The ninety-one types of isogonal tilings in the plane
HTML articles powered by AMS MathViewer
- by Branko Grünbaum and G. C. Shephard PDF
- Trans. Amer. Math. Soc. 242 (1978), 335-353 Request permission
Erratum: Trans. Amer. Math. Soc. 249 (1979), 446-446.
Abstract:
A tiling of the plane by closed topological disks of isogonal if its symmetries act transitively on the vertices of the tiling. Two isogonal tilings are of the same type provided the symmetries of the tiling relate in the same way every vertex in each to its set of neighbors. Isogonal tilings were considered in 1916 by A. V. Šubnikov and by others since then, without obtaining a complete classification. The isogonal tilings are vaguely dual to the isohedral (tile transitive) tilings, but the duality is not strict. In contrast to the existence of 81 isohedral types of planar tilings we prove the following result: There exist 91 types of isogonal tilings of the plane in which each tile has at least three neighbors.References
- E. S. Fedorov, Načala učeniya o figurah, Izdat. Akad. Nauk SSSR, Moscow, 1953 (Russian). MR 0062061 —, Systems of planigons as typical isohedra in the plane, Bull. Acad. Imp. Sci. Ser. (6) 10 (1916), 1523-1534. (Russian)
- Branko Grünbaum and G. C. Shephard, The eighty-one types of isohedral tilings in the plane, Math. Proc. Cambridge Philos. Soc. 82 (1977), no. 2, 177–196. MR 448233, DOI 10.1017/S0305004100053810
- Branko Grünbaum and Geoffrey C. Shephard, Tilings by regular polygons. Patterns in the plane from Kepler to the present, including recent results and unsolved problems, Math. Mag. 50 (1977), no. 5, 227–247. MR 500451, DOI 10.2307/2689529 F. Haag, Die regelmässigen Planteilungen. Z. Kristallographie 49 (1911), 360-369; Die regelmässigen Planteilungen und Punktsysteme, Ibid. 58 (1923), 478-489; Die Planigone von Fedorow, Ibid. 63 (1926), 179-186; Strukturformeln für Ebenenteilungen, Ibid. 83 (1932), 301-307. H. Heesch, Über topologisch gleichwertige Kristallbindungen, Z. Kristallographie 84 (1933), 399-407. R. Sauer, Ebene gleicheckige Polygongitter, Jber. Deutsch. Math.-Verein. 47 (1937), 115-124. A. Subnikov, To the question of the structure of crystals, Bull. Acad. Imp. Sci. Ser. (6) 10 (1916), 755-779. (Russian) A. V. Šubnikov and V. A. Kopcik, Symmetry in science and art, Nauka, Moscow, 1972. (Russian); English transl. A. V. Shubnikov and V. A. Koptsik, Symmetry in science and art, Plenum Press, New York and London, 1974.
Additional Information
- © Copyright 1978 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 242 (1978), 335-353
- MSC: Primary 05B45; Secondary 52A45
- DOI: https://doi.org/10.1090/S0002-9947-1978-0496813-3
- MathSciNet review: 496813