A coloring‐book approach to finding coordination sequences. Issue 1 (18th December 2018)

• Record Type: Journal Article
• Authors: Goodman-Strauss, C.; Sloane, N. J. A.
• Is Part Of: Acta crystallographica. Volume 75:Issue 1(2019:Jan.)
• Page Start: 121
• Page End: 134
• Publication Date: 2018-12-18
• Subjects: coordination sequences -- uniform tiling -- dual tiling -- Cairo tiling -- tetravalent vertices -- trivalent vertices; Crystallography -- Periodicals; Condensed matter -- Periodicals; 548
• DOI: 10.1107/S2053273318014481
• Languages: English
• ISSNs: 2053-2733

Abstract

Abstract : This article presents a simple method for finding formulas for coordination sequences, based on coloring the underlying graph according to certain rules. It is illustrated by applying it to several uniform tilings and their duals. Abstract : An elementary method is described for finding the coordination sequences for a tiling, based on coloring the underlying graph. The first application is 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. The authors thought that such a simple fact should have a simple proof, and this article is the result. The method is also used 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, and the snub‐632 tiling. In several cases the results provide proofs for previously conjectured formulas.