Occupy Math has talked before about complex systems. Few human systems are more complex than the economy of a large country like the United States of America. This post reacts to the recent proposed tax bill in the United States — a very complex collection of modifications to a super-complicated system. Since the US economy is pretty much beyond human comprehension, Occupy Math will start with a much simpler complex system. This relatively simple example builds on an earlier post Evolution Can Do Math That People Can’t. That post was about evolving cellular automata. The way cellular automata work is that we start with a row of zeros, except the middle three numbers in the row are 1,2,1. The numbers can go from 0 to 7. We sum blocks of five adjacent numbers — getting a sum from 0 to 35 — and look up the new numbers using the table of 36 numbers:
So, for example, if the five numbers in a block sum 15, the new value for the number in the center of the block is “3” because the 15th number (starting counting at zero in the list above) is “3”. We apply the rule over and over to generate rows of new numbers — mapping the numbers to colors to draw a picture. The color map is zero;white, one;red; two;green, three;blue, four;yellow, five;violet; six;cyan; and seven;black. The picture for the rule above is:
Here is a simple experiment in complex systems: change one number in the list of numbers that forms the rule for the cellular automata and look at the picture that the new rule draws. There are 36 examples below the fold.