# Math 113: Studies in Mathematics

## Information for Students

Course syllabusTutorial:

Tutorial sections are mandatory. Tutorial is not "homework help" (although you can get help with your homework there) but will often cover new material not done in lectures, and review the material that was covered in the lecture. As you may have noticed, lectures move very quickly. Tutorial helps you find out if you've caught everything.

Bring your book! You will never need to bring it to lecture, but you will need it in tutorial.

Your tutor doesn't do most of the talking in tutorial, you do!! You, yes you, will be up there writing on the board like a PRO.

## Additional references

Permutation groups Supplement to chapter 12.Frieze groups table . Here are some friezes for you to practice on. As seen in tutorial.

Wallpaper groups

## Homework

HW due 1/11HW due 1/18 Reading 1: Classifying groups

A reading on the (recently completed) epic program to understand and classify the most fundamental finite groups. Starring group theory superstar Daniel Gorenstein.

We'll discuss this reading in class on Friday.

HW due 1/23

HW due 1/30

Reading 2: Proofs and Refutations

A beautiful and fascinating piece by Imre Lakatos on the process of mathematical proof. Also widely influential: google scholar counts 2643 citations!

Reading this is homework for next Monday, February 6. You may wish to do it sooner, if you have the time. It is also fairly long, but more than worthwhile.

HW due 2/6

Guidelines for your Independent project

HW due 2/13

HW due 2/20

HW due 2/27

Reading 3: Penrose Tilings

A reference on Penrose tiles and aperiodic tilings from Martin Gardner's book "Penrose Tiles to Trapdoor Ciphers"

HW due 3/5

Reading 4: Hyperbolic and Spherical geometries

This reading (with some really delightful pictures) supplements the material covered in class during 10th week. It does not rigorously cover definitions, but should help you a great deal with a conceptual understanding of spherical and hyperbolic geometry.

[p.s. Just in case you forgot that math was delicious. Remember this gingerbread polyhedron that was mentioned on one of your assignments. Well, here is a shortbread Poincare disc.]

Just for fun: Make your own hyperbolic fish

Print this out, cut it out and then make extra cuts to separate any two triangles that happen to be touching. Print an extra copy so you can cut out more squares and triangles to fill in the gaps in the first. Now start taping the pieces together so that there are exactly three squares and three triangles (alternating) at each vertex. The result will be a wavy, kind of frilly piece of paper. Voila: you have a model of hyperbolic space, covered in fish just like Escher's Circle Limit III For extra artistic effect, use some colors.

Just for fun: Make your own hyperbolic space

For a non-tessellated version of hyperbolic space, print out this triangle paper. Notice how it is flat, and there are 6 triangles around each vertex. If you cut it up and taped it back together so there were 5 triangles around each vertex, you'd get a (roughly-)spherical icosahedron. To make a hyperbolic space, cut it up and tape it back together so that there are 7 triangles around each vertex. Print more paper if you like - unlike the icosahedron, hyperbolic space goes on forever.