## Thursday, November 1, 2012

### Cayley Tables!

Long ago I posted about a mathematical structure called a group. In particular I described the group S3. If you would like a refresher, read this post. This post picks up where that one left off.

S3 can be thought of as the set of rotations and reflections on equilateral triangle. Doing that gives the following table called a Cayley Table:

 * I R1 R2 F1 F2 F3 I I R1 R2 F1 F2 F3 R1 R1 R2 I F2 F3 F1 R2 R2 I R1 F3 F1 F2 F1 F1 F3 F2 I R2 R1 F2 F2 F1 F3 R1 I R2 F3 F3 F2 F1 R2 R1 I

Here's the thing though: This table is too busy to read. Sure, if you wanted to know what R1*F1 was, you could look it up (F2), but it's hard to take in all this information at once. Basically, it's hard to see.

But you know what's easy to see? Color!

 * I R1 R2 F1 F2 F3 I I R1 R2 F1 F2 F3 R1 R1 R2 I F2 F3 F1 R2 R2 I R1 F3 F1 F2 F1 F1 F3 F2 I R2 R1 F2 F2 F1 F3 R1 I R2 F3 F3 F2 F1 R2 R1 I

Two simplifications are in order: First, the top row and far left column are redundant, so we can dispense with them. Secondly, the labels are more distracting than helpful, so lets dispense with them as well. This leaves us with just a square grid:

 The Cayley Table for S3

This allows us to more easily see symmetries (the whites are symmetrical along the main diagonal) as well as asymmetries (notice the pattern of blues, greens and purples in the top right corner vs the same colors in bottom left).

As I mentioned, the group we have been considering is called S3. There are other groups too:

 The Cayley Table for S4

 The Cayley Table for S5

 The Cayley Table for S6

All of the pictures above are of what mathematicians call Symmetric Groups. But not all groups are symmetric groups. For example, there are the Alternating groups:
 The Cayley Table for A4

 The Cayley Table for A5

 The Cayley Table for A6
And the Cyclic groups:
 The Cayley Table for Z60

 The Cayley Table for Z60 with the elements arranged by their order
Hopefully I will post some explanations of these groups soon. In the mean time, try your luck with Wikipedia.

The groups are generated by a Python script I wrote. The Pictures are generated by Processing. The list of all possible ways four people can stand in line was generated by Matlab. Altogether there's about 450 lines of code going into these pictures. Anyone who wants access to this code is welcome to leave a comment.