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This table shows the 11 convex uniform tilings (regular and semiregular) of the Euclidean plane, and their dual tilings. There are three regular and eight semiregular tilings in the plane. The semiregular tilings form new tilings from their duals, each made from one type of irregular face. Uniform tilings are listed by their vertex configuration, the sequence of faces that exist on each vertex. For example ''4.8.8'' means one square and two octagons on a vertex. These 11 uniform tilings have 32 different ''uniform colorings''. A uniform coloring allows identical sided polygons at a vertex to be colored differently, while still maintaining vertex-uniformity and transformational congruence between vertices. (Note: Some of the tiling images shown below are ''not'' color-uniform) In addition to the 11 convex uniform tilings, there are also 14 nonconvex tilings, using star polygons, and reverse orientation vertex configurations. The dual tilings of these tilings have vertices at the centers of the regular polygons, and edges connecting centers of regular polygons that share an edge. The tiles of the dual tiling are not necessarily regular themselves, but each vertex has edges evenly spaced around it. These dual tilings are listed by their face configuration, the number of faces at each vertex of a face. For example ''V4.8.8'' means isosceles triangle tiles with one corner with four triangles, and two corners containing eight triangles. In the 1987 book, ''Tilings and Patterns'', Branko Grünbaum calls the vertex-uniform tilings ''Archimedean'' in parallel to the Archimedean solids, and their dual tilings ''Laves tilings'' in honor of crystallographer Fritz Laves. John Conway calls the duals ''Catalan tilings'', in parallel to the Catalan solid polyhedra.〔The Symmetries of things, Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, Euclidean Plane Tessellations, p. 288〕 == Convex uniform tilings of the Euclidean plane == All reflectional forms can be made by Wythoff constructions, represented by Wythoff symbols, or Coxeter-Dynkin diagrams, each operating upon one of three Schwarz triangle (4,4,2), (6,3,2), or (3,3,3), with symmetry represented by Coxeter groups: (), (), or . Alternated forms such as the snub can also be represented by special markups within each system. Only one uniform tiling can't be constructed by a Wythoff process, but can be made by an elongation of the triangular tiling. An orthogonal mirror construction () also exists, seen as two sets of parallel mirrors making a rectangular fundamental domain. If the domain is square, this symmetry can be doubled by a diagonal mirror into the () family. Families: * (4,4,2), , () * (6,3,2), , 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「List of convex uniform tilings」の詳細全文を読む スポンサード リンク
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