Wednesday, July 02, 2014

Cracks in the crystal

In the late 1980s, I consumed reams of paper printing out (using my HP 75C) evolutions of elementary cellular automaton rule #193 (which in binary is 11000001, suggesting that the central cells of 111, 110, and 000 remain or become 1 in the next generation). This is the mirrored complement of the more famous rule #110. In rule #193, the ones form right "triangles" with a jagged hypotenuse facing southwest. Back then, in order to conserve ink I suppressed the printing of triangles of size-3 which (in evolutions of random starts) soon predominate. Looking at the time history of an evolution, the staggered size-3 triangles form a sort of "crystal" based on the stable configuration of the following cell (left and right are joined, time moves down):


Yesterday I spent some time programming this old plaything in Mathematica. Here is how a larger configuration of crystal cells appears in my implementation:

The lighter-blue ones of the size-3 triangles are meant to blend in with the darker-blue zeros, providing contrast for the other-sized triangles (in orange) of a typical evolution, out of which I have cropped this small detail:

The orange "particles" are recognized as defects, or cracks, in the crystal. They move left or right or just stand still. Colliding particles obviously conserve the sum of the defect offset numbers that individual particles may be said to possess, as they interact and regroup, or occasionally disappear. You can surmise from the (final) four particles at the very bottom of my example that the size of my space is a multiple of 14, the necessary space-size of a left-right-joined perfect crystal. (The right-moving particles will crash into the left-moving particles and disappear.)


  1. GGGGG... You have an interesting hobby, using Mathematica. I remember purchasing Mathematica 3.0 for Mac with a student discount when I was working on my Ph.D. That's more than 15 years ago, I think. I needed it just in order to draw a normal distribution curve. That was it. I don't think I ever used it again. You must be a math-oriented person to Mathematica.

    Tom Bluewater

    1. Hey, Tom. Yes, recreational arithmetic (finding numbers with certain properties) is one of my lifelong hobbies. And Mathematica makes it easy.