Recipe for the week of October 23

The Cook Book

Recipe for the week of October 23 - 29

Meet Max, the sensitive space-filler

Here is a remarkable fellow named Max, the smallest known configuration for Conway's Game of Life that spreads out to fill the lattice with density 1/2. Apparently there is no known finite lattice animal that grows on an empty background with density greater than 1/2. Max was discovered by Hartmut Holzwart in the Fall of 1993 using some rather sophisticated algorithmic search techniques. As an exercise, compute the number of cells in Holzwart's construction.

Max is one of a very large number of exotic seeds for Life that have been collected over the past few years by aficionados such as Alan Hensel and David Bell. These have now been cataloged in a terrific new resource: Callahan's Life page, one of several recent additions to our Kitchen Sink. Look under the lwssgun-memory listing of his alphabetical Browsable Pattern Archive to find the following biography:

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MAX, the smallest known example of a "spacefiller"

This is the fastest-growing known pattern in Conway's Game of Life (possibly the fastest possible). It fills space to a density of 1/2, conjectured to be the maximum density, and does it at a speed of c/2 in each of the 4 directions, which has been proven to be the maximum possible speed.

Population growth is [(t+19)^2+463]/4 for t divisible by 4; [(t+19)^2+487]/4 for t even, not div. by 4; [(t+18)^2+639]/4 for t odd.

Original construction, top/bottom stretchers by Hartmut Holzwart; Size optimization, left/right stretchers by David Bell; Original idea, middle part, l/r stretcher connection by Al Hensel. This spacefiller by David Bell, September 1993.
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Just for fun, we have aimed a Conway glider toward Max' midsection. This week's soup shows the state of affairs shortly after the moment of impact. Such sensitive dependence on initial conditions is one illustration that finite and infinite nonlinear systems can behave very differently, especially when the dynamics are non-monotone.

One more remark: with a screen capture utility you can simply cut out the thumbnail .gif patterns from Calahan's archive, convert them to .bmp format, and then paste them into WinCA. I tried this with ak47, a glider gun, to test that its period was 47 as claimed. Worked first try, no sweat.