A Coloring Book Approach to Finding Coordination Sequences
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.
Keywords
Cite
@article{arxiv.1803.08530,
title = {A Coloring Book Approach to Finding Coordination Sequences},
author = {C. Goodman-Strauss and N. J. A. Sloane},
journal= {arXiv preprint arXiv:1803.08530},
year = {2019}
}
Comments
25 pages, 17 figures, 1 table. Apr 3 2018: Added a comment, several references, acknowledgments. Jan 31, 2019: Final accepted version