The Cook Book

Recipe for the week of June 10 - 16

Big Bang Majority Maze

Recent versions of the Netscape and Microsoft browsers support animated gifs, little cartoons that are spreading over the web like wildfire. We couldn't resist spicing up the Kitchen with a few, such as the Larger than Life period 5 strobe on our title page and the excitable CA spiral core (run in reverse) that leads back to that page from any other. A combination of blinking gizmos, garish billboards and self-congratulatory award logos quickly turns any attractive site into a virtual Las Vegas eyesore, so we'll try to resist the temptation of glitz for its own sake.

That said, a welcome application of the new format is to show short inline snippets of CA dynamics. This week let us consider the first few updates of one of the simplest and most interesting of all CA rules: majority vote, which has been mentioned briefly in our discussion of plurality vote and three-party majority voting, but not yet featured in its own right. Our soup this week features the range 4 Box version, in which case a cell switches allegiance (Democrat vs. Republican, 'for' vs. 'against', etc.) whenever the majority of its 81 neighbors hold the opposite opinion. Starting from a 50-50 completely random mix of opinions, and with wrap-around boundary, the state after 4 updates is shown. Using the 'level set' color scheme of the past several weeks, 0's are colored black, whereas 1's are colored red, purple, green, yellow or blue according to whether the cell last became a 1 at time 0,1,2,3 or 4, respectively. The movie below begins with the same time 4 configuration, then restarts to show the rapid self-organization from noise, looping through this sequence repeatedly.

Notice that by time 4 the space is divided into well-separated components of 0's and 1's. In fact, one can check that once there is a certain minimal distance between any two distinct components of the same type, then that minimal distance is preserved at all future times. Moreover, with one important proviso that will be mentioned next week, any component that is completely surrounded by the opposite type shrinks and vanishes under the dynamics. Because of the edge condition, our soup is a kind of random maze on a torus. See if you can find a loop through 1's connecting the top to the bottom, and another loop of 1's connecting the left side to the right. Together these guaranatee that 1's constitute a background for the entire array, in which case the level set coloring scheme yields spectacular graphics if continued until a final state is reached. Our next recipe will feature some of these most excellent soups, and dig a little deeper into the mathematics of majority vote on finite arrays.

To be continued ...

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