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The Fine Structure of the Box Triangle

The center of the box triangle is two-thirds black (Triangularity).

We can now completely describe the structure of the non-black part.

Third of Center: ''For any {$0<r<\frac12$}, all the triples on the line between {$\left(r,\frac12-r,\frac12\right)$} and {$\left(\frac13,\frac13,\frac13\right)$} (the dashed line in the picture below) are colored the same. In particular, either for all the boxes the paths are finite or for all they are infinite.

This proposition gives us a detailed picture of the box triangle.

Note that in the non-black part of the center we have drawn only three different colors. That's a corollary of the fine structure theorem plus the Destinations of Paths on Triangular Boxes proposition which limits the destinations on triangular boxes. That is,

Corollary: If {$a,b<c<a+b$} then the path on the {$a\times b\times c$} box must end at one of the corners {$101$}, {$110$}, or {$011$}.

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Page last modified on May 28, 2013, at 10:41 AM