Six triangle picture

Here's a possible picture to illustrate the triangles.

pic

First draft of 1 or 3.

Here's 1 or 3 so far.

pdf
tex

Master document; proof of "no backfill" for exactly 1

Here is the LaTeX and PDF for the "master document" (i.e., the document containing all of the sections). I included section titles for 14 and 174 (M) and 13 and 134 (H) as placeholders.

The proof of "no backfill" for Exactly 1 is at the top of page 4. The other new thing is that I introduce the claim of "no backfill" in the third paragraph of Section 2: Exactly 1 (page 2).

A couple of things to consider for the "no backfill" proof for Exactly 1:

• I showed it for the rotated version; should we bother to show afterward that it also holds for the unrotated version?
• In the second paragraph of the proof (the "assume holds for n and show holds for n+1" part), I may need to say a bit more. For example, I may need to show that the sites outside B_N_(n+1) cannot affect sites in B_N_(n+1) because of Lemma 2 (analogous to the statement in the previous paragraph in which I show that sites outside B_3 do not affect sites in B_3 because of the boundary of empty cells formed by Lemma 2). Or is what I have written enough?
  

Introduction

I started working on the portion of the introduction that proves properties of the transformation T. I am struggling with the second part of the proof that the T^kC partition Z^2. Here is the LaTeX, PDF.

Consequences of the change of notation

Here is something to consider: if we change the subscript t's to superscript t's, we must change the following:

1. A_t
2. B_t
3. D_t
4. Q_n
5. a_n
6. q_n

...and there may be more that I can't think of right now. Do we really want to make a change in notation that makes 6 objects look somewhat awkward just so that A_I^t looks like tau_I^t? I think it'd be okay to have A_t^I but tau_I^t. After all, we already had rho_I for the asymptotic density, which bucks the trend that we had of subscript t for objects 1 through 6.

Section on Binary Rule, T, Subsitution Rule

I worked on this writeup some more, and it's almost done. PDF, LaTeX.

One question that came to mind: Our current wording of the Binary Rule holds only for the first quadrant; do we want to throw some absolute values in there to extend it to the other 3 quadrants?

Another question to consider: split up Corollary 1 into two corollaries or keep it as one corollary?
1) A_infty = UT^(2k)C
2) A_infty satisfies the "complementary Binary Rule"

updated trivials and 14

here are updated documents for 1or4 and 1or2. I will add the wolfram 174 thingy and have something "tangible" for 134 rule by monday.