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 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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