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Rhombitrihexagonal tiling. In geometry, the rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. There are one triangle, two squares, and one hexagon on each vertex. It has Schläfli symbol of rr {3,6}. John Conway calls it a rhombihexadeltille. [1]
Uniform colorings. There are a total of 32 uniform colorings of the 11 uniform tilings: Triangular tiling – 9 uniform colorings, 4 wythoffian, 5 nonwythoffian. Square tiling – 9 colorings: 7 wythoffian, 2 nonwythoffian. Hexagonal tiling – 3 colorings, all wythoffian. Trihexagonal tiling – 2 colorings, both wythoffian.
The (6,4,2) triangular hyperbolic tiling that inspired Escher. Escher became interested in tessellations of the plane after a 1936 visit to the Alhambra in Granada, Spain, [3] [4] and from the time of his 1937 artwork Metamorphosis I he had begun incorporating tessellated human and animal figures into his artworks.
The symmetry group of the tiling is the (2,3,7) triangle group, and a fundamental domain for this action is the (2,3,7) Schwarz triangle.This is the smallest hyperbolic Schwarz triangle, and thus, by the proof of Hurwitz's automorphisms theorem, the tiling is the universal tiling that covers all Hurwitz surfaces (the Riemann surfaces with maximal symmetry group), giving them a triangulation ...
Properties. Vertex-transitive, edge-transitive, face-transitive. In geometry, the order-6 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,6}.
Three of them – (7 3 2), (5 4 2), and (4 3 3) – and no others, are minimal in the sense that if any of their defining numbers is replaced by a smaller integer the resulting pattern is either Euclidean or spherical rather than hyperbolic; conversely, any of the numbers can be increased (even to infinity) to generate other hyperbolic patterns.
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M22 is one of the 26 sporadic groups and was introduced by Mathieu (1861, 1873). It is a 3-fold transitive permutation group on 22 objects. The Schur multiplier of M 22 is cyclic of order 12, and the outer automorphism group has order 2. There are several incorrect statements about the 2-part of the Schur multiplier in the mathematical literature.