 200.0448 880.1353 m 200.0033 855.7625 l 199.9618 831.3898 l 175.6721 880.1768 l 200.0448 880.1353 L s U u 200.0441 880.1356 m 200.0029 855.7627 l 199.9618 831.3898 l 224.417 880.0944 l 200.0441 880.1356 L s U U u u 3 w 175.6705 880.1784 m 151.38 928.9663 l 127.0895 977.7542 l 224.4583 904.4689 l 175.6705 880.1784 L s U u 1 w 166.0364 948.4409 m 146.563 963.0975 l 127.0895 977.7542 l 151.3798 928.9674 l 166.0364 948.4409 L s U u 190.3268 899.6542 m 170.8533 914.3108 l 151.3798 928.9674 l 175.6702 880.1807 l 190.3268 899.6542 L s U u 190.3266 899.6533 m 170.8532 914.3104 l 151.3798 928.9674 l 204.9836 919.1267 l 190.3266 899.6533 L s U U u u 3 w 273.0393 806.8932 m 248.7488 855.6811 l 224.4583 904.4689 l 321.8272 831.1837 l 273.0393 806.8932 L s U u 1 w 263.4053 875.1557 m 243.9318 889.8123 l 224.4583 904.4689 l 248.7487 855.6822 l 263.4053 875.1557 L s U u 287.6956 826.3689 m 268.2222 841.0256 l 248.7487 855.6822 l 273.039 806.8955 l 287.6956 826.3689 L s U u 287.6954 826.3681 m 268.2221 841.0252 l 248.7487 855.6822 l 302.3524 845.8415 l 287.6954 826.3681 L s U U u u 3 w 224.2515 782.6027 m 273.0393 806.8932 l 321.8272 831.1837 l 248.542 733.8148 l 224.2515 782.6027 L s U u 1 w 292.514 792.2367 m 307.1706 811.7102 l 321.8272 831.1837 l 273.0404 806.8934 l 292.514 792.2367 L s U u 243.7272 767.9464 m 258.3839 787.4199 l 273.0404 806.8934 l 224.2537 782.603 l 243.7272 767.9464 L s U u 243.7264 767.9466 m 258.3834 787.42 l 273.0404 806.8934 l 263.1997 753.2896 l 243.7264 767.9466 L s U U U U u u u u 3 w 540.8536 635.8249 m 492.0657 611.5344 l 443.2779 587.2439 l 565.1441 587.037 l 540.8536 635.8249 L s U u 1 w 492.0234 587.1608 m 467.6506 587.2023 l 443.2779 587.2439 l 492.0649 611.5336 l 492.0234 587.1608 L s U u 540.8105 611.4505 m 516.4377 611.492 l 492.0649 611.5336 l 540.852 635.8232 l 540.8105 611.4505 L s U u 540.8108 611.4513 m 516.4379 611.4924 l 492.0649 611.5336 l 540.7696 587.0784 l 540.8108 611.4513 L s U U u u 3 w 467.5684 538.456 m 516.3562 562.7465 l 565.1441 587.037 l 443.2779 587.2439 l 467.5684 538.456 L s U u 1 w 516.3986 587.1201 m 540.7713 587.0786 l 565.1441 587.037 l 516.357 562.7473 l 516.3986 587.1201 L s U u 467.6115 562.8304 m 491.9843 562.7889 l 516.357 562.7473 l 467.57 538.4577 l 467.6115 562.8304 L s U u 467.6112 562.8296 m 491.9841 562.7884 l 516.357 562.7473 l 467.6524 587.2025 l 467.6112 562.8296 L s U U u u 3 w 467.5684 538.456 m 418.7805 514.1655 l 369.9927 489.875 l 443.2779 587.2439 l 467.5684 538.456 L s U u 1 w 399.3059 528.822 m 384.6493 509.3485 l 369.9927 489.875 l 418.7794 514.1654 l 399.3059 528.822 L s U u 448.0927 553.1123 m 433.436 533.6389 l 418.7794 514.1654 l 467.5661 538.4557 l 448.0927 553.1123 L s U u 448.0935 553.1121 m 433.4365 533.6388 l 418.7794 514.1654 l 428.6201 567.7691 l 448.0935 553.1121 L s U U u u 3 w 540.8536 635.8249 m 492.0657 611.5344 l 443.2779 587.2439 l 516.5631 684.6127 l 540.8536 635.8249 L s U u 1 w 472.5911 626.1909 m 457.9345 606.7174 l 443.2779 587.2439 l 492.0646 611.5342 l 472.5911 626.1909 L s U u 521.3779 650.4812 m 506.7212 631.0077 l 492.0646 611.5342 l 540.8514 635.8246 l 521.3779 650.4812 L s U u 521.3787 650.481 m 506.7217 631.0076 l 492.0646 611.5342 l 501.9054 665.138 l 521.3787 650.481 L s U U u u 3 w 565.1441 587.037 m 540.8536 635.8249 l 516.5631 684.6127 l 613.932 611.3275 l 565.1441 587.037 L s U u 1 w 555.5101 655.2995 m 536.0366 669.9561 l 516.5631 684.6127 l 540.8535 635.826 l 555.5101 655.2995 L s U u 579.8004 606.5128 m 560.327 621.1694 l 540.8535 635.826 l 565.1438 587.0393 l 579.8004 606.5128 L s U u 579.8002 606.5119 m 560.3268 621.169 l 540.8535 635.826 l 594.4572 625.9853 l 579.8002 606.5119 L s U U U U u u u u 3 w 540.5616 465.2124 m 516.105 416.5077 l 491.6483 367.8029 l 589.2664 440.7558 l 540.5616 465.2124 L s U u 1 w 530.695 396.9832 m 511.1717 382.3931 l 491.6483 367.8029 l 516.1048 416.5066 l 530.695 396.9832 L s U u 555.1515 445.6869 m 535.6282 431.0967 l 516.1048 416.5066 l 540.5613 465.2102 l 555.1515 445.6869 L s U u 555.1512 445.6877 m 535.628 431.0971 l 516.1048 416.5066 l 569.7418 426.1645 l 555.1512 445.6877 L s U U u u 3 w 540.3531 343.3462 m 564.8097 392.051 l 589.2664 440.7558 l 491.6483 367.8029 l 540.3531 343.3462 L s U u 1 w 550.2198 411.5755 m 569.7431 426.1657 l 589.2664 440.7558 l 564.8099 392.0522 l 550.2198 411.5755 L s U u 525.7632 362.8718 m 545.2866 377.462 l 564.8099 392.0522 l 540.3534 343.3485 l 525.7632 362.8718 L s U u 525.7635 362.871 m 545.2867 377.4616 l 564.8099 392.0522 l 511.1729 382.3942 l 525.7635 362.871 L s U U u u 3 w 540.3531 343.3462 m 515.8965 294.6415 l 491.4398 245.9367 l 491.6483 367.8029 l 540.3531 343.3462 L s U u 1 w 491.5229 294.6822 m 491.4813 270.3094 l 491.4398 245.9367 l 515.8956 294.6407 l 491.5229 294.6822 L s U u 515.9787 343.3862 m 515.9372 319.0134 l 515.8956 294.6407 l 540.3515 343.3446 l 515.9787 343.3862 L s U u 515.9795 343.3865 m 515.9376 319.0136 l 515.8956 294.6407 l 491.6066 343.4284 l 515.9795 343.3865 L s U U u u 3 w 540.5616 465.2124 m 516.105 416.5077 l 491.6483 367.8029 l 491.8569 489.6691 l 540.5616 465.2124 L s U u 1 w 491.7314 416.5484 m 491.6899 392.1757 l 491.6483 367.8029 l 516.1042 416.5069 l 491.7314 416.5484 L s U u 516.1873 465.2524 m 516.1457 440.8796 l 516.1042 416.5069 l 540.56 465.2108 l 516.1873 465.2524 L s U u 516.1881 465.2527 m 516.1461 440.8798 l 516.1042 416.5069 l 491.8152 465.2946 l 516.1881 465.2527 L s U U u u 3 w 589.2664 440.7558 m 540.5616 465.2124 l 491.8569 489.6691 l 613.723 489.4606 l 589.2664 440.7558 L s U u 1 w 540.6024 489.586 m 516.2296 489.6276 l 491.8569 489.6691 l 540.5608 465.2133 l 540.6024 489.586 L s U u 589.3064 465.1302 m 564.9336 465.1717 l 540.5608 465.2133 l 589.2648 440.7574 l 589.3064 465.1302 L s U u 589.3067 465.1294 m 564.9337 465.1713 l 540.5608 465.2133 l 589.3486 489.5023 l 589.3067 465.1294 L s U U U U u u u u 3 w 564.6012 270.1848 m 589.0579 318.8896 l 613.5145 367.5944 l 515.8965 294.6415 l 564.6012 270.1848 L s U u 1 w 574.4679 338.414 m 593.9912 353.0042 l 613.5145 367.5944 l 589.058 318.8907 l 574.4679 338.414 L s U u 550.0114 289.7104 m 569.5347 304.3006 l 589.058 318.8907 l 564.6016 270.1871 l 550.0114 289.7104 L s U u 550.0116 289.7096 m 569.5348 304.3001 l 589.058 318.8907 l 535.421 309.2328 l 550.0116 289.7096 L s U U u u 3 w 564.8097 392.051 m 540.3531 343.3462 l 515.8965 294.6415 l 613.5145 367.5944 l 564.8097 392.051 L s U u 1 w 554.9431 323.8218 m 535.4198 309.2316 l 515.8965 294.6415 l 540.353 343.3451 l 554.9431 323.8218 L s U u 579.3996 372.5254 m 559.8763 357.9353 l 540.353 343.3451 l 564.8095 392.0488 l 579.3996 372.5254 L s U u 579.3994 372.5263 m 559.8762 357.9357 l 540.353 343.3451 l 593.9899 353.0031 l 579.3994 372.5263 L s U U u u 3 w 564.8097 392.051 m 589.2664 440.7558 l 613.723 489.4606 l 613.5145 367.5944 l 564.8097 392.051 L s U u 1 w 613.64 440.715 m 613.6815 465.0878 l 613.723 489.4606 l 589.2672 440.7566 l 613.64 440.715 L s U u 589.1841 392.011 m 589.2257 416.3838 l 589.2672 440.7566 l 564.8114 392.0526 l 589.1841 392.011 L s U u 589.1833 392.0108 m 589.2253 416.3837 l 589.2672 440.7566 l 613.5562 391.9688 l 589.1833 392.0108 L s U U u u 3 w 564.6012 270.1848 m 589.0579 318.8896 l 613.5145 367.5944 l 613.306 245.7282 l 564.6012 270.1848 L s U u 1 w 613.4315 318.8488 m 613.473 343.2216 l 613.5145 367.5944 l 589.0587 318.8904 l 613.4315 318.8488 L s U u 588.9756 270.1449 m 589.0172 294.5176 l 589.0587 318.8904 l 564.6029 270.1864 l 588.9756 270.1449 L s U u 588.9748 270.1446 m 589.0167 294.5175 l 589.0587 318.8904 l 613.3477 270.1026 l 588.9748 270.1446 L s U U u u 3 w 515.8965 294.6415 m 564.6012 270.1848 l 613.306 245.7282 l 491.4398 245.9367 l 515.8965 294.6415 L s U u 1 w 564.5605 245.8112 m 588.9332 245.7697 l 613.306 245.7282 l 564.602 270.184 l 564.5605 245.8112 L s U u 515.8565 270.2671 m 540.2293 270.2255 l 564.602 270.184 l 515.898 294.6398 l 515.8565 270.2671 L s U u 515.8562 270.2679 m 540.2292 270.226 l 564.602 270.184 l 515.8143 245.895 l 515.8562 270.2679 L s U U U U u u u u 3 w 443.069 465.3808 m 467.3589 416.5922 l 491.6487 367.8036 l 394.2805 441.0909 l 443.069 465.3808 L s U u 1 w 452.702 397.1177 m 472.1753 382.4607 l 491.6487 367.8036 l 467.359 416.5911 l 452.702 397.1177 L s U u 428.4123 445.9051 m 447.8856 431.2481 l 467.359 416.5911 l 443.0693 465.3785 l 428.4123 445.9051 L s U u 428.4125 445.906 m 447.8858 431.2485 l 467.359 416.5911 l 413.7551 426.4327 l 428.4125 445.906 L s U U u u 3 w 442.8601 343.5138 m 418.5703 392.3024 l 394.2805 441.0909 l 491.6487 367.8036 l 442.8601 343.5138 L s U u 1 w 433.2272 411.7769 m 413.7538 426.4339 l 394.2805 441.0909 l 418.5702 392.3035 l 433.2272 411.7769 L s U u 457.5169 362.9894 m 438.0435 377.6465 l 418.5702 392.3035 l 442.8598 343.5161 l 457.5169 362.9894 L s U u 457.5166 362.9886 m 438.0434 377.646 l 418.5702 392.3035 l 472.1741 382.4618 l 457.5166 362.9886 L s U U u u 3 w 442.8601 343.5138 m 467.15 294.7252 l 491.4398 245.9367 l 491.6487 367.8036 l 442.8601 343.5138 L s U u 1 w 491.5237 294.6825 m 491.4817 270.3096 l 491.4398 245.9367 l 467.1508 294.7244 l 491.5237 294.6825 L s U u 467.2347 343.4702 m 467.1927 319.0973 l 467.1508 294.7244 l 442.8618 343.5122 l 467.2347 343.4702 L s U u 467.2339 343.4706 m 467.1923 319.0975 l 467.1508 294.7244 l 491.6069 343.429 l 467.2339 343.4706 L s U U u u 3 w 443.069 465.3808 m 467.3589 416.5922 l 491.6487 367.8036 l 491.8576 489.6706 l 443.069 465.3808 L s U u 1 w 491.7326 416.5494 m 491.6906 392.1765 l 491.6487 367.8036 l 467.3597 416.5914 l 491.7326 416.5494 L s U u 467.4436 465.3372 m 467.4016 440.9643 l 467.3597 416.5914 l 443.0707 465.3791 l 467.4436 465.3372 L s U u 467.4428 465.3375 m 467.4012 440.9645 l 467.3597 416.5914 l 491.8158 465.296 l 467.4428 465.3375 L s U U u u 3 w 394.2805 441.0909 m 443.069 465.3808 l 491.8576 489.6706 l 369.9906 489.8795 l 394.2805 441.0909 L s U u 1 w 443.1118 489.7545 m 467.4847 489.7126 l 491.8576 489.6706 l 443.0698 465.3816 l 443.1118 489.7545 L s U u 394.324 465.4654 m 418.6969 465.4235 l 443.0698 465.3816 l 394.2821 441.0926 l 394.324 465.4654 L s U u 394.3237 465.4646 m 418.6967 465.4231 l 443.0698 465.3816 l 394.3653 489.8377 l 394.3237 465.4646 L s U U U U u u u u 3 w 564.5182 221.438 m 588.8081 172.6494 l 613.0978 123.8608 l 515.7296 197.1481 l 564.5182 221.438 L s U u 1 w 574.1511 153.1749 m 593.6245 138.5178 l 613.0978 123.8608 l 588.8081 172.6483 l 574.1511 153.1749 L s U u 549.8614 201.9623 m 569.3348 187.3053 l 588.8081 172.6483 l 564.5185 221.4357 l 549.8614 201.9623 L 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@llx 446 @lly 384 @urx 622 @ury 4500 @rwi @setspecial %%BeginDocument: dots4.ps %AI3_ColorUsage: Black&White %AI3_TemplateBox: 306 396 306 396 %AI3_TileBox: 30 31 582 761 %AI3_DocumentPreview: Header userdict /Adobe_packedarray 5 dict dup begin put /initialize % - initialize - { /packedarray where { pop } { Adobe_packedarray begin Adobe_packedarray { dup xcheck { bind } if userdict 3 1 roll put } forall end } ifelse } def /terminate % - terminate - { } def /packedarray % arguments count packedarray array { array astore readonly } def /setpacking % boolean setpacking - { pop } def /currentpacking % - setpacking boolean { false } def currentdict readonly pop end Adobe_packedarray /initialize get exec currentpacking true setpacking userdict /Adobe_cmykcolor 4 dict dup begin put /initialize % - initialize - { /setcmykcolor where { pop } { userdict /Adobe_cmykcolor_vars 2 dict dup begin put /_setrgbcolor /setrgbcolor load def /_currentrgbcolor /currentrgbcolor load def Adobe_cmykcolor begin Adobe_cmykcolor { dup xcheck { bind } if pop pop } forall end end Adobe_cmykcolor begin } ifelse } def /terminate % - terminate - { currentdict Adobe_cmykcolor eq { end } if } def /setcmykcolor % cyan magenta yellow black setcmykcolor - { 1 sub 4 1 roll 3 { 3 index add neg dup 0 lt { pop 0 } if 3 1 roll } repeat Adobe_cmykcolor_vars /_setrgbcolor get exec pop } def /currentcmykcolor % - currentcmykcolor cyan magenta yellow black { Adobe_cmykcolor_vars /_currentrgbcolor get exec 3 { 1 sub neg 3 1 roll } repeat 0 } def currentdict readonly pop end setpacking currentpacking true setpacking userdict /Adobe_cshow 3 dict dup begin put /initialize % - initialize - { /cshow where { pop } { userdict /Adobe_cshow_vars 1 dict dup begin put /_cshow % - _cshow proc {} def Adobe_cshow begin Adobe_cshow { dup xcheck { bind } if userdict 3 1 roll put } forall end end } ifelse } def /terminate % - terminate - { } def /cshow % proc string cshow - { exch Adobe_cshow_vars exch /_cshow exch put { 0 0 Adobe_cshow_vars /_cshow get exec } forall } def currentdict readonly pop end setpacking currentpacking true setpacking userdict /Adobe_customcolor 5 dict dup begin put /initialize % - initialize - { /setcustomcolor where { pop } { Adobe_customcolor begin Adobe_customcolor { dup xcheck { bind } if pop pop } forall end Adobe_customcolor begin } ifelse } def /terminate % - terminate - { currentdict Adobe_customcolor eq { end } if } def /findcmykcustomcolor % cyan magenta yellow black name findcmykcustomcolor object { 5 packedarray } def /setcustomcolor % object tint setcustomcolor - { exch aload pop pop 4 { 4 index mul 4 1 roll } repeat 5 -1 roll pop setcmykcolor } def /setoverprint % boolean setoverprint - { pop } def currentdict readonly pop end setpacking currentpacking true setpacking userdict /Adobe_IllustratorA_AI3 61 dict dup begin put % initialization /initialize % - initialize - { % 47 vars, but leave slack of 10 entries for custom Postscript fragments userdict /Adobe_IllustratorA_AI3_vars 57 dict dup begin put % paint operands /_lp /none def /_pf {} def /_ps {} def /_psf {} def /_pss {} def /_pjsf {} def /_pjss {} def /_pola 0 def /_doClip 0 def % paint operators /cf currentflat def % - cf flatness % typography operands /_tm matrix def /_renderStart [/e0 /r0 /a0 /o0 /e1 /r1 /a1 /i0] def /_renderEnd [null null null null /i1 /i1 /i1 /i1] def /_render -1 def /_rise 0 def /_ax 0 def % x character spacing (_ax, _ay, _cx, _cy follows awidthshow naming convention) /_ay 0 def % y character spacing /_cx 0 def % x word spacing /_cy 0 def % y word spacing /_leading [0 0] def /_ctm matrix def /_mtx matrix def /_sp 16#020 def /_hyphen (-) def /_fScl 0 def /_cnt 0 def /_hs 1 def /_nativeEncoding 0 def /_useNativeEncoding 0 def /_tempEncode 0 def /_pntr 0 def /_tDict 2 dict def % typography operators /Tx {} def /Tj {} def % compound path operators /CRender {} def % printing /_AI3_savepage {} def % color operands /_gf null def /_cf 4 array def /_if null def /_of false def /_fc {} def /_gs null def /_cs 4 array def /_is null def /_os false def /_sc {} def /_i null def Adobe_IllustratorA_AI3 begin Adobe_IllustratorA_AI3 { dup xcheck { bind } if pop pop } forall end end Adobe_IllustratorA_AI3 begin Adobe_IllustratorA_AI3_vars begin newpath } def /terminate % - terminate - { end end } def % definition operators /_ % - _ null null def /ddef % key value ddef - { Adobe_IllustratorA_AI3_vars 3 1 roll put } def /xput % key value literal xput - { dup load dup length exch maxlength eq { dup dup load dup length 2 mul dict copy def } if load begin def end } def /npop % integer npop - { { pop } repeat } def % marking operators /sw % ax ay string sw x y { dup length exch stringwidth exch 5 -1 roll 3 index 1 sub mul add 4 1 roll 3 1 roll 1 sub mul add } def /swj % cx cy fillchar ax ay string swj x y { dup 4 1 roll dup length exch stringwidth exch 5 -1 roll 3 index 1 sub mul add 4 1 roll 3 1 roll 1 sub mul add 6 2 roll /_cnt 0 ddef {1 index eq {/_cnt _cnt 1 add ddef} if} forall pop exch _cnt mul exch _cnt mul 2 index add 4 1 roll 2 index add 4 1 roll pop pop } def /ss % ax ay string matrix ss - { 4 1 roll { % matrix ax ay char 0 0 {proc} - 2 npop (0) exch 2 copy 0 exch put pop gsave false charpath currentpoint 4 index setmatrix stroke grestore moveto 2 copy rmoveto } exch cshow 3 npop } def /jss % cx cy fillchar ax ay string matrix jss - { 4 1 roll { % cx cy fillchar matrix ax ay char 0 0 {proc} - 2 npop (0) exch 2 copy 0 exch put gsave _sp eq { exch 6 index 6 index 6 index 5 -1 roll widthshow currentpoint } { false charpath currentpoint 4 index setmatrix stroke }ifelse grestore moveto 2 copy rmoveto } exch cshow 6 npop } def % path operators /sp % ax ay string sp - { { 2 npop (0) exch 2 copy 0 exch put pop false charpath 2 copy rmoveto } exch cshow 2 npop } def /jsp % cx cy fillchar ax ay string jsp - { { % cx cy fillchar ax ay char 0 0 {proc} - 2 npop (0) exch 2 copy 0 exch put _sp eq { exch 5 index 5 index 5 index 5 -1 roll widthshow } { false charpath }ifelse 2 copy rmoveto } exch cshow 5 npop } def % path construction operators /pl % x y pl x y { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add exch itransform } def /setstrokeadjust where { pop true setstrokeadjust /c % x1 y1 x2 y2 x3 y3 c - { curveto } def /C /c load def /v % x2 y2 x3 y3 v - { currentpoint 6 2 roll curveto } def /V /v load def /y % x1 y1 x2 y2 y - { 2 copy curveto } def /Y /y load def /l % x y l - { lineto } def /L /l load def /m % x y m - { moveto } def } {%else /c { pl curveto } def /C /c load def /v { currentpoint 6 2 roll pl curveto } def /V /v load def /y { pl 2 copy curveto } def /Y /y load def /l { pl lineto } def /L /l load def /m { pl moveto } def }ifelse % graphic state operators /d % array phase d - { setdash } def /cf {} def % - cf flatness /i % flatness i - { dup 0 eq { pop cf } if setflat } def /j % linejoin j - { setlinejoin } def /J % linecap J - { setlinecap } def /M % miterlimit M - { setmiterlimit } def /w % linewidth w - { setlinewidth } def % path painting operators /H % - H - {} def /h % - h - { closepath } def /N % - N - { _pola 0 eq { _doClip 1 eq {clip /_doClip 0 ddef} if newpath } { /CRender {N} ddef }ifelse } def /n % - n - {N} def /F % - F - { _pola 0 eq { _doClip 1 eq { gsave _pf grestore clip newpath /_lp /none ddef _fc /_doClip 0 ddef } { _pf }ifelse } { /CRender {F} ddef }ifelse } def /f % - f - { closepath F } def /S % - S - { _pola 0 eq { _doClip 1 eq { gsave _ps grestore clip newpath /_lp /none ddef _sc /_doClip 0 ddef } { _ps }ifelse } { /CRender {S} ddef }ifelse } def /s % - s - { closepath S } def /B % - B - { _pola 0 eq { _doClip 1 eq % F clears _doClip gsave F grestore { gsave S grestore clip newpath /_lp /none ddef _sc /_doClip 0 ddef } { S }ifelse } { /CRender {B} ddef }ifelse } def /b % - b - { closepath B } def /W % - W - { /_doClip 1 ddef } def /* % - [string] * - { count 0 ne { dup type (stringtype) eq {pop} if } if _pola 0 eq {newpath} if } def % group operators /u % - u - {} def /U % - U - {} def /q % - q - { _pola 0 eq {gsave} if } def /Q % - Q - { _pola 0 eq {grestore} if } def /*u % - *u - { _pola 1 add /_pola exch ddef } def /*U % - *U - { _pola 1 sub /_pola exch ddef _pola 0 eq {CRender} if } def /D % polarized D - {pop} def /*w % - *w - {} def /*W % - *W - {} def % place operators /` % matrix llx lly urx ury string ` - { /_i save ddef 6 1 roll 4 npop concat userdict begin /showpage {} def false setoverprint pop } def /~ % - ~ - { end _i restore } def % color operators /O % flag O - { 0 ne /_of exch ddef /_lp /none ddef } def /R % flag R - { 0 ne /_os exch ddef /_lp /none ddef } def /g % gray g - { /_gf exch ddef /_fc { _lp /fill ne { _of setoverprint _gf setgray /_lp /fill ddef } if } ddef /_pf { _fc fill } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /G % gray G - { /_gs exch ddef /_sc { _lp /stroke ne { _os setoverprint _gs setgray /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /k % cyan magenta yellow black k - { _cf astore pop /_fc { _lp /fill ne { _of setoverprint _cf aload pop setcmykcolor /_lp /fill ddef } if } ddef /_pf { _fc fill } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /K % cyan magenta yellow black K - { _cs astore pop /_sc { _lp /stroke ne { _os setoverprint _cs aload pop setcmykcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /x % cyan magenta yellow black name gray x - { /_gf exch ddef findcmykcustomcolor /_if exch ddef /_fc { _lp /fill ne { _of setoverprint _if _gf 1 exch sub setcustomcolor /_lp /fill ddef } if } ddef /_pf { _fc fill } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /X % cyan magenta yellow black name gray X - { /_gs exch ddef findcmykcustomcolor /_is exch ddef /_sc { _lp /stroke ne { _os setoverprint _is _gs 1 exch sub setcustomcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def % locked object operator /A % value A - { pop } def currentdict readonly pop end setpacking % annotate page operator /annotatepage { } def %AI3-Grid.0 18 18 3 0 0 0 3 Adobe_cmykcolor /initialize get exec Adobe_cshow /initialize get exec Adobe_customcolor /initialize get exec Adobe_IllustratorA_AI3 /initialize get exec 0 A u u 0 R 0 G 0 i 0 J 0 j 1 w 4 M []0 d %AI3_Note: 0 D %DVIPSCommandLine: dvips -B -o fault2.ps fault2 %DVIPSParameters: dpi=400, comments removed %DVIPSSource: TeX output 1994.06.23:1108 /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{ /nn 8 dict N nn begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N string /base X array /BitMaps X /BuildChar{CharBuilder}N /Encoding IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /df{ /sf 1 N /fntrx FMat N df-tail}B /dfs{div /sf X /fntrx[sf 0 0 sf neg 0 0] N df-tail}B /E{pop nn dup definefont setfont}B /ch-width{ch-data dup length 5 sub get}B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{ 128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N /rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 sub]{ch-image}imagemask restore}B /D{/cc X dup type /stringtype ne{]} if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 2 index S get sf div put}if put /ctr ctr 1 add N}B /I{ cc 1 add D}B /bop{userdict /bop-hook known{bop-hook}if /SI save N @rigin userdict /offset-hook known { offset-hook } if 0 0 moveto /V matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore showpage userdict /eop-hook known{eop-hook}if}N /@start{userdict /start-hook known{start-hook}if pop /VResolution X /Resolution X 1000 div /DVImag X /IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize X 65781.76 div /hsize X}N /p{show}N /RMat[1 0 0 -1 0 0]N /BDot 260 string N /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V {}B /RV statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(Display)eq exch 0 4 getinterval(NeXT)eq or}{pop false} ifelse}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale rulex ruley false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave newpath transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail {dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail}B /c{-4 M} B /d{-3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{3 M}B /k{ 4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w}B /q{ p 1 w}B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{3 2 roll p a}B /bos{/SS save N}B /eos{SS restore}B end TeXDict begin /SDict 200 dict N SDict begin /@SpecialDefaults{/hs 612 N /vs 792 N /ho 0 N /vo 0 N /hsc 1 N /vsc 1 N /ang 0 N /CLIP 0 N /rwiSeen false N /rhiSeen false N /letter{}N /note{}N /a4{}N /legal{}N}B /@scaleunit 100 N /@hscale{@scaleunit div /hsc X}B /@vscale{@scaleunit div /vsc X}B /@hsize{/hs X /CLIP 1 N}B /@vsize{/vs X /CLIP 1 N}B /@clip{ /CLIP 2 N}B /@hoffset{/ho X}B /@voffset{/vo X}B /@angle{/ang X}B /@rwi{ 10 div /rwi X /rwiSeen true N}B /@rhi{10 div /rhi X /rhiSeen true N}B /@llx{/llx X}B /@lly{/lly X}B /@urx{/urx X}B /@ury{/ury X}B /magscale true def end /@MacSetUp{userdict /md known{userdict /md get type /dicttype eq{userdict begin md length 10 add md maxlength ge{/md md dup length 20 add dict copy def}if end md begin /letter{}N /note{}N /legal{} N /od{txpose 1 0 mtx defaultmatrix dtransform S atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{itransform lineto} }{6 -2 roll transform 6 -2 roll transform 6 -2 roll transform{ itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll curveto}}{{ closepath}}pathforall newpath counttomark array astore /gc xdf pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack}if}N /txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1 -1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop 90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr 2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4 -1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S TR}if}N /cp {pop pop showpage pm restore}N end}if}if}N /normalscale{Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale}if 0 setgray} N /psfts{S 65781.76 div N}N /startTexFig{/psf$SavedState save N userdict maxlength dict begin /magscale false def normalscale currentpoint TR /psf$ury psfts /psf$urx psfts /psf$lly psfts /psf$llx psfts /psf$y psfts /psf$x psfts currentpoint /psf$cy X /psf$cx X /psf$sx psf$x psf$urx psf$llx sub div N /psf$sy psf$y psf$ury psf$lly sub div N psf$sx psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub TR /showpage{}N /erasepage{}N /copypage{}N /p 3 def @MacSetUp}N /doclip{ psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2 roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath moveto}N /endTexFig{end psf$SavedState restore}N /@beginspecial{SDict begin /SpecialSave save N gsave normalscale currentpoint TR @SpecialDefaults count /ocount X /dcount countdictstack N}N /@setspecial {CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR }{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury lineto closepath clip}if /showpage{}N /erasepage{}N /copypage{}N newpath }N /@endspecial{count ocount sub{pop}repeat countdictstack dcount sub{ end}repeat grestore SpecialSave restore end}N /@defspecial{SDict begin} N /@fedspecial{end}B /li{lineto}B /rl{rlineto}B /rc{rcurveto}B /np{ /SaveX currentpoint /SaveY X N 1 setlinecap newpath}N /st{stroke SaveX SaveY moveto}N /fil{fill SaveX SaveY moveto}N /ellipse{/endangle X /startangle X /yrad X /xrad X /savematrix matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix}N end TeXDict begin 40258431 52099146 1200 400 400 (/tmp_mnt/d6/faculty/radin/graphics/fault2.dvi) @start end TeXDict begin 2 0 bop 332 2933 a @beginspecial 0 @llx 0 @lly 460 @urx 727 @ury 2784 @rwi @setspecial /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /l {lineto} bind def /m {moveto} bind def /s {stroke} bind def /n {newpath} bind def /gs {gsave} bind def /gr {grestore} bind def /clp {closepath} bind def /graycol {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul setrgbcolor} bind def /col-1 {} def /col0 {0 0 0 setrgbcolor} bind def /col1 {0 0 1 setrgbcolor} bind def /col2 {0 1 0 setrgbcolor} bind def /col3 {0 1 1 setrgbcolor} bind def /col4 {1 0 0 setrgbcolor} bind def /col5 {1 0 1 setrgbcolor} bind def /col6 {1 1 0 setrgbcolor} bind def /col7 {1 1 1 setrgbcolor} bind def /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y translate xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix } def end /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def $F2psBegin 0 setlinecap 0 setlinejoin -47.0 759.0 translate 0.720 -0.720 scale 0.500 setlinewidth %%223.9613 572.9138 m %%142.253 572.9162 L %%167.5005 495.2061 L %%208.3558 524.8881 L %%223.961 572.9151 L %%223.9613 572.9138 L %%s %%U %%u 130.000 572.0000 m 130.000 572.0000 L 381.4209 572.0000 L s U u % Ellipse %%n 140.000 572.9162 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr %%gs col-1 s gr n 130.000 572.0000 4 4 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr n 143.000 572.0000 2 2 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr n 166.000 572.0000 2 2 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr n 381.4209 572.000 4 4 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr n 370.0000 572.0000 2 2 0 360 DrawEllipse gs 0.00 setgray fill gr gs col-1 s gr %%n 315.3148 447.176 1 1 0 360 DrawEllipse gs 0.00 setgray fill gr %%gs col-1 s gr %%n 274.4607 572.9162 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr %%gs col-1 s gr %%n 381.4209 650.6225 1 1 0 360 DrawEllipse gs 0.00 setgray fill gr %%gs col-1 s gr %intermediate points %%n 155.47377 572.91583 5 5 0 360 DrawEllipse gs 0.00 setgray fill gr %%gs col-1 s gr $F2psEnd @endspecial eop end userdict /end-hook known{end-hook}if %%EndDocument @endspecial eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF %%zz4 showpage %%Trailer EndOfTheIncludedPostscriptMagicCookie \closepsdump % Finally, here is text12.tex: \magnification 1200 \global\let\figures\relax % % ??/epsf.tex (written by Radical Eye Software and copied below) % defines the macro \epsfbox with one argument, % the encapsulated PostScript file to include. % Invoking it causes a \vbox with the natural size of the drawing % to be inserted at the point of invocation. % Usually figures are meant to be centered and set off, and possibly % to have a title and/or a figure number. The macros below do that. % You should assign values that please you to the variables % \abovefigskip, \belowfigskip, figtitleskip, figtitlefont,... % AFTER \inputting the present file: \input /u2/kbi/tex/epsfig % and BEFORE the first invocation of any of the macros below. % RESET AT YOUR PLEASURE THE VARIABLES AT THE VERY BOTTOM! % % For convenience we make a dimension for figures: \newdimen\FigSize \FigSize=.9\hsize % alter at your convenience % % For a SCALED HORIZONTALLY CENTERED FIGURE use \epsfig. % First argument is the horizontal width of the figure, given % in any way TeX can understand. % The second argument is an encapsulated PostScript file name (filnam.eps). % Note the mandatory semicolons between arguments in this example!: % \epsfig .8\hsize; example.ps; % will put a centered scaled \vbox of width .8\hsize suitably offset % at the point of invocation \newskip\abovefigskip \newskip\belowfigskip \gdef\epsfig#1;#2;{\par\vskip\abovefigskip\penalty -500 {\everypar={}\epsfxsize=#1\noindent \centerline{\epsfbox{#2}}}% \vskip\belowfigskip}% % % SCALED TITLED EPSFIG HORIZONTALLY CENTERED: \tepsfig. % First argument is the horizontal width of the figure, % second an encapsulated PostScript file name, % third a title for the figure. % Note the mandatory semicolons between arguments! % example: \tepsfig5truein; example.ps;{This is a figure} \newskip\figtitleskip \gdef\tepsfig#1;#2;#3{\par\vskip\abovefigskip\penalty -500 {\everypar={}\epsfxsize=#1\noindent \vbox {\centerline{\epsfbox{#2}}\vskip\figtitleskip \centerline{\figtitlefont#3}}}% \vskip\belowfigskip}% % % SCALED NUMBERED TITLED EPSFIG HORIZONTALLY CENTERED: \nepsfig % The figure number is automatically increased for every % invocation of \nepsfig or \nipsfig. \newcount\FigNr \global\FigNr=0 \gdef\nepsfig#1;#2;#3{\global\advance\FigNr by 1 \tepsfig#1;#2;{Figure\space\the\FigNr.\space#3}}% % % % Often you would rather have TeX decide where to put the figure % by using \midinsert. Here are macros that do that % (mnemonics ``ipsfig'' is for ``midInsert PS FIGure'') % % TeX-PLACED SCALED EPSFIG HORIZONTALLY CENTERED: \ipsfig \gdef\ipsfig#1;#2;{%\goodbreak ?? \midinsert{\everypar={}\epsfxsize=#1\noindent \centerline{\epsfbox{#2}}}% \endinsert}% % % TeX-PLACED SCALED TITLED EPSFIG HORIZONTALLY CENTERED: \tipsfig \gdef\tipsfig#1;#2;#3{\midinsert {\everypar={}\epsfxsize=#1\noindent \vbox{\centerline{\epsfbox{#2}}% \vskip\figtitleskip \centerline{\figtitlefont#3}}}\endinsert}% % % TeX-PLACED SCALED NUMBERED TITLED EPSFIG HORIZONTALLY CENTERED: \nipsfig % example: \nipsfigd.9\hsize;example.ps;{This is an example figure} \gdef\nipsfig#1;#2;#3{\global\advance\FigNr by1% \tipsfig#1;#2;{Figure\space\the\FigNr.\space#3}}% % % ================================================================ % old: % \ifx\Input\undefined\let\next\input\else\let\next\Input\fi % \next /usr/local/lib/tex/inputs/epsf.tex %On Suns %% \next /usr/lib/tex/inputs/epsf.tex %On NexTs % % to make sure no new version of epsf.tex corrupts the above, % I copy here that file in toto: % % EPSF.TEX macro file: % Written by Tomas Rokicki of Radical Eye Software, 29 Mar 1989. % Revised by Don Knuth, 3 Jan 1990. % Revised by Tomas Rokicki to accept bounding boxes with no % space after the colon, 18 Jul 1990. % % TeX macros to include an Encapsulated PostScript graphic. % Works by finding the bounding box comment, % calculating the correct scale values, and inserting a vbox % of the appropriate size at the current position in the TeX document. % % To use with the center environment of LaTeX, preface the \epsffile % call with a \leavevmode. (LaTeX should probably supply this itself % for the center environment.) % % To use, simply say % \input epsf % somewhere early on in your TeX file % \epsfbox{filename.ps} % where you want to insert a vbox for a figure % % Alternatively, you can type % % \epsfbox[0 0 30 50]{filename.ps} % to supply your own BB % % which will not read in the file, and will instead use the bounding % box you specify. % % The effect will be to typeset the figure as a TeX box, at the % point of your \epsfbox command. By default, the graphic will have its % `natural' width (namely the width of its bounding box, as described % in filename.ps). The TeX box will have depth zero. % % You can enlarge or reduce the figure by saying % \epsfxsize= \epsfbox{filename.ps} % (or % \epsfysize= \epsfbox{filename.ps}) % instead. Then the width of the TeX box will be \epsfxsize and its % height will be scaled proportionately (or the height will be % \epsfysize and its width will be scaled proportiontally). The % width (and height) is restored to zero after each use. % % A more general facility for sizing is available by defining the % \epsfsize macro. Normally you can redefine this macro % to do almost anything. The first parameter is the natural x size of % the PostScript graphic, the second parameter is the natural y size % of the PostScript graphic. It must return the xsize to use, or 0 if % natural scaling is to be used. Common uses include: % % \epsfxsize % just leave the old value alone % 0pt % use the natural sizes % #1 % use the natural sizes % \hsize % scale to full width % 0.5#1 % scale to 50% of natural size % \ifnum#1>\hsize\hsize\else#1\fi % smaller of natural, hsize % % If you want TeX to report the size of the figure (as a message % on your terminal when it processes each figure), say `\epsfverbosetrue'. % \newread\epsffilein % file to \read \newif\ifepsffileok % continue looking for the bounding box? \newif\ifepsfbbfound % success? \newif\ifepsfverbose % report what you're making? \newdimen\epsfxsize % horizontal size after scaling \newdimen\epsfysize % vertical size after scaling \newdimen\epsftsize % horizontal size before scaling \newdimen\epsfrsize % vertical size before scaling \newdimen\epsftmp % register for arithmetic manipulation \newdimen\pspoints % conversion factor % \pspoints=1bp % Adobe points are `big' \epsfxsize=0pt % Default value, means `use natural size' \epsfysize=0pt % ditto % \def\epsfbox#1{\global\def\epsfllx{72}\global\def\epsflly{72}% \global\def\epsfurx{540}\global\def\epsfury{720}% \def\lbracket{[}\def\testit{#1}\ifx\testit\lbracket \let\next=\epsfgetlitbb\else\let\next=\epsfnormal\fi\next{#1}}% % \def\epsfgetlitbb#1#2 #3 #4 #5]#6{\epsfgrab #2 #3 #4 #5 .\\% \epsfsetgraph{#6}}% % \def\epsfnormal#1{\epsfgetbb{#1}\epsfsetgraph{#1}}% % \def\epsfgetbb#1{% % % The first thing we need to do is to open the % PostScript file, if possible. % \openin\epsffilein=#1 \ifeof\epsffilein\errmessage{I couldn't open #1, will ignore it}\else % % Okay, we got it. Now we'll scan lines until we find one that doesn't % start with %. We're looking for the bounding box comment. % {\epsffileoktrue \chardef\other=12 \def\do##1{\catcode`##1=\other}\dospecials \catcode`\ =10 \loop \read\epsffilein to \epsffileline \ifeof\epsffilein\epsffileokfalse\else % % We check to see if the first character is a % sign; % if not, we stop reading (unless the line was entirely blank); % if so, we look further and stop only if the line begins with % `%%BoundingBox:'. % \expandafter\epsfaux\epsffileline:. \\% \fi \ifepsffileok\repeat \ifepsfbbfound\else \ifepsfverbose\message{No bounding box comment in #1; using defaults}\fi\fi }\closein\epsffilein\fi}% % % Now we have to calculate the scale and offset values to use. % First we compute the natural sizes. % \def\epsfsetgraph#1{% \epsfrsize=\epsfury\pspoints \advance\epsfrsize by-\epsflly\pspoints \epsftsize=\epsfurx\pspoints \advance\epsftsize by-\epsfllx\pspoints % % If `epsfxsize' is 0, we default to the natural size of the picture. % Otherwise we scale the graph to be \epsfxsize wide. % \epsfxsize\epsfsize\epsftsize\epsfrsize \ifnum\epsfxsize=0 \ifnum\epsfysize=0 \epsfxsize=\epsftsize \epsfysize=\epsfrsize % % We have a sticky problem here: TeX doesn't do floating point arithmetic! % Our goal is to compute y = rx/t. The following loop does this reasonably % fast, with an error of at most about 16 sp (about 1/4000 pt). % \else\epsftmp=\epsftsize \divide\epsftmp\epsfrsize \epsfxsize=\epsfysize \multiply\epsfxsize\epsftmp \multiply\epsftmp\epsfrsize \advance\epsftsize-\epsftmp \epsftmp=\epsfysize \loop \advance\epsftsize\epsftsize \divide\epsftmp 2 \ifnum\epsftmp>0 \ifnum\epsftsize<\epsfrsize\else \advance\epsftsize-\epsfrsize \advance\epsfxsize\epsftmp \fi \repeat \fi \else\epsftmp=\epsfrsize \divide\epsftmp\epsftsize \epsfysize=\epsfxsize \multiply\epsfysize\epsftmp \multiply\epsftmp\epsftsize \advance\epsfrsize-\epsftmp \epsftmp=\epsfxsize \loop \advance\epsfrsize\epsfrsize \divide\epsftmp 2 \ifnum\epsftmp>0 \ifnum\epsfrsize<\epsftsize\else \advance\epsfrsize-\epsftsize \advance\epsfysize\epsftmp \fi \repeat \fi % % Finally, we make the vbox and stick in a \special that dvips can parse. % \ifepsfverbose\message{#1: width=\the\epsfxsize, height=\the\epsfysize}\fi \epsftmp=10\epsfxsize \divide\epsftmp\pspoints \vbox to\epsfysize{\vfil\hbox to\epsfxsize{% \special{PSfile=#1 llx=\epsfllx\space lly=\epsflly\space urx=\epsfurx\space ury=\epsfury\space rwi=\number\epsftmp}% \hfil}}% \epsfxsize=0pt\epsfysize=0pt}% % % We still need to define the tricky \epsfaux macro. This requires % a couple of magic constants for comparison purposes. % {\catcode`\%=12 \global\let\epsfpercent=%\global\def\epsfbblit{%BoundingBox}}% % % So we're ready to check for `%BoundingBox:' and to grab the % values if they are found. % \long\def\epsfaux#1#2:#3\\{\ifx#1\epsfpercent \def\testit{#2}\ifx\testit\epsfbblit \epsfgrab #3 . . . \\% \epsffileokfalse \global\epsfbbfoundtrue \fi\else\ifx#1\par\else\epsffileokfalse\fi\fi}% % % Here we grab the values and stuff them in the appropriate definitions. % \def\epsfgrab #1 #2 #3 #4 #5\\{% \global\def\epsfllx{#1}\ifx\epsfllx\empty \epsfgrab #2 #3 #4 #5 .\\\else \global\def\epsflly{#2}% \global\def\epsfurx{#3}\global\def\epsfury{#4}\fi}% % % We default the epsfsize macro. % \def\epsfsize#1#2{\epsfxsize}% % % Finally, another definition for compatibility with older macros. % \let\epsffile=\epsfbox % ================================================================ % execution: why not set \epsfverbosetrue % reset at your pleasure \abovefigskip=\baselineskip % reset at your pleasure \belowfigskip=\baselineskip % reset at your pleasure \global\let\figtitlefont\bf % reset at your pleasure \global\figtitleskip=.5\baselineskip % reset at your pleasure \let\nd\noindent % NOINDENT \def\NL{\hfill\break}% NEWLINE \def\qed{\hbox{\hskip 6pt\vrule width6pt height7pt depth1pt \hskip1pt}} \def\natural{{\rm I\kern-.18em N}} \def\N{{\bf N}} \def\integer{{\rm Z\kern-.32em Z}} \def\chix{{\raise.5ex\hbox{$\chi$}}} \def\Z{{\bf Z}} \def\ep{\epsilon} \def\real{{\rm I\kern-.2em R}} \def\R{{\bf R}} \def\complex{\kern.1em{\raise.47ex\hbox{ $\scriptscriptstyle |$}}\kern-.40em{\rm C}} \def\C{{\bf C}} \def\vs#1 {\vskip#1truein} \def\hs#1 {\hskip#1truein} \hsize=6truein \hoffset=.25truein %was \hoffset=1.2truein \vsize=8.8truein %\voffset=1truein \pageno=1 \baselineskip=12pt \parskip=6 pt \parindent=20pt \overfullrule=0pt \lineskip=0pt \lineskiplimit=0pt \hbadness=10000 \vbadness=10000 % REPORT ONLY BEYOND THIS BADNESS %\nopagenumbers \pageno=0 \font\bit=cmbxti10 \footline{\ifnum\pageno=0\hss\else\hss\tenrm\folio\hss\fi} \hbox{} \vskip 1truein\centerline{{\bf THE ISOPERIMETRIC PROBLEM FOR PINWHEEL TILINGS}} \vskip .5truein\centerline{by} \centerline{Charles Radin${}^1$ and Lorenzo Sadun${}^2$} \footnote{}{1\ Research supported in part by NSF Grant No. DMS-9304269 and \vs-.1 \hs.15 Texas ARP Grant 003658-113\hfil} %\footnote{}{} \footnote{}{2\ Research supported in part by an NSF Mathematical Sciences Postdoctoral \vs-.1 \hs.15 Fellowship and Texas ARP Grant 003658-037 \hfil} \vskip .2truein\centerline{Mathematics Department} \centerline{University of Texas} \centerline{Austin, TX\ \ 78712} \vs.1 \centerline{radin@math.utexas.edu and sadun@math.utexas.edu} \vs.5 \centerline{{\bf Abstract}} \vs.1 \nd In aperiodic ``pinwheel'' tilings of the plane there exist unions of tiles with ratio (area)/(perimeter)${}^2$ arbitrarily close to that of a circle. Such approximate circles can be constructed with arbitrary center and any sufficiently large radius. The existence of such circles follows from the metric on pinwheel space being almost Euclidean at large distances; if $P$ and $Q$ are points separated by large Euclidean distance $R$, then the shortest path along tile edges from $P$ to $Q$ has length $R + o(R)$. \vs.8 \centerline{January 1994} \vs.2 %\centerline{Subject Classification:\ \ 52B60, 52C20, 05B45, 82B05} \vfill\eject \nd {\bf I. Introduction and Statement of Results} The classic isoperimetric problem in the plane, which asks for the curve of least length enclosing some fixed area, has stimulated much important mathematics. One generalization which has developed within geometric measure theory treats spaces less symmetric than the Euclidean plane, such as spaces representing the structure of crystals. Due to the periodic arrangement of their atoms such structures are, on a macroscopic scale, invariant under translations but not rotations. This has easily observed consequences for crystals; for quartz or table salt one can literally see an optimal polyhedral shape, a shape which solves a version of the isoperimetric problem that can be described as follows. There is a ``cost function'' $f( \vec n)$ associated with variable normal directions $\vec n$ of planes in space. (Physically, $f( \vec n)$ is the energy per unit area needed to separate a crystal into two parts along a plane with normal $\vec n$). The problem is to imagine integrating $f$ over each possible surface enclosing a region of fixed volume $V$, and then to find the surface which minimizes this integral. In 1901 Wulff gave a simple construction for such optimal surfaces (see [Wul], [Tay]) which is still used to produce the polyhedral ``Wulff shapes'' for common crystals. In this paper we generalize this isopermetric problem to geometries associated with quasicrystals. Quasicrystals are an exotic class of solids, usually metallic alloys, with an internal structure more complicated than that of ordinary periodic crystals [S-O]. A common technique for modeling such materials utilizes aperiodic tilings of space by polyhedra; the best known examples are three dimensional versions of Penrose's polygonal tilings of the plane [Gar]; Fig.\ 1. For tilings of the plane there is a natural isoperimetric problem yielding the Wulff shape, as follows: Given a tiling of the plane, find that collection of tiles which covers a fixed area $A$ with boundary of minimal length. In order to obtain the necessary perspective of macroscopic shape compared to atomic substructure, it is appropriate to seek an optimal shape asymptotic in $A\to \infty$. There are no real difficulties determining the Wulff shapes for some such structures. For instance, it is not hard to see that the Wulff shape for Penrose tilings is a regular decagon. However, there are natural analogs of the Penrose tilings, exhibiting more unusual symmetries, for which the Wulff shape is not so obvious. In this paper we consider a certain ``pinwheel'' tiling of the Euclidean plane made by congruent copies of a $1, 2, \sqrt 5$ right triangle (see Fig.\ 2), and determine the shape of that region of fixed area which can be covered by a portion of the tiling with least perimeter. (See [Rad2], [Rad3] for discussions of pinwheel tilings and [B-R], [Rad1], [Rad5] for the relation to possible quasicrystals.) We solve the isoperimetric problem asymptotically for large area, and prove that the optimal limiting shape is a smooth circle, in that there are regions for which $\hbox{area}/\hbox{(perimeter)}^2$ is arbitrarily close to $(4\pi)^{-1}$. In fact we prove that large approximate circles are ubiquitous: \nd {\bf Theorem 1.} Given $\ep >0$ there is a distance $R$ such that, for any disk $C$ with radius $r>R$ there is a region $D$, whose boundary follows the edges of triangles in the tiling, that approximates $C$ in the sense that $$|\hbox{(perimeter of $D$)}-2\pi r| < \epsilon r, \qquad \hbox{(area of $C\backslash D$)} + \hbox{(area of $D\backslash C$)} < \epsilon r^2. \eqno(1)$$ In particular, (area of $D$)/(perimeter of $D)^2 > (4\pi)^{-1} - \ep$. This follows from an approximate metric on the space. For large separations, the length of the shortest path between two points $P,Q$ is approximately the Euclidean distance $\|P-Q\|$. More precisely, \nd {\bf Theorem 2.} Given any $\epsilon > 0$ there exists $R$ such that, for any two points $P,Q$ on the boundaries of triangles with $\|P-Q\|>R$, there is a path $h$ along the boundaries of these triangles, connecting $P$ to $Q$, with length $|h|$ satisfying $${|h|-\|P-Q\|\over \|P-Q\|}<\epsilon. \eqno(2)$$ \vs.2 \nd {\bf II. The Details} The pinwheel tiling is constructed as follows. We start with a $1, 2, \sqrt 5$ right triangle, divide it into five similar triangles as in Fig.\ 3, and then expand the collection of five triangles by the factor $\sqrt 5$. We now have five triangles congruent to the original. Move and rotate this collection so that the middle triangle coincides with the triangle we started with. Now subdivide each of the five triangles to produce a collection of twenty-five triangles and again rescale by $\sqrt{5}$. Move and rotate this collection so the middle five triangles, highlighted in Fig.\ 3, coincide with the five triangles from the previous stage. Repeat this process of subdivision, rescaling and movement {\it ad infinitum}, with the central fifth of each new pattern always coinciding with the entire previous pattern. The end result is the pinwheel tiling. As a first step towards Theorem 2, we prove a similar result about subdividing triangles without expanding or moving. Take the basic $1,2,\sqrt{5}$ right triangle, which we denote $T$, subdivide it into five similar triangles, subdivide each of those triangles, and so on. (See Fig.\ 4 for the result of 5 subdivisions.) Let $X$ be the space of pairs of points on the boundary of $T$, with the product topology. For any $p=(P,Q) \in X$, let $f(p,n)$ be the minimum distance along a path from $P$ to $Q$ along edges of triangles in the $n$-th subdivision. \nd {\bf Theorem 3.} For any $p = (P,Q) \in X$, $\lim_{n \to \infty} f(p,n)=\|P-Q\|$. \vs.1 \nd Proof. When $P=Q$ the result is trivial, so we henceforth only consider pairs of distinct points. Let $g(p,n) = f(p,n) / \|p\|$, where $\|p\|$ is the Euclidean distance between the points in $p$. Let $F(p)=\inf_{n\ge 0} f(p,n)$ and let $G(p)=\inf_{n\ge 0} g(p,n)= F(p)/\|p\|$. $F$ is continuous on $X$, so $G$ is continuous where defined. We first show that $G$ attains its supremum on the compact subset $X'=\{p\in X\,|\,\|p\|\ge 0.01\}$. We then show that this supremum is 1. Since $G \ge 1$, this shows that $G$ is identically 1, which is tantamount to Theorem 3. The points in $X\backslash X'$ are of two types. Either both points lie on the same edge, in which case $G(p)=g(p,0)=1$, or $P$ and $Q$ are close to, but on opposite sides of, one of the three vertices. The theorem clearly holds for the first type, so we consider the second type. Let $p_n$ be the pair of points $5^n$ times farther from the given vertex than $P$ and $Q$. For pairs near the acute angles, subdividing twice shows that $g(p,n)=g(p_1,n-2)$, and so $G(p)=G(p_1)=G(p_2)$, {\it etc}. Eventually one of the pairs $p_n$ will lie in $X'$, so $G(p) \le \sup_{p'\in X'}G(p')$. Finally, subdividing once shows that, for pairs near the right angle, $G(p)$ is bounded by a weighted average of $G(p')$ for pairs $p'$ near the two acute angles, and therefore by $\sup_{p'\in X'}G(p')$. Thus $\sup_{p\in X}G(p)=\sup_{p\in X'}G(p)$. Since $X'$ is compact, $\sup_{p\in X'}G(p)=\max_{p\in X'}G(p)$. We will prove this equals 1 by contradiction. So assume there is a $\tilde p\in X'$ such that $\max_{p\in X'}G(p)=G(\tilde p)=1+\delta$ with $\delta >0$. If $\tilde P,\tilde Q$ are the points in $\tilde p$, let $\tilde H$ be the straight line joining $\tilde P$ to $\tilde Q$. Perform $m$ subdivisions on the original triangle $T$ so that $T$ contains a triangle $T'$ intersected by $\tilde H$, with this intersection in the middle third of the line. (Further requirements on $m$ will appear later.) Fix $n$ such that for each line determined by a $p\in X$ there is contained in the triangle of level $n$ some triangle of level 0 for which the small edge makes an angle $\theta$ with respect to the line, with $1/\cos(\theta)< 1+\delta/2$ (this is possible from [Rad4]). Now subdivide the triangle $T$ an additional $n$ times, obtaining a triangle $T''$ within $T'$ for which the short edge makes an angle of $\theta$ with respect to $\tilde H$, with $1/\cos(\theta)< 1+\delta/2$. Let $L_1$ be the straight line joining $\tilde P$ to $t_1$, the vertex on the short edge of $T''$ closer to $\tilde P$, let $L_2$ be that short edge of $T''$, and let $L_3$ be the straight line joining $t_2$, the vertex on the short edge of $T''$ closer to $\tilde Q$, to $\tilde Q$. Let $\ell_1$ (resp. $\ell_2$, resp. $\ell_3$) be the projection of $L_1$ (resp. $L_2$, resp. $L_3$) onto $\tilde H$. We denote the length of a line $L$ by $|L|$; in particular, for $L_2$, which is a short edge obtained by subdividing $m+n$ times the triangle $T$, we have $|L_2|= 5^{-(m+n)/2}$. Since $T'$ was obtained by subdividing $m$ times, it follows that $$ \eqalign{|L_1| &\le (|\ell_1|^2+ 5^{1-m})^{1/2}\cr &= |\ell_1|(1+{5^{1-m}\over |\ell_1|^2})^{1/2}\cr &< |\ell_1|+{5^{1-m}\over |\ell_1|}.\cr}\eqno(3)$$ \nd So $$|L_1|-|\ell_1| < {5^{1-m}\over |\ell_1|} < 3{5^{1-m}\over |\tilde H|}, \eqno(4) $$ \nd since $|\ell_1|>|\tilde H|/ 3$. By the same reasoning the same inequality holds for $L_3$, namely $$|L_3|-|\ell_3|< 3{5^{1-m}\over |\tilde H|}.\eqno(5)$$ \nd Now $|L_2|=|\ell_2|/\cos(\theta)< |\ell_2|(1+\delta/2) < 2 |\ell_2|$, so $$|\ell_2|(1+\delta)-|L_2|>|\ell_2|{\delta\over 2} > |L_2|{\delta\over 4} = 5^{-(n+m)/2}{\delta\over 4}. \eqno(6)$$ \nd But, if we choose $m$ such that $$5^{-(m+n)/2}{\delta\over 4}> 6{5^{1-m}\over |\tilde H|}, \eqno(7)$$ \nd which is certainly possible, then we get a contradiction with the optimality of $\tilde p$ by the following argument. By assumption there is a path, along edges, connecting the endpoints of $L_1$ (resp.\ $L_3$) no longer than $|L_1|(1+\delta)$ (resp.\ $|L_3|(1+\delta)$), so from the above there is such a path, along edges, connecting the endpoints of $\tilde H$ which is no longer than $|L_1|(1+\delta)+ |L_2|+ |L_3|(1+\delta)$. From (4)-(7) this is less than $|\tilde H|(1+\delta)$, which is the contradiction. \qed We now return to the pinwheel tiling. We call the 1, 2, $\sqrt{5}$ triangles that make up the tiling ``level-0 triangles''. Each such triangle, combined with four of its neighbors, forms a $\sqrt{5}, 2 \sqrt{5}, 5$ ``level-1 triangle''. Each level-1 triangle, together with four of its neighbors, forms a $5,10,5\sqrt{5}$ ``level-2 triangle'', and so on; see Fig.\ 5. If $P$ and $Q$ are points on the boundary of a level-$n$ triangle, then the shortest path from $P$ to $Q$ along level-0 triangles has length $\|P-Q\| g(n,p)$, where $p$ is the pair of points on the basic $1,2,\sqrt{5}$ triangle that correspond to $(P,Q)$ under similarity. Theorem 3 states that $g(n,\cdot)$ approaches 1 pointwise as $n \to \infty$. If $P$ and $Q$ are points that are separated by Euclidean distance $|L|$, then the straight line $L$ from $P$ to $Q$ will cut across several level-$n$ triangles, crossing the edges of the level-$n$ triangles at points $a,b,\ldots z$. This is illustrated in Fig.\ 6, in which the triangles are level-$n$, the smaller-scale tilings are not shown, and points $P,a,b,z$ and $Q$ are indicated by disks. The idea of the proof is to choose $n$ large, so $g(n,\cdot)$ is close to 1 and the shortest-path distance from each pair $(r,s)$ of consecutive points is close to the Euclidean distance. We then require $|L|$ to be large compared to $5^{n/2}$, making the effect of the end segments $P a$ and $zQ$ negligible compared to $|L|$. The difficulty is that the convergence $g(\cdot,n) \to 1$ is not uniform; for any $n$ there is a set of $p$'s for which $g(n,p)$ is not close to 1. We must control the contribution of this set, and this requires knowledge of the number of level-$n$ segments in the straight line from $P$ to $Q$. \nd {\bf Lemma.} Let $N$ be the number of level-$n$ segments in the straight line from $P$ to $Q$. If $|L|>(10)5^{n/2}$, then $5^{-(n+1)/2} |L|\le N \le 5^{-(n-2)/2} |L|$. \nd Proof: The diameter of a level-$n$ triangle is $5^{(n+1)/2}$. This gives an upper bound on the length of each segment, and so a lower bound on the number of segments. Now, if a level-$n$ triangle intersects $L$, it lies entirely in the set of points within distance $5^{(n+1)/2}$ of $L$. This region has area $5^{(n+1)/2}|L| + 5^{n+1} \pi$. Since each level-$n$ triangle has area $5^n$, there can be at most $5^{(1-n)/2}|L| + 5\pi < 5^{(2-n)/2} |L|$ such triangles. \qed \nd Proof of Theorem 2: Pick $\delta = \ep/75$ and let $X'$ be the set of pairs of points on the edges of the basic triangle that are separated by a Euclidean distance $\ge \delta$. $X'$ is compact, so we can find an $n$ such that $g(\cdot,n)< 1 + \ep/3$ on $X'$. On $X \backslash X'$ we have only the basic estimate $g(\cdot,n) \le g(\cdot,0) \le 5$. Finally, pick $R > (30)5^{(n+1)/2}/\ep$. For any $P$ and $Q$ with $\|P-Q\| >R$, we find a path from $P$ to $Q$ as follows. Draw the straight line $L$ from $P$ to $Q$. This defines points $a,\ b,\ \cdots,\ z$ where $L$ crosses the edges of level-$n$ triangles, as before. Take the shortest path from $P$ to $a$, followed by the shortest path from $a$ to $b$, and so on. Some of the pairs $(r,s)$ will correspond to points in $X'$. The total length of the resultant paths is then at most $|L|(1+\ep/3)$. Some (at most $5^{(2-n)/2}|L|$) of the pairs will correspond to points in $X\backslash X'$; the paths that result from these segments have total length at most $25 \delta |L| = |L| \ep/3$. Finally, the paths from $P$ to $a$, and from $z$ to $Q$, have total length at most $(10) 5^{(n+1)/2} < |L|\ep/3$. Thus the total path from $P$ to $Q$ has length less than $|L|(1+\ep)$. \qed Finally we return to our original claim (Theorem 1) about the existence of approximate circles. Given $\ep>0$ there is an $N$ such that a regular $N$-gon inscribed in a circle of radius $R$ approximates it, for both length and area, to within a fraction $\ep/2$, in the sense that $$2\pi R-\hbox{ (perimeter of $N$-gon)} < {\epsilon\over 2} R,\qquad \pi R^2 - \hbox{(area of $N$-gon)} < {\epsilon\over 2} R^2. \eqno(8)$$ Pick a length $L$ for which the metric distance used above is Euclidean to within a fraction $\ep^2$ and set $K=NL/(2\pi)$. Then, for any center point and any $R >K$ we build our approximate circle as follows. Within the circle of radius $R$ around the point we inscribe a regular $N$-gon in the circle, then move each of the vertices a distance $<1/2$ to place them on triangle edges. Connecting these points by shortest paths along triangle edges gives us an approximate $N$-gon of which we need show that the area and perimeter are accurate to within $\ep/2$, which would thus give us the desired approximation of the circle to within $\ep$. By assumption each approximation to a leg of the $N$-gon (we ignore corrections due to the small shifts of the vertices, as they are much smaller) has length accurate to within $\ep^2$, and so the perimeter is also accurate to within $\ep^2<\ep/2$. We can overestimate the error in area by assuming each approximate leg is a circular arc spanning the leg. To lowest order in the angles $\alpha$ of these arcs, the excess area is $N^2L^2{\alpha\over 12N}$ and in this notation the excess perimeter is $NL{\alpha^2\over 24}$. From this we see that ${\alpha^2\over 24}=\ep^2$ and therefore ${\alpha\over 12N}={\ep\over \sqrt{6}N}<{\ep\over 2}$. \qed The techniques we have used above also suffice to analyze a variety of similar problems. For example, consider the following signal-propagation problem on the vertices of the pinwheel tiling. Suppose that initially a single vertex $P$ is colored red, with all the other vertices being black. At each time step color red the nearest neighbors of the red vertices. After a large number of steps, what will be the size and shape of the red region? In light of Theorem 2, the answer should be no surprise: \nd {\bf Theorem 4.} Given $\ep>0$, there exists an $R$ such that every vertex $Q$ with $\|P-Q\|>R$ gets colored in at most $\|P-Q\| (1+\ep)/\sqrt{5}$ steps. The problem can be restated as if the color red moved with various speeds along the different edges: $\sqrt{5}$ along hypotenuses, 1 along short legs and 2 along long legs. Since the signal cannot propagate at a speed faster than $\sqrt{5}$, the theorem shows that the red region is asymptotically a round disk of radius $\sqrt{5} n + o(n)$, where $n$ is the number of steps. (The analogous result in which one assumes the same speed on all edges is simply a restatement of Theorem 2.) The proof of Theorem 4 is almost identical to the proof of Theorem 2 (via Theorem 3), and is essentially due to the existence of hypotenuses lying arbitrarily close to the desired direction. For the proof just assume that along hypotenuses the red color moves with speed $\sqrt{5}$ and along legs at speed 1. (Our use of a lower speed along long legs could only raise the estimate on the number of steps.) The only subtlety is that, in order to prove the analog of Theorem 3, we must only consider subdividing by even numbers of times since only even-order subdivisions decompose hypotenuses into smaller hypotenuses and legs into smaller legs. \vs.2 \nd {\bf III. Conclusion} In many ways, the structure of aperiodic tilings is intermediate between periodic grids (such as cubic lattices) and stochastic grids as analyzed in percolation [Kes]. For the isoperimetric problem on pinwheel space we have shown that the stochastic features dominate. There is enough variation in the position and orientation of triangles to allow the construction of good approximate circles of arbitrary size and arbitrary center. The only essential features of the pinwheel tiling used in our proof were its hierarchical nature, which is common in the aperiodic tilings used to model quasicrystals [S-O], and its ``statistical'' rotational symmetry [Rad3] -- the existence of edges lying close to every direction. Similar results are to be expected from any tiling with these properties. \vfill\eject \centerline{References}\vs.1 \nd [B-R] Berend, D.\ and Radin, C.: Are there chaotic tilings? Comm.\ Math.\ Phys.\ {\bf 152}, 215-219 (1993) \vs.05 \nd [Gar] Gardner, M.: Extraordinary nonperiodic tiling that enriches the theory of tiles. Sci.\ Amer., 116-119 (January 1977) \vs.05 \nd [Kes] Kesten, H.: Percolation theory and ergodic theory of infinite particle systems. New York: Springer-Verlag 1987. \vs.05 \nd [Rad1] Radin, C.: Global order from local sources. Bull.\ Am.\ Math.\ Soc.\ {\bf 25}, 335-364 (1991) \vs.05 \nd [Rad2] Radin, C.: The pinwheel tilings of the plane. Annals of Math.\ {\bf 139}, 661-702 (1994) \vs.05 \nd [Rad3] Radin, C.: Symmetry and tilings. Notices Am.\ Math.\ Soc.\ {\bf 42}, 26-31 (1995) \vs.05 \nd [Rad4] Radin, C.: Space tilings and substitutions, Geometriae Dedicata (to appear) [Obtainable from the electronic archive: mp\underbar{ }arc@math.utexas.edu] \vs.05 \nd [Rad5] Radin, C.: Miles of tiles. Preprint, Univ.\ of Texas, (1994) [Obtainable from the electronic archive: mp\underbar{ }arc@math.utexas.edu] \vs.05 \nd [S-O] Steinhardt, P.\ J., and Ostlund, S.: The physics of quasicrystals. Singapore: World Scientific 1987 \vs.05 \nd [Tay] Taylor, J.\ E.: Crystalline variational problems. Bull.\ Am.\ Math.\ Soc.\ {\bf 84}, 568-588 (1978) \vs.05 \nd [Wul] Wulff, G.: Zur frage der geschwindigkeit des wachsthums und der aufl\"osung der krystallflachen, Zeitschrift f\"ur Krystallographie und Mineralogie {\bf 34}, 449-530 (1901) \vfill \eject \nopagenumbers %\lett %\nd \hbox{} \epsfig 1.15\hsize; Fig1.ps; \vs-.5 \centerline{Figure 1.\ \ \ A Penrose tiling} \vfill \eject \hbox{} \vs.1 \hbox{}\hs-1.5 \vbox{\epsfxsize=7.5truein\epsfbox{Fig2.ps}} \vs.1 \centerline{Figure 2.\ \ \ A pinwheel tiling} \vfill \eject \hbox{}\vs2 \epsfig .9\hsize; Fig3.ps; \vs.5 \centerline{Figure 3. Two iterations of the pinwheel construction, without rotations.} \vfill \eject \hbox{}\vbox{\epsfxsize=4truein\epsfbox{Fig4.ps}} \centerline{Figure 4.\ \ \ Result of 5 subdivisions} \vfill\eject \hbox{} \vbox{\epsfxsize=6truein\epsfbox{Fig5.ps}} \vs.1 \centerline{Figure 5.\ \ \ Two levels of triangles} \vfill\eject \hbox{} \vs.1 \hbox{}\hs-1.5 \vbox{\epsfxsize=7.5truein\epsfbox{Fig6.ps}} \vs.1 \centerline{Figure 6\ \ \ Points $P,\ a,\ b,\ z$ and $Q$} \vfill \bye