M316L: Foundations of Geometry, Statistics and Probability, Fall 2009

Basic Information


Instructor: Michael B. Williams (call me Mike)
Office: RLM 12.132
Office hours: W 10-12, Th 2:30-3:20 (note the change!), by appointment
Email: mwilliams [at] math [dot] utexas [dot] edu
Unique #: 57635 / 57640
Time: MWF 9-10 am / 12-1 pm
Location: RAS 310 / 215

Here is the first day handout.

Here is more detailed information about assignments and exams:

Announcement: attend the lecture by Jeff Weeks, and you will get some extra credit (details on the extra credit coming soon). Here is more information on the lecture.

Announcement: the first exam is October 12. More information has been posted below.

Announcement: the first project has been posted and is due October 21. More information is below.

Announcement: attend the lecture by Thomas Banchoff, and you will get some extra credit (find me after the lecture to get the extra credit). Here is more information on the lecture.

Announcement: Based on the responses in Journal 3, most people have only a vague idea of what a mathematician is/does. To figure this out, do the following:

  1. look at the UT Mathematics faculty page
  2. select either a Professor (could be Associate or Assistant also), or a "Lecturer-RTG" that perhaps doesn't fit in with the stereotypical mathematician you described in the journal
  3. contact that person and politely request a (brief) interview
  4. interview that person to find out what mathematicians do
  5. submit, on or before November 23, a written description of what mathematicians do, based on this interview

Announcement: the second exam is November 16. More information has been posted below.

Announcement: you can now access all of your grades for the class on eGradebook (available from UT Direct). It is your responsiblity to check them for accuracy.

Announcement: the second project has been posted and is due December 4. More information is below.

Announcement: you can earn extra credit by attending this exhibition at the Bob Bullock Texas State History Museum. The exhibit is about quilts. To get extra credit, you must show me your program and describe two different quilts that you liked. You should talk about mathematics, e.g., the symmetries or geometric construction of the quilts in these descriptions.

You may take advantage of (at most) two of the four extra credit opportunities listed above.

Here's a picture of all the polyhedra from earlier in the semester:

Homework Assignments

Homework will be collected at the beginning of class. What you submit should be neat, and stapled if necessary.

  1. Due Friday, Sept. 4: § 8.1, #2, 4, 7, 8, 10, 11 a-c, 18 (only the first three in each part)
  2. Due Friday, Sept. 11: § 8.2, #7, 8 (a, b, c, d, Geoboard only), 10, 13 (b, d), 16 (a, b, c, d), 24 (Hint: try #4 first), 27, 29 (a, b [sketch them], c), 39, 41
  3. Due Friday, Sept. 18: § 8.3, #1, 3, 8, 10, 11, 13, 16, 17, 22, 23, 25,
    Also: Turn in Part 4 of Exploration 8.19. Your package must have at least one side that is not a rectangle. Turn in the package, a drawing of a corresponding net, the explanation described in 3c. Be creative!
  4. Due Friday, Sept. 25: § 9.1: 4, 9, 10 (1) a-f (draw the dots, plus a line to represent the mirror), 11 (draw the first figure, plus a line to represent the mirror), 15, 22, 24, 28,
    Also: You will construct either a cuboctahedron or a truncated octahedron. There are three conditions:
    1. You must make the polyhedron larger than what was described by the nets obtained in class (or downloaded). For the cuboctahedron, each square must have side length of at least 3 inches. For the truncated octahedron, each square must have side length of at least 2 inches.
    2. You must use paper that is higher quality than plain printer paper. For example: construction paper, poster board, cereal box, etc.
    3. The faces of your polyhedron must be colored in some way, either by using colored paper or by decorating it yourself. Be creative!
    Further details of the construction are left up to you: whether or not to use a whole net, how edges are joined, etc.
  5. Due Friday, Oct. 2: § 9.2, #2, 5, 8, 10, 11, 12, 20, 22 a, 24 a, 29, 31.
  6. Due Friday, Oct. 9:
    • § 8.2, #32, 33, 34 c,d (Yes, this section is correct!)
    • § 9.3, #1, 3, 4, 5, 7, 9, 10, 13
    • Also: Draw a parallelogram ABCD and a point P not on or in the parallelogram. Draw the image of ABCD under a dilation with scale factor 2 and center of dilation P. How does the perimeter of the new parallogram relate to that of ABCD? Why? How many copies of ABCD will fit inside the new parallelogram? Why?
  7. Due Monday, Oct. 26: (moved by popular demand)
    • § 10.1, #4, 7, 9, 10, 14, 16, 18, 23
    • Also: Draw all of the letters in the English alphabet (both uppercase and lowercase). Now sort them into groups (not groups in the project sense!), such that all letters in any group are topologically equivalent, and any two letters from different groups are not topologically equivalent. (See the October 7 exploration description below for more on topological equivalence.)
    • Also: Is a t-shirt topologically equivalent to a pair of pants? Why or why not?
  8. Due Friday, Oct. 30:
    • Write a 1-page biography of Karen Uhlenbeck, with spacing no more than 1.5. You should cite at least two sources, perhaps from the internet (but Wikipedia should not be one of them).
    • Do and turn in Part 3 of the Fractal Exploration.
  9. Due Friday, Nov. 6:
    • § 10.2, #1, 8, 11, 18, 36, 43, 52
    • § 10.3, #1, 2, 5, 15
  10. Due Friday, Nov. 13:
    • § 10.2, #9, 15, 20, 33, 35, 51
    • § 10.3, #7 (try $12 trillion!), 12 (cassettes, lol), 20, 23, 26, 30
  11. Due Friday, Nov. 20: § 7.1, #1, 4, 8, 9, 10, 13, 16, 19

Journals
  1. Due Wednesday, Sept. 2: Journal #1. You will need to read some of this paper.
  2. Due Wednesday, Sept. 23: Journal #2.
  3. Due Friday, Oct. 16: Journal #3.
  4. Due Monday, Nov. 9: Journal #4.
  5. Due Monday, Nov. 30: Journal #5

Reading Assignments
  1. For Friday, Aug. 28: Ch 8.1, p. 498-508
  2. For Monday, Aug. 31: Ch 8.1, remaining pages
  3. For Wednesday, Sept. 2: Ch 8.1, remaining pages
  4. For Friday, Sept. 4: Ch 8.2 (just start looking at it)
  5. For Wednesday, Sept. 9: Ch 8.2, p. 521-539
  6. For Friday, Sept. 11: Ch 8.2, remaining pages
  7. For Monday, Sept. 14: Read through Exploration 8.10 (we will discuss it in class)
  8. For Wednesday, Sept. 16: Ch 8.3 (the whole section; it isn't too long)
  9. For Monday, Sept. 21: Ch 9.1
  10. For Monday, Sept. 27: Ch 9.2, and read through Exploration 9.6
  11. For Wednesday, Oct 7: Ch 9.3
  12. For Friday, Oct 9: Study for the exam on Monday!
  13. For Wednesday, Oct. 14: Ch 10.1
  14. For Monday, Oct. 19: Ch 10.2
  15. For Monday, Nov. 2: Ch 10.3
  16. For Friday, Nov. 13: Ch 7.1
  17. For Friday, Nov. 23: Ch 7.2

Explorations
  1. Friday, Aug. 28: Exploration 8.1. If you didn't finish Part 2 of the Exploration in class, you can use the "Geoboard paper" in the appendix of your textbook to complete any remaining exercises.
  2. Monday, Aug. 31: Exploration 8.5. Proof is the essence of mathematics. Without proof, it is a hollow collection of unrelated ideas. Hopefully this exploration gave you the opportunity to see that proofs can be a fun, engaging, as well as challenging.
  3. Wednesday, Sept. 2: Exploration 8.4. Geometry comes up all over in everyday situations; this was a good opportunity to use some knowledge of geometry to understand why certain things (e.g. manhole covers) are they way they are. Be sure you can quantitatively describe how large a 'lip' should be to keep a square manhole cover from falling in. Also keep thinking about how to cut out the parallelogram with only one cut.
  4. Friday, Sept. 4: Exploration 8.7. The point of the first part was to see help you break apart figures into various components, to see the relationships between those components, and to recognize the various properties of the components. Everyone did a great job of replicating complex figures from memory! I hope it is clear that recognizing patterns and symmetry (and using the ideas just described) is very important in doing this.
  5. Wednesday, Sept. 9: Exploration 8.9. Effectively communicating mathematical ideas takes practice. Even for a reasonable-sounding definition of a familiar object, it is often easy to find an unnecessary restriction, or a loop-hole that allows other things than the desired object. Think about your definitions, and we will continue to discuss them on Friday.
  6. Friday, Sept. 11, Exploration 8.8. Mathematics as a process is emphasized here. You need to be able to think about large tasks like this (finding all shapes possible) in a systematic way, rather than just randomly trying different things.
  7. Monday, Sept. 14: Exploration 8.10. We quickly discussed a few ideas about adding up the measures of interior angles of polygons. There were some neat ideas here, and they will probably show up again this semester. Also, Exploration 8.14. This was our introduction to the three-dimensional analogs of polygons, called polyhedra. You will read more about their properties in section 8.3.
  8. Wednesday, Sept. 16: Exploration 8.19. The exercises with nets hopefully allowed you to see connections between the 2D objects we'be been studying, and 3D objects we are now studying. Finding patterns and using systematic, methodical thinking are themes that come up again and again...
  9. Friday, Sept. 18: We watched the film Flatland. You can find more information about the film here. The complete text of the original book is here.
  10. Monday, Sept. 21: Exploration 9.1. We looked at various notions of transformations of objects in the plane, as well as what sort of transformations preserve certain types of symmetries of those objects.
  11. Wednesday, Sept. 23: Exploration 9.4. We looked at how order matters when reflecting figures across two different lines. First, there was some confusion regarding just what it means to reflect across a line. Second, there was some confusion about that is means by saying that two transformation are "the same." Be sure you understand these points! Also important to notice is that it is often possible to represent any single transformation as a combination several other transformations. For example, reflecting twice across perpendicular lines was just a rotation by 180 degrees about the point of intersection, and reflecting across two parallel lines was just translation.
  12. Friday, Sept. 25: Exploration 9.5. In this exploration we saw folding paper as a way to visualize reflections and reflective symmetry. This kind of symmetry is required to cut shapes out of paper folded in regular ways, as in the exploration. Note that it is possible to cut out an equilateral triangle as in Part 3 (I may have incorrectly told some people that it was impossible.) Also, just to be safe, read through Exploration 9.6.
  13. Monday, Sept. 28: Exploration 9.7. This was actually a good warm up for the first project. Get used to thinking of a geometric figure as having a set of associated symmetries, which in our case are congruence transformations (rotations, reflections, translations). The symmetries are themselves objects of interest.
  14. Wednesday, Sept. 30: Exploration 9.10. Quilts, which we briefly studied at the beginning of the semester, provide a great way to explore symmetry (and non-symmetry!). Additionally, one can easily tie in lessons from history in such discussions, as quilting was a hugely influential passtime in this countries formative years.
  15. Friday, Oct. 2: Exploration 9.11. Using the pattern blocks is a good way to get a feel for similarity. One main lesson take away is that to create a similar 2-dimensional figure, one must increase the width and length by the same factor (called the scale factor). As a result, when you compute the area of the similar figure, the result is the scale factor squared, times the area of the original figure.
  16. Monday, Oct 5: Dilation handout. This covers material that is not included in the textbook. Namely, we think of congruent figures resulting from certain transformations (e.g., rotations, reflections, translations), but what transformations give similar figures? These are called dilations and contractions, and are explored in the handout.
  17. Wednesday, Oct. 7: Topology! Briefly leaving the world of geometry (which deals with distance and angle), we discussed continuous deformations. We are allowed to bend, stretch, shrink, or twist, but NOT to tear, puncture, or cut. We call two objects "topologically equivalent" if we can continuously deform each into the other. It is hard to overstate the fundamental importance of topology in modern mathematics, but sadly we don't have any more time to discuss it!
  18. Friday, Oct. 9: No official Exploration. We discussed the project and answered a few questions regarding the exam.
  19. Wednesday, Oct. 14. Introduction to Chapter 10: Geometry as Measurement. We listed a bunch of different measurable quantities (like distance, time, temperature, etc.) and some units used to express measurements of those quantities (like meters, seconds, degrees Celcius, etc.). We also discussed the imporant of measurement in the real world, talked about the metric system, and watched the video "Powers of Ten."
  20. Friday, Oct. 16: Exploration 10.4. It takes some effort for most people to visualize very large quantities of things. However, it can be done: you just need to find a good way to estimate a smaller quantity (e.g., 50 pennies is 3 inches tall, and 3/4 inches in diameter), scale that quantity to the right amount, and then put it into units that make sense (e.g., 75000 inches is 1.18 miles).
  21. Monday, Oct. 19: Area. We discussed the notion of area, and talked about a few ways to prove area formulas for various polygons. The actual exploration dealt with estimating area of irregularly shaped objects. We traced our hands on grid papers of different sized squares, and found that the finer grids lead to more accurate measurements. The idea of taking find and finer grids leads to the idea of a "limit", which is the basis of calculus.
  22. Wednesday, Oct. 21: Exploration 10.12. This exploration dealt with perimeter and area, and the relationship between the two. It turns out the this relationship is not as nice as one would expect. For example, doubling the area of an object doesn't necessarily result in the perimeter being doubled.
  23. Friday, Oct. 23: Fractals! This worksheet started with a simple fractal, Koch's curve. We found the patterns in its construction, and discovered that the total length of the object goes to inifinity as the number of iterations increases! We'll continue with part 2 on Monday: Sierpinski's triangle.
  24. Monday, Oct. 26: part 2 of the fractal exploration. We constructed Sierpinski's triangle, and saw that the total area left in the triangle goes to zero as the number of iterations goes to infinity. The homework had further investigations.
  25. Wednesday, Oct. 28: Technically, Exploration 10.7. We discussed a few topics regarding area, including possible areas of squares on a Geoboard, and how to prove some area formulas for polygons. The main idea here was to break up the shape into simpler pieces, whose areas are easy to compute.
  26. Friday, Oct 30: Exploration 10.14. Here we began moving from two to three dimensions. However, the quantity we looked at, surface area, is still a two-dimensional quantity. It just happens that this quantity belongs to a three-dimensional object. We constructed some nets for a few objects, and used them to compute the surface area of the corresponding objects. This was pretty simple: just add up the areas of all the two-dimensional shapes in the net.
  27. Monday, Nov. 2: We discussed the notion of volume. We looked at a number of different three-dimensional objects, and talked about various methods for finding their volumes. Some of the ideas we've used before for area still worked, but some didn't. One very general strategy still applied: break up a complex object into simple pieces whenever possible.
  28. Wednesday, Nov. 4: We discussed the first project. The goal of this project was for you to grapple with a fundamental --and accessible-- topic in modern mathematics, group theory. You saw how the notion of a group relates and unifies many things that you alread know about, from geometry, arithmetic, and algebra.
  29. Friday, Nov. 6: Donald in Mathmagic Land.
  30. Monday, Nov. 9: Volume Exploration. We investigated what goes into constructing containers (a box and a cylinder) of given volumes, and how the surface area can change when the volume changes. As with area and perimeter, the surface area is not related to volume in a simple way.
  31. Wednesday, Nov. 11: Volume, continued. Also, introduction to Chapter 7: statistics and probability.
  32. Friday, Nov. 13: Exploration 7.1. We discussed data, how to represent it, and what information we can get from these representations. We looked at various ways of thinking about census data, from raw population numbers, to net increase and percent increase from decade to decade. There were things that some graphs showed that weren't apparent from others.
  33. Monday, Nov. 16: Exam 2. Totally sweet.

Exam Information
  1. Monday, Oct. 12. The exam will cover book sections 8.1 through 9.2. You must be familiar with all of this material, but emphasis will be on:
    • topics that we actually discussed in class,
    • things that appeared in the homework,
    • ideas that arose in the Explorations.
    You may omit the part of § 8.2 on Special Line Segments in Triangles, and the sections in § 9.2 on Symmetries of Strip Patterns, and on Tesselations. There will be a variety of questions, including True/False and free resonponse. The difficulty of the problems will be (roughly) at the level of homework problems, although their length will be more suitable for an in-class exam. A good way to study is to
    • carefully reread the relevant book sections,
    • review your completed homework problems,
    • do the Review Exercises from chapters 8 and 9,
    • read through the relevant Explorations,
    • talk to me about any questions!
  2. Monday, Nov. 16. The exam will cover book sections 9.3 through 10.3. No parts of these book sections will be omitted this time. However, there were topics covered in class (and in the project) that did not appear in the book, but which might appear on the exam: topological transformations and equivalence, dilations, fractals, group tables. The same advice from the first exam applies (with the appropriate changes to the book chapters)! Also, you will not have to memorize any formulas from section 10.3.

Projects
  1. Due Wednesday, Oct. 21: Here is the project description. You should begin working on this as soon as possible (in particular, you will definitely not be able to finish it the night before it is due!). Please see me if you have any questions.

    Note: I modified the project description on 10/8. Part 1 was shortened by removing #2. Also, I will soon post a cover page, which you must attach to your submitted project. It will include more details on how the project will be graded.

    Update (10/12): Here is the cover page that you must attach to your submitted project. I ask that you type your projects, but you may write in any mathematics by hand, if necessary.

    Update (10/14): I modified Part 2, #1 b slightly. The table in the Exploration writes elements as either the identity, a rotation, or a product of a rotation and a distinguished reflection m. In your table, I want you to just write out each individual element (the identity, three nontrivial rotations, four different reflections).
  2. Due Friday, Dec. 4: Here is the project description. Tell me who is your partner as soon as possible. You will also need to read this book chapter.

Detailed Class Calendar

This is tentative and may change at any time (but I hope not by much).

Month
Day Reading Exploration Assignment Due
August W 26



F 28 8.1, p. 498-508 8.1, Parts 1,2

M 31 8.1, remaining pages 8.5
September W 2 8.1, remaining pages 8.4 Journal 1

F 4 8.2 8.7, Parts 1,2
 HW

W 9 8.2, p. 521-539 8.9

F 11 8.2, remaining pages 8.8
 HW

M 14 read Exp. 8.10 8.10, Part 1; 8.14, Part 1

W 16 all of 8.3 8.19

F 18 8.3 Flatland HW

M 21 9.1 9.1, Parts 1,2

W 23 9.1 9.4 Journal 2

F 25 9.1 9.5, Parts 2,3 HW

M 28 all of 9.2 9.7

W 30 9.2 9.10
October F 2 9.3 9.11 HW

M 5 9.3 Dilations

W 7 9.3 Topology discussion

F 9 Project intro HW

M 12

Exam 1

W 14 10.1 Measurement intro

F 16 10.1 10.4 Journal 3

M 19 10.2 Area worksheet

W 21 10.2 10.12 Project 1

F 23 10.2 Fractals, Part 1

M 26 10.2 Fractals, Part 2 HW

W 28 10.2 10.7

F 30 10.3 10.14 HW
November M 2 10.3 Volume discussion

W 4 10.3 Project 1 discussion

F 6 10.3 Donald in Mathmagic Land HW

M 9 10.3 Volume Exploration Journal 4

W 11 7.1 Statistics intro

F 13 7.1 7.1 HW

M 16

Exam 2

W 18 7.1


F 20 7.2
HW

M 23 7.2

W 25 7.2

M 30 7.3 Journal 5
December W 2 7.3

F 4 7.3
Project 2, HW

Other Links

Here is other information that is (hopefully) relevant to our discussions from the following class days:

last modified: 11/7/2009