The Cook Book
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 spacetime diagrams of additive onedimensional 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 ...
