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Transactions of the American Mathematical Society

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The ninety-one types of isogonal tilings in the plane

Authors: Branko Grünbaum and G. C. Shephard
Journal: Trans. Amer. Math. Soc. 242 (1978), 335-353
MSC: Primary 05B45; Secondary 52A45
Erratum: Trans. Amer. Math. Soc. 249 (1979), 446-446.
MathSciNet review: 496813
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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.

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Keywords: Tiling, isogon, isogonal tiling, isohedral tiling, unform tiling, adjacency symbol, vertex symbol, vertex transitivity, symmetry, tessellation, type of tiling, classification of tilines
Article copyright: © Copyright 1978 American Mathematical Society