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Recipe for the week of November 27 - December 3

A Holiday Chestnut: Pascal's Triangle mod N

Some rather pleasing patterns can be made by painting Pascal's Triangle of binomial coefficients modulo N. In other words, if N is 2 or more, color a number k in the triangle one of N colors (using your favorite palette) according to the remainder when k is divided by N. In our scheme, a remainder of 0 yields dark blue, remainder 1 green, etc. (Black is the background for the checkerboard lattice.)

Note that you need not compute actual values in the triangle to determine a pattern; for given N, the modular arithmetic defines a simple rule whereby any interior color is a consequence of the two colors located just above. In other words, these are space-time diagrams of additive one-dimensional multitype cellular automata, with time running down the screen. The two color case was mentioned last June 5. It generates a Sierpinski Lattice, the discrete counterpart to a familiar fractal known as the Sierpinski gasket. Additivity amounts to a superposition property, a kind of linearity, that makes these models quite amenable to rigorous mathematical analysis.

This week's soup shows the first 40 rows of the case N = 13. Several years ago I used such a design, derived from the educational software Graphical Aids for Stochastic Processes (GASP) that I wrote with Bob Fisch, as a holiday greeting card. Here are a few more examples -- the Sierpinski case and some of my favorites:
2 Colors
7 Colors
9 Colors
11 Colors

By the way, the combinatorial structure named after Pascal can be traced to China several hundred years before the French philosopher/mathematician's birth, and to India more than one thousand years ago where it was used to count the number of ways to make curries from a given number of spices. Once again, knowledge begins in the kitchen ...

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