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Recipe for the week of March 20 - 26

Critical Threshold Growth from a Large Box

Last week we featured a linear random growth rule known as Richardson's model. This week's offering depicts the nonlinear evolution of a range 51 box Threshold Growth CA started from a 101 by 101 cell square at time 0. Janko Gravner and I are currently studying the geometry of minimal size nucleating droplets for Threshold Growth. The present experiment is used to obtain upper bounds on the threshold for which droplets of size s can nucleate in a threshold s rule.

As it turns out, the central box here spreads without bound for thresholds less than 4330, but fixates in a final finite configuration for thresholds of 4330 or more. Our graphic displays the successively added sites, each update in a new color, for the first 131 steps of the threshold 4329 process. Note the complex pattern of layers as the droplet evolves toward its asymptotic shape while trying to spread past the corners of the original square. Later this spring we hope to offer a corresponding lower bound graphic that will shed more light on the actual geometry of critical minimal droplets.

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