Exactly 1 Diamond Writeup

Re-writing the Exactly 1 Diamond section using the order of topics below (namely, with the Binary Rule at the end, sortof as an afterthought) will be harder than I anticipated. We used the Binary Rule extensively to prove things listed before it on the outline below (especially item 1). Anyway, here's my first stab at it: PDF, LaTeX.

Updated Exactly 1 Diamond Outline

Here is the PDF and LaTeX of the updated outline, which incorporates David's suggestions.

I altered the order somewhat to the (hopefully more logical) following: transformation T,
cloning process, proving density = 2/3 two ways (cloning process, transformation T), the Binary Rule, and the Substution rule.

Formulas for the population counts of diamond rules

Do we want to find the formulas for the population counts of the diamond rules at time (2^n)-1, like we did for the box rules? Or just the asymptotic densities?

Exactly 1 Diamond Rule

First I'm working on what to include in the section of the paper about the Exactly 1 rule on the Diamond Neighborhood and how to organize it. I wrote a list of things that could be included in LaTeX; you can get the PDF here and, if you would like to modify it, the LaTeX file here.

Fortunately, we have a rough draft of each of the items I mention. The first step I'd like to do, however, is to get your input on what to include and what organization to use (order of items, depth into which I write about each item, etc.). Then the next step would be to polish and improve the rough drafts of each item.

174

The blurb Charlie sent looks good. I agree with Mike that this argument should be included in the discussion of 1or4 (the example in NKS). Then one can describe the dynamics in a little more detail by noting that 1or4 is "solid" at all times in the sense that it equals Exactly1 at the same time but "filled in," and that 174 at that time consists of the filled in region intersected with the checkerboard C.

A New Kind of Science

As far as I can tell, the only mention of "1 or X" box/diamond rules in ANKS is on pages 170-172. Wolfram shows pictures of the evolution of 1 or 4 diamond rule (code 942) on page 171, states that the rule "yields an intricate, if very regular, pattern of growth," shows a 3D space-time display of the evolution on page 172 -- and that's about the extent of it.

The only other remotely close rule he describes is the "3 or 5" box rule (not counting self) (code = 175850) on page 177, the pattern of which he asserts has "seemingly random irregularities, at least on a small scale" but on a larger scale it follows a "rather smooth curve."

The missing page!

I've replaced PackardOriginal.pdf with an even funkier photocopy that has the missing page.