The Cook Book

Recipe for the week of July 15 - 21

Multicolor Replicators

Before describing the science behind this week's soup, I beg the reader's indulgence during a digression about the trials of popularization. The immediate trigger is our Kitchen's selection for the WebGuide section in Issue 7 of an electronic and print publication from Underground Online, in the Amusements category. What's my gripe with
(highest rating), you ask? Well, the fine print goes on to say:

Contrary to the title, don't expect to find great recipes here. This site was created by a math professor to display his computer-generated images in random patterns, not his mother's chili recipe. The prof uses "Garden-variety PCs and dedicated Cellular Automation Machines" as mixmasters for the colorful images he displays weekly. He blames his food-fixation, among other cerebral ramblings, for the unrelated title. The site is mostly eye-candy, but as visual stimulation, it's first-rate.

Ok, I confess to the food thing, and an occasional tendency to ramble. But the Chef encourages all PSK visitors to occasionally read the odd recipe, to perhaps discover thereby that the title of the premises is NOT unrelated to its content, and that there is intellectual content in addition to the weekly fix of optical cocaine. For instance, that's why our new POSTCARD SERVER (try it, you'll like it!) is accessed by clicking on the little envelope icons of selected recipe pages -- so folks will be forced to confront a few words along with the pictures.

We don't mean to sound ungrateful for publicity, which is always welcome. Most recently, at the academic end of the spectrum, PSK has been honored by the Canadian Math Society as a Link of the Braid. Then again, Quantitative Data Systems bestowed us with their coveted Strange Site designation, in the distinguished company of such Web gems as The Adam Ant Interactive Ant Farm, Codpiece International, Stark's Museum of Vacuum Cleaners, and one of my all time favorites: Strawberry Pop-Tart Blow-Torches. Until recently, our debt to Glenn Davis for an early selection as Cool Site of the Day was gratefully acknowledged on the main Kitchen page, but see Suck, November 6, 1995 for a rather devastating critique of the subsequent proliferation of such services.

Enough already; back to mathematics. This week we feature the additive, two-dimensional counterpart of Pascal's Triangle Mod N. At each update, the new color index of a site is obtained by adding the previous color values over its neighbor set and calculating the remainder after division by N. Our soup uses a nearest neighbor Box neighborhood and 16 colors. Some elementary number theory shows that when N is a power of 2, then any initial configuration is replicated exactly at suitable times which are powers of 2. At most large times the state is more or less indistinguishable from N-color noise, whereas some times that are powers of 2 yield designs on the border between pattern and disorder. The thumbnail above shows our initial design after a couple of updates, while this week's soup shows the result of 128 iterations.

One possible use for such intermittent emergence of information from noise is for encryption, although nonlinear rules such as Fredkin's Reversible Cellular Automata are better suited for that purpose. On a more theoretical note, in the context of our discussion of two weeks ago, it would be interesting to collect examples of near-critical multitype dynamics and suitable initial configurations which generate multicolor counterparts of the Replicating Skeeters and other two-type replicators.

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