Recipe for the week of April 24

The Cook Book

Recipe for the week of April 24 - 30

Life Without Death (revisited)

Recall our virtual snowflake from the week of February 13, generated by the CA rule that we now call Life Without Death: an empty site becomes occupied if there are exactly 3 occupied sites in its range 1 Box neighborhood. In other words, this is Conway's Game without the transitions from occupied to empty. Starting with a radius 50 lattice ball, the evolution produced a complex pattern reminiscent of ice crystals. We mentioned at the time that we had no idea whether or not the growth would continue in all radial directions.

In an attempt to gain further insight into the remarkably intricate dynamics of this simple rule, we have conducted a number of additional Life Without Death experiments starting from large balls. Kellie Evans grew this week's soup, using WinCA, by starting from a ball of 'radius' 80. In order to capture as large a flake as possible she appealed to the lattice symmetry and positioned one quarter of the initial disc at the extreme upper left of the array. Our graphic shows the result after 6000 updates. We have reduced the image to 1/9 its original area for easy viewing; those blessed with large desktops and lots of memory should see the original 1500 by 1300 hugesoup.gif.

Note the nearly periodic structure extending in the diagonal directions. Thus there seems to be some preliminary evidence that this growth is increasingly 'healthy' as time goes on. But the subtlety of the survival question is indicated by the fact that, even for large radii, small changes in the size of the initial ball can produce radically different trajectories. We have not yet attempted a detailed empirical study of this sensitive dependence on initial conditions to see whether any quantifiable regularity can be identified. But surely Life Without Death is one of the very simplest two-dimensional CA growth models with exemplary chaotic dynamics.