Monday, June 1, 2009

Math Art

This picture represents a good deal of what I accomplished my senior year of college. I didn't make this picture; that credit belongs to my former Professor, Jacob Siehler, who is quite amazing at using computers to represent mathematical data.

Basically this picture represents a result of my senior thesis. Here's more or less what I did:

Take a positive integer m. Make a table with m columns and infinitely many rows. Label the columns 0, 1, ... , up through m - 1. Now place a ball in the 0 column of some row, say row x. Then define a sequence of rotation remainders as follows:
  • When the ball is in row x, you move it x spaces to the right, and if there aren't that many spaces left in the row, just wrap around to the next row (just like reading a page left to right, top to bottom)
  • After each move, write down the column position the ball is in
This sequence of column positions is called the sequence of rotation remainders, which is a term I made up, and it's what I titled my thesis. Surely I could have picked a more exciting and/or cute name for it, but what can you do...

So you might ask how the above picture represents this sequence at all. In fact, each row of pixels in the picture represents an integer (1, 2, 3, 4, 5, ...) and the pixels in a given row represent the numbers in the rotation remainder sequence corresponding to x. In this picture, m = 63.

Isn't it pretty? I promise there's a lot more complicated stuff behind it, too. I'm sure you'd be very impressed.

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