A Coloring Book Approach to Finding Coordination Sequences
| Autor: | Neil J. A. Sloane, C. Goodman-Strauss |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
Uniform tiling
Condensed Matter Physics Mathematical proof Biochemistry Graph Vertex (geometry) Inorganic Chemistry Combinatorics 05A15 05B45 05C30 Structural Biology FOS: Mathematics Mathematics - Combinatorics General Materials Science Combinatorics (math.CO) Physical and Theoretical Chemistry Simple fact Mathematics |
| DOI: | 10.48550/arxiv.1803.08530 |
| Popis: | 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. Comment: 25 pages, 17 figures, 1 table. Apr 3 2018: Added a comment, several references, acknowledgments. Jan 31, 2019: Final accepted version |
| Databáze: | OpenAIRE |
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