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arXiv · 1803.08530

A Coloring Book Approach to Finding Coordination Sequences

C. Goodman-Strauss·N. J. A. Sloane
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Abstract

An elementary method is described for finding the coordination sequences for a tiling, based on coloring the underlying graph. We illustrate the method by first applying it to the two kinds of vertices (tetravalent and trivalent) in the Cairo (or dual-3^2.4.3.4) tiling. The coordination sequence for a tetravalent vertex turns out, surprisingly, to be 1, 4, 8 ,12, 16, ..., the same as for a vertex in the familiar square (or 4^4) tiling. We thought that such a simple fact should have a simple proof, and this article is the result. We also apply the method to obtain coordination sequences for the 3^2.4.3.4, 3.4.6.4, 4.8^2, 3.12^2, and 3^4.6 uniform tilings, as well as the snub-632 and bew tilings. In several cases the results provide proofs for previously conjectured formulas.

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C. Goodman-Strauss, N. J. A. Sloane. 2019-01-31. A Coloring Book Approach to Finding Coordination Sequences. https://doi.org/10.1107/s2053273318014481

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