A coloring-book approach to finding coordination sequences.

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-32.4.3.4) 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 44) 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 32.4.3.4, 3.4.6.4, 4.82, 3.122 and 34.6 uniform tilings, and the snub-632 tiling. In several cases the results provide proofs for previously conjectured formulas.

[1]  P. D. L. Harpe,et al.  Séries de croissance et séries d'Ehrhart associées aux réseaux de racines , 1997 .

[2]  M. O'Keefe,et al.  Plane nets in crystal chemistry , 1980, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[3]  Gorjan Alagic,et al.  #p , 2019, Quantum information & computation.

[4]  David B. A. Epstein,et al.  The Use of Knuth-Bendix Methods to Solve the Word Problem in Automatic Groups , 1991, J. Symb. Comput..

[5]  P. Harpe,et al.  Conjugacy growth series of some infinitely generated groups , 2016, 1603.07943.

[6]  M. Benson Growth series of finite extensions of ℤn are rational , 1983 .

[7]  J. Eon Topological density of lattice nets. , 2013, Acta crystallographica. Section A, Foundations of crystallography.

[8]  J. Eon Topological features in crystal structures: a quotient graph assisted analysis of underlying nets and their embeddings. , 2016, Acta crystallographica. Section A, Foundations and advances.

[9]  G. C. Shephard,et al.  Tilings and Patterns , 1990 .

[10]  Chaim Goodman-Strauss,et al.  Regular production systems and triangle tilings , 2009, Theor. Comput. Sci..

[11]  A. P. Shevchenko,et al.  Applied Topological Analysis of Crystal Structures with the Program Package ToposPro , 2014 .

[12]  N. Sloane,et al.  Algebraic Description of Coordination Sequences and Exact TopologicalDensities for Zeolites , 1996 .

[13]  Neil J. A. Sloane,et al.  Low–dimensional lattices. VII. Coordination sequences , 1997, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[14]  A. Shutov The Number of Words of a Given Length in the Planar Crystallographic Groups , 2005 .

[15]  M. O'keeffe,et al.  The Reticular Chemistry Structure Resource (RCSR) database of, and symbols for, crystal nets. , 2008, Accounts of chemical research.

[16]  N. J. A. Sloane,et al.  The On-Line Encyclopedia of Integer Sequences , 2003, Electron. J. Comb..

[17]  J. Eon Algebraic determination of generating functions for coordination sequences in crystal structures. , 2002, Acta crystallographica. Section A, Foundations of crystallography.

[18]  D. Knuth,et al.  Simple Word Problems in Universal Algebras , 1983 .

[19]  Richard L Gregory,et al.  Zap! , 1991, IEEE Spectrum.

[20]  Branko Grünbaum,et al.  Tilings by Regular Polygons , 1977 .

[21]  John J. Cannon,et al.  The Magma Algebra System I: The User Language , 1997, J. Symb. Comput..

[22]  Coordination sequences for root lattices and related graphs , 1997, cond-mat/9706122.

[23]  J. Eon,et al.  Infinite geodesic paths and fibers, new topological invariants in periodic graphs. , 2007, Acta crystallographica. Section A, Foundations of crystallography.

[24]  J. Conway,et al.  The Symmetries of Things , 2008 .

[25]  La Harpe,et al.  Topics in Geometric Group Theory , 2000 .

[26]  D. L. Johnson Presentations of groups , 1976 .

[27]  Jean-Guillaume Eon,et al.  Symmetry and Topology: The 11 Uninodal Planar Nets Revisited , 2018, Symmetry.

[28]  H. Coxeter,et al.  Generators and relations for discrete groups , 1957 .

[29]  J. Eon Topological density of nets: a direct calculation. , 2004, Acta crystallographica. Section A, Foundations of crystallography.

[30]  Darrah Chavey,et al.  Tilings by regular polygons—II: A catalog of tilings , 1989 .

[31]  M. O'keeffe,et al.  Coordination sequences for lattices , 1995 .