FAR EASTERN BRANCH OF THE RUSSIAN ACADEMY OF SCIENCES
INSTITUTE OF APPLIED MATHEMATICS KHABAROVSK DIVISION
Far Eastern Mathematical Journal
On a number of polyhex plane tilings |
Shutov A.V., Kolomeykina E.V. |
2017, issue 2, P. 257-265 |
Abstract |
| A tiling is called a lattice tiling if there is a group of translations which acts on the set of the tiles transitively. In the paper the low and upper bounds for the number of lattice tilings of plane with centrally symmetrical polyhexes are found |
Keywords: tilings, lattice tilings, polyhexes, self-avoiding walks |
Download the article (PDF-file) |
References |
| [1] D. Klarner, “A Cell growth problems”, Cand. J. Math., 19, (1967), 851–863. [2] M. Gardner, Puteshestvie vo vremeni, Mir, M, 1990. [3] S. Golomb, Polimino, Mir, M, 1975. [4] J. Myers, “Polyomino, polyhex and polyiamond tiling”, http://www.srcf.ucam.org/ jsm28/tiling/. [5] A.V. Maleev, “Algorithm and computer-program search for variants of polyhex packing in plane”, Crystallography Reports, 60:6, (2015), 986–992. [6] H. Fukuda, N. Mutoh, G. Nakamura, D. Schattschneider, “Enumeration of Polyominoes, Polyiamonds and Polyhexes for Isohedral Tilings with Rotational Symmetry”, KyotoCGGT LNCS 4535 H. Ito et al. (Eds.), Springer-Verlag, Berlin, 2008, 68–78. [7] M. Gardner, “Ch. 11. Polyhexes and Polyaboloes”, Mathematical Magic Show, New York, 1978, 146–159. [8] J.V. Knop, K. Szymanski, Z. Jericevi?c, N. Trinajsti?c, “On the total number of polyhexes”, Match: Commun. Math. Chem., 16, (1984), 119–134. [9] D.A. Klarner, R.L. Rivest, “A procedure for improving the upper bound for the number of n-ominoes”, Canad. J. Math., 25, (1973), 585–602. [10] F. Harary, “The cell growth problem and its attempted solutions”, Beitrage zur Grathen- theorie, Teubner, Leipzig, 1968, 49–60. [11] R.C. Read, “Contributions to the cell-growth problem”, Canad. J. Math., 14, (1962), 1–20. [12] H. Fukuda, N. Mutoh, G. Nakamura, D. Schattschneider, “A Method to Generate Polyominoes and Polyiamonds for Tilings with Rotational Symmetry”, Graphs and Combinatorics, 23, (2007), 259–267. [13] J.R. Dias, “A Periodic Table for Polycyclic Aromatic Hydrocarbons. 1. Isomer Enumeration of Fused Polycyclic Aromatic Hydrocarbon”, J. Chem. Inf. Comput. Sci., 22, (1982), 15-22. [14] J.R. Dias, “A Periodic Table for Polycyclic Aromatic Hydrocarbons. 2. Polycyclic Aromatic Hydrocarbons Conteining Tetragonal, Pentagonal”, J. Chem. Inf. Comput. Sci., 22, (1982), 15-22. [15] M. Gardner, Matematicheskie golovolomki i razvlecheniia, Mir, M, 1999. [16] M. Gardner, Matematicheskie dosugi, Mir, M, 1972. [17] M. Gardner, Matematicheskie novelly, Mir, M, 1974. [18] G.C. Rhoads, “Planar tilings by polyominoes, polyhexes, and polyiamonds”, Journal of Computational and Applied Mathematics, 174, (2005), 329–353. [19] A.V. Maleev, A.V. Shutov, “O chisle transliatsionnykh razbienii ploskosti na polimino”, Trudy IX Vserossiiskoi nauchnoi shkoly "Matematicheskie issledovaniia v estestvennykh naukakh", Apatity, 2013, 101–106. [20] S. Brlek, A. Frosini, S. Rinaldi, L. Vuillon, “Tilings by translation: enumeration by a rational language approach”, The electronic journal of combinatorics, 13, (2006). [21] A.V. Shutov, E.V. Kolomeikina, “Otsenka chisla reshetchatykh razbienii ploskosti na tsentral'no-simmetrichnye polimino zadannoi ploshchadi”, Modelirovanie i Analiz Informatsionnykh Sistem, 20, (2014), 148–157. [22] D. Schattschneider, “Will it Tile? Try the Conway Criterion!”, Mathematics Magazine, 53:4, (1980), 224–233. [23] H. Duminil-Copin, S. Smirnov, “The connective constant of the honeycomb lattice equals *, Annals of Mathematics, 175:3, (2012), 1653–1665. [24] Iu.V. Nesterenko, A.I. Galochkin, A.B. Shidlovskii, Vvedenie v teoriiu chisel, Izdatel'stvo Moskovskogo Universiteta, M, 1984. |