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- Uniform_tiling abstract "In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-transitive.Uniform tilings can exist in both the Euclidean plane and hyperbolic plane. Uniform tilings are related to the finite uniform polyhedra which can be considered uniform tilings of the sphere.Most uniform tilings can be made from a Wythoff construction starting with a symmetry group and a singular generator point inside of the fundamental domain. A planar symmetry group has a polygonal fundamental domain and can be represented by the group name represented by the order of the mirrors in sequential vertices.A fundamental domain triangle is (p q r), and a right triangle (p q 2), where p, q, r are whole numbers greater than 1. The triangle may exist as a spherical triangle, a Euclidean plane triangle, or a hyperbolic plane triangle, depending on the values of p, q and r.There are a number of symbolic schemes for naming these figures, from a modified Schläfli symbol for right triangle domains: (p q 2) → {p, q}. The Coxeter-Dynkin diagram is a triangular graph with p, q, r labeled on the edges. If r = 2, the graph is linear since order-2 domain nodes generate no reflections. The Wythoff symbol takes the 3 integers and separates them by a vertical bar (|). If the generator point is off the mirror opposite a domain node, it is given before the bar.Finally tilings can be described by their vertex configuration, the sequence of polygons around each vertex.All uniform tilings can be constructed from various operations applied to regular tilings. These operations as named by Norman Johnson are called truncation (cutting vertices), rectification (cutting vertices until edges disappear), and Cantellation (cutting edges). Omnitruncation is an operation that combines truncation and cantellation. Snubbing is an operation of Alternate truncation of the omnitruncated form. (See Uniform polyhedron#Wythoff construction operators for more details.)".
- Uniform_tiling thumbnail Tiling_Dual_Semiregular_V4-8-8_Tetrakis_Square-2-color-zoom.svg?width=300.
- Uniform_tiling wikiPageExternalLink uniftil.htm.
- Uniform_tiling wikiPageExternalLink sici?sici=0080-4614%2819540513%29246%3A916%3C401%3AUP%3E2.0.CO%3B2-4.
- Uniform_tiling wikiPageExternalLink tilings.html.
- Uniform_tiling wikiPageExternalLink tessel.htm.
- Uniform_tiling wikiPageExternalLink www.tess-elation.co.uk.
- Uniform_tiling wikiPageExternalLink ue2.
- Uniform_tiling wikiPageID "12126635".
- Uniform_tiling wikiPageLength "22155".
- Uniform_tiling wikiPageOutDegree "211".
- Uniform_tiling wikiPageRevisionID "666266647".
- Uniform_tiling wikiPageWikiLink Alternation_(geometry).
- Uniform_tiling wikiPageWikiLink Apeirogon.
- Uniform_tiling wikiPageWikiLink Apeirogonal_antiprism.
- Uniform_tiling wikiPageWikiLink Apeirogonal_prism.
- Uniform_tiling wikiPageWikiLink Branko_Grünbaum.
- Uniform_tiling wikiPageWikiLink Cantellation.
- Uniform_tiling wikiPageWikiLink Cantellation_(geometry).
- Uniform_tiling wikiPageWikiLink Category:Uniform_tilings.
- Uniform_tiling wikiPageWikiLink Coxeter-Dynkin_diagram.
- Uniform_tiling wikiPageWikiLink Coxeter_diagram.
- Uniform_tiling wikiPageWikiLink Coxeter_group.
- Uniform_tiling wikiPageWikiLink Coxeter–Dynkin_diagram.
- Uniform_tiling wikiPageWikiLink Dihedron.
- Uniform_tiling wikiPageWikiLink Elongated_triangular_tiling.
- Uniform_tiling wikiPageWikiLink Euclidean_plane.
- Uniform_tiling wikiPageWikiLink Euclidean_tilings_of_convex_regular_polygons.
- Uniform_tiling wikiPageWikiLink File:Order-3_heptakis_heptagonal_tiling.png.
- Uniform_tiling wikiPageWikiLink Frieze_group.
- Uniform_tiling wikiPageWikiLink Fundamental_domain.
- Uniform_tiling wikiPageWikiLink G.C._Shephard.
- Uniform_tiling wikiPageWikiLink Geoffrey_Colin_Shephard.
- Uniform_tiling wikiPageWikiLink Geometry.
- Uniform_tiling wikiPageWikiLink Great_rhombitriheptagonal_tiling.
- Uniform_tiling wikiPageWikiLink Great_rhombitrihexagonal_tiling.
- Uniform_tiling wikiPageWikiLink Great_rhombitrioctagonal_tiling.
- Uniform_tiling wikiPageWikiLink H.S.M._Coxeter.
- Uniform_tiling wikiPageWikiLink H._S._M._Coxeter.
- Uniform_tiling wikiPageWikiLink Harold_Scott_MacDonald_Coxeter.
- Uniform_tiling wikiPageWikiLink Heptagonal_tiling.
- Uniform_tiling wikiPageWikiLink Hexagonal_tiling.
- Uniform_tiling wikiPageWikiLink Hyperbolic_space.
- Uniform_tiling wikiPageWikiLink Isogonal_figure.
- Uniform_tiling wikiPageWikiLink J._C._P._Miller.
- Uniform_tiling wikiPageWikiLink JSTOR.
- Uniform_tiling wikiPageWikiLink List_of_convex_uniform_tilings.
- Uniform_tiling wikiPageWikiLink List_of_uniform_tilings.
- Uniform_tiling wikiPageWikiLink M._S._Longuet-Higgins.
- Uniform_tiling wikiPageWikiLink Michael_S._Longuet-Higgins.
- Uniform_tiling wikiPageWikiLink Non-Wythoffian.
- Uniform_tiling wikiPageWikiLink Norman_Johnson_(mathematician).
- Uniform_tiling wikiPageWikiLink Octagonal_tiling.
- Uniform_tiling wikiPageWikiLink Omnitruncation.
- Uniform_tiling wikiPageWikiLink Orbifold_notation.
- Uniform_tiling wikiPageWikiLink Order-3_heptagonal_tiling.
- Uniform_tiling wikiPageWikiLink Order-3_octagonal_tiling.
- Uniform_tiling wikiPageWikiLink Order-7_triangular_tiling.
- Uniform_tiling wikiPageWikiLink Order-8_triangular_tiling.
- Uniform_tiling wikiPageWikiLink Poincaré_disk_model.
- Uniform_tiling wikiPageWikiLink Pythagorean_tiling.
- Uniform_tiling wikiPageWikiLink Rectification_(geometry).
- Uniform_tiling wikiPageWikiLink Regular_polygon.
- Uniform_tiling wikiPageWikiLink Regular_tiling.
- Uniform_tiling wikiPageWikiLink Rhombitrihexagonal_tiling.
- Uniform_tiling wikiPageWikiLink Schlafli_symbol.
- Uniform_tiling wikiPageWikiLink Schläfli_symbol.
- Uniform_tiling wikiPageWikiLink Simplex.
- Uniform_tiling wikiPageWikiLink Small_rhombitrihexagonal_tiling.
- Uniform_tiling wikiPageWikiLink Snub_hexagonal_tiling.
- Uniform_tiling wikiPageWikiLink Snub_square_tiling.
- Uniform_tiling wikiPageWikiLink Snub_trihexagonal_tiling.
- Uniform_tiling wikiPageWikiLink Sphere.
- Uniform_tiling wikiPageWikiLink Spherical_triangle.
- Uniform_tiling wikiPageWikiLink Spherical_trigonometry.
- Uniform_tiling wikiPageWikiLink Square_tiling.
- Uniform_tiling wikiPageWikiLink Star_polygon.
- Uniform_tiling wikiPageWikiLink Symmetry_group.
- Uniform_tiling wikiPageWikiLink Tessellation.
- Uniform_tiling wikiPageWikiLink Triangular_tiling.
- Uniform_tiling wikiPageWikiLink Triheptagonal_tiling.
- Uniform_tiling wikiPageWikiLink Trihexagonal_tiling.
- Uniform_tiling wikiPageWikiLink Trioctagonal_tiling.
- Uniform_tiling wikiPageWikiLink Truncated_hexagonal_tiling.
- Uniform_tiling wikiPageWikiLink Truncated_square_tiling.
- Uniform_tiling wikiPageWikiLink Truncated_triheptagonal_tiling.
- Uniform_tiling wikiPageWikiLink Truncated_trihexagonal_tiling.
- Uniform_tiling wikiPageWikiLink Truncated_trioctagonal_tiling.
- Uniform_tiling wikiPageWikiLink Truncation_(geometry).
- Uniform_tiling wikiPageWikiLink Two-dimensional_space.
- Uniform_tiling wikiPageWikiLink Uniform_polyhedron.
- Uniform_tiling wikiPageWikiLink Uniform_polytope.
- Uniform_tiling wikiPageWikiLink Uniform_tilings_in_hyperbolic_plane.
- Uniform_tiling wikiPageWikiLink Vertex-transitive.
- Uniform_tiling wikiPageWikiLink Vertex_configuration.
- Uniform_tiling wikiPageWikiLink Vertex_figure.
- Uniform_tiling wikiPageWikiLink Wythoff_construction.
- Uniform_tiling wikiPageWikiLink Wythoff_symbol.
- Uniform_tiling wikiPageWikiLink File:Order-3_octakis_octagonal_tiling.png.
- Uniform_tiling wikiPageWikiLink File:Order-4_bisected_pentagonal_tiling.png.