A good exercise and remark for the paper

We discovered today that Packard-Wolfram has a supposed solidification rule that is in fact Exactly 1 modified so that a 1 that is completely surrounded becomes a 0. This can be analyzed as another perturbation of Exactly 1. We have the tools to show that the final configuration is simply a checkerboard (and so has density 1/2) and that the state at any point in time is that checkerboard intersected with the corresponding state of the 1or2 rule. Work out the details and remember to include a remark in the write up.

Outline for paper for Diamond rules

1. Introduction / History / Notation, Lightcone/Sierpinski, 12x rules, T and its properties
2. 1 (C)
3. 14 and 174 (M)
4. 13 and 134 (H)
5. Binary Rule, A = Union of iterates of T, and Subsitution rule (C)
6. Asymptotic Density (M)
   Finite seed invariance
   Macro dynamics - von Koch schemes
7. References

Also - Pictures!!!

Wolfram's papers

N.H. Packard and S. Wolfram: Journal of Statistical Physics, 38 (March 1985) 901-946

S. Wolfram: Scientific American, 251 (September 1984) 188-203

A New Kind of Science (online version)

Let's continue to use my file space

http://psoup.math.wisc.edu/CURL/

as one place for documents. My papers with Janko are called PackardSnowflakes and Snowfakes there. I've also put a preprint version of the first Packard paper there under the name PackardOriginal. Most of the other resources I asked you to gather should be available on the web.

Background Research

We need to find and read the following articles:

Packards original paper in World Scientific, "Theory and Application of CA" 1986 pp. 305-310
Article by Wolfram in Scientific American vol 251, 1984
Paper by Packard and Wolfram jointly in Journal of Statistical Physics vol 38 #5,6 1985
A New Kind of Science by Wolfram
Advances in Applied Math vol 21 pp. 241-304 article by Griffeath 1998

We also need to think about proving that any finite seed has the same density as a singleton (for the Diamond Rules)