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Right triangle domains can have as few as 1 face type, making regular forms, while general triangles have at least 2 triangle types, leading at best to a quasiregular tiling. There are different notations for expressing these uniform solutions, Wythoff symbol , Coxeter diagram , and Coxeter's t-notation.
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If a tiling made of 2 apeirogons is also counted, the total can be considered 39 uniform tilings. In 1981, Grünbaum, Miller, and Shephard, in their paper Uniform Tilings with Hollow Tiles, list 25 tilings using the first two expansions and 28 more when the third is added (making 53 using Coxeter et al.'s definition). When the fourth is added ...
Order-8 square tiling Poincaré disk model of the hyperbolic plane: Type: Hyperbolic regular tiling: Vertex configuration: 4 8: Schläfli symbol {4,8} Wythoff symbol: 8 | 4 2 Coxeter diagram: Symmetry group [8,4], (*842) Dual: Order-4 octagonal tiling: Properties: Vertex-transitive, edge-transitive, face-transitive
An example of uniform tiling in the Archeological Museum of Seville, Sevilla, Spain: rhombitrihexagonal tiling Regular tilings and their duals drawn by Max Brückner in Vielecke und Vielflache (1900) This table shows the 11 convex uniform tilings (regular and semiregular) of the Euclidean plane , and their dual tilings.
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The triangular tiling has Schläfli symbol of {3,6}. English mathematician John Conway called it a deltille , named from the triangular shape of the Greek letter delta (Δ). The triangular tiling can also be called a kishextille by a kis operation that adds a center point and triangles to replace the faces of a hextille .