MA562: Topics in Geometry
Schedule, Fall 2004

(Assignments may change up until the date listed.)
9/13/04 Introduction, Classification The game Set
  1. Give an example of a way in which you classify or sort things in your everyday life.
  2. List a few of the ways mathematicians or scientists use classification systems and explain why you think these are useful to them in their work.
  3. Imagine a game of Set in which there were four different numbers, colors and shadings of objects on the cards. How many cards should there be in the deck? Why? How many cards would you need to deal out to ensure that there is at least one set present? Why?
9/20/04 Symmetry and Transformations Transformations using Kali and the Geometer's Sketchpad
  1. Sort the lowercase letters in the alphabet by symmetry. (Note: lowercase letters are less symmetric than uppercase.)
  2. Explain in your own words why the angles between mirrors in the rosette patterns must evenly divide 360 degrees.
  3. Imagine a game in which you are given a starting object and a same-size (but possibly reflected) copy of the object and are asked to use only reflections to transform the starting object into the transformed copy. Can you always do this using three or fewer reflections? Why or why not?
9/27/04 Identifying Symmetries of Patterns Identifying Symmetries of Patterns This assignment is due 10/18/04
  1. Draw (by hand or using a computer "paint" tool) a repeating pattern with symmetry type either 3*3 or 4*2.
  2. List all the type symbols with a cost of less than $2. These are the symmetry types of repeating patterns on spheres.
    Note: This list will include infinite families of symbols like 22, 33, ... nn. Symbols like 34 or *34, with two unequal gyrations or kaleidoscopic points are disallowed for reasons we'll discuss on October 25.
10/4/04 Proof that there are exactly 17 repeating patterns in the plane.
Invariants (knot equivalence example) and the Euler Characteristic.
Quiz 1 (symmetry types)
Game introducing invariants.
Calculation of Euler Characteristics.
No homework assigned this week. If you're bored, try some of the games at Try to determine what invariant is illustrated by each game.
10/18/04 Euler's Theorem and fractional Euler characteristic
Illustration of two kaledoscopic points on a sphere: *22.
Euler's Theorem on a bagel and for 4*2 Write approximately one paragraph in your own words explaining why it doesn't matter what graph you choose to compute the Euler characteristic of any surface. (Theorem 2, pp. 72-73.)
10/25/04 Definition of Orbifold Orbifolds of Repeating Patterns (bring scissors and tape)
  1. Describe the orbifold of a pattern with symmetry type *2222.
  2. Describe the orbifold of a pattern with symmetry type 3*3.
  3. Describe, as best you can, the orbifold of a pattern with symmetry type *x.
11/1/04 Turning orbifolds into patterns, and a hyperbolic pattern Quiz 2 (orbifolds)
Making patterns by paper cutting.
This assignment is due 11/15/04
  1. In your own words, explain why there are exactly seven symmetry types for frieze patterns.
  2. Describe (or demonstrate) how you would fold paper in order for a cutout to create a pattern with symmetry type *2222 when the paper is unfolded.
11/8/03 Classification of Surfaces: Klein Bottles and games on the torus and Klein bottle. Presentations by:

Jessica Lampert "Match Game"

Lino Cabral "The Mystery of the m and b in y=mx+b".

Will Kellogg "4x4 Set" (C code)

Quiz 3 (take-home) In five pages or fewer, summarize the complete proof that there are exactly 17 symmetry types for repeating patterns in the plane.
11/15/03 Logo Commands Presentations by:

Pamela Walsh "Chaos and Fractals"

Paul Canniff "Construction of a Pentagon"

Getting started with Logo

Print out or write down a logo program that draws a simple design then returns the turtle to its initial position and heading (use PRINT POS and PRINT HEADING to check that you're back to the home position) without using commands like HOME or CLEARSCREEN.

If you're looking for more things for your turtle to do, try Logo for Kids. You may wish to follow the instructions at for installing a logo.bat program in your Logo directory; if you use this program to start Logo you will be able to use notepad as your program editor rather than Jove.

11/22/03 Drawing a frieze pattern with Logo. Presentations by:

Patty Watters

Nicole LoPilato

Michelle LeBlanc

Follow the outline provided in class to redo Quiz 3 for an improved grade.
11/29/03 More Logo commands, a little bit about groups, and generators of frieze groups Presentations by:

Jen Pineau "Fractals"

Amy Cortright "Fibonacci Numbers and Spirals" (see web site)

Coloring Frieze Patterns

Quiz 4 (Take-home) Write a Logo program to display the seven different frieze patterns on your screen. Be prepared to run them during class. (If you can't fit all the frieze patterns on your screen, try experimenting with a combination of WAIT and CLEARSCREEN.)
12/6/03 Colored Frieze Patterns Coloring Frieze Patterns
  1. The symmetry groups of frieze patterns with type *∞∞ and 22∞ each have two generators. Choose one of these frieze patterns and describe the possible two colorings. For full credit, be sure to include an argument for why no other colorings are possible.
  2. The symmetry group of a frieze pattern with type * also has two generators. Compare and contrast the symmetries of this frieze pattern with those of the pattern you worked on in the previous problem.
  3. Choose a frieze pattern symmetry type. Write a Logo program to create a plane pattern with that symmetry type.
12/13/03 Review of the semester Finding type symbols, coloring 2*∞, constructing orbifolds, and matching orbifolds to type symbols. No Homework Assigned
12/20/03 Discussion of Final Projects Quiz 5 Quiz on classification systems, finding symmetry types, invariants and the Euler Characteristic and colored patterns. Happy Holidays!