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Vertex-transitive tiling of the plane by regular polygons
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
Uniform_tiling
Symmetric subdivision in hyperbolic geometry
In hyperbolic geometry, a uniform hyperbolic tiling (or regular, quasiregular or semiregular hyperbolic tiling) is an edge-to-edge filling of the hyperbolic
Uniform tilings in hyperbolic plane
Uniform_tilings_in_hyperbolic_plane
hexagonal tiling and triangular tiling. (3,3,3), A ~ 2 {\displaystyle {\tilde {A}}_{2}} , [3[3]] There are a total of 32 uniform colorings of the 11 uniform tilings:
List of Euclidean uniform tilings
List_of_Euclidean_uniform_tilings
Regular tiling of a two-dimensional space
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex
Hexagonal_tiling
Subdivision of the plane into polygons that are all regular
tiling Regular polyhedron (the Platonic solids) Semiregular polyhedron (including the Archimedean solids) Tessellation Tiling with rectangles Uniform
Euclidean tilings by convex regular polygons
Euclidean_tilings_by_convex_regular_polygons
Uniform tiling of the plane with regular polygons
In geometry of the Euclidean plane, the 3-4-3-12 tiling is one of 20 2-uniform tilings of the Euclidean plane by regular polygons, containing regular
3-4-3-12_tiling
k-uniform tiling is a tiling of tilings of the plane by convex regular polygons, connected edge-to-edge, with k types of vertices. The 1-uniform tiling
List_of_k-uniform_tilings
Covering by shapes without overlaps or gaps
wallpaper groups. A tiling that lacks a repeating pattern is called "non-periodic". An aperiodic tiling uses a small set of tile shapes that cannot form
Tessellation
Regular tiling of the plane
regular tilings of the plane. The other two are the square tiling and the hexagonal tiling. There are 9 distinct uniform colorings of a triangular tiling. (Naming
Triangular_tiling
Semiregular tiling of the Euclidean plane
degenerate into edges, a triangular tiling results, constructed as a snub triangular tiling, . There is one related 2-uniform tiling, having hexagons dissected
Rhombitrihexagonal_tiling
Convex polygon which can tile the plane by itself
exists a 13-8 slab tiling and a 14-10 non-clock tiling. Finally, there are 7-5 tilings using all clock planigons: Each uniform tiling corresponds to a circle
Planigon
Semiregular tiling of the Euclidean plane
In geometry, the snub hexagonal tiling (or snub trihexagonal tiling) is a semiregular tiling of the Euclidean plane. There are four triangles and one hexagon
Snub_trihexagonal_tiling
Uniform tiling of the Euclidean plane
truncated trihexagonal tiling has three related 2-uniform tilings, one being a 2-uniform coloring of the semiregular rhombitrihexagonal tiling. The first dissects
Truncated_trihexagonal_tiling
Tiling of the plane by pentagons
geometry, a pentagonal tiling is a tiling of the plane where each individual piece is in the shape of a pentagon. A regular pentagonal tiling on the Euclidean
Pentagonal_tiling
Tiling of a plane by regular hexagons and equilateral triangles
In geometry, the trihexagonal tiling is one of 11 uniform tilings of the Euclidean plane by regular polygons. It consists of equilateral triangles and
Trihexagonal_tiling
Spatial tiling of convex uniform polyhedra
unique honeycombs from the square tiling, but all 6 tiling truncations are listed below for completeness, and tiling images are shown by colors corresponding
Convex_uniform_honeycomb
Regular tiling of the Euclidean plane
In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane consisting of four squares around every vertex
Square_tiling
Uniform tiling of the plane using regular polygons
In geometry of the Euclidean plane, a 33344-33434 tiling is one of two of 20 2-uniform tilings of the Euclidean plane by regular polygons. They contains
33344-33434_tiling
Tiling by squares of two sizes
A Pythagorean tiling or two squares tessellation is a tiling of the Euclidean plane by squares of two different sizes, in which each square touches four
Pythagorean_tiling
Uniform Tiling
In geometry of the Euclidean plane, the 3-4-6-12 tiling is one of 20 2-uniform tilings of the Euclidean plane by regular polygons, containing regular
3-4-6-12_tiling
Semiregular tiling of the plane
tiling is a semiregular tiling of the Euclidean plane. There are three triangles and two squares on each vertex. It is named as a triangular tiling elongated
Elongated_triangular_tiling
Tiling of the hyperbolic plane
In geometry, the heptagonal tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of {7,3}, having three regular heptagons
Heptagonal_tiling
Uniform tiling of the hyperbolic plane
Commons has media related to Uniform tiling 4-6-4-6. Square tiling Tilings of regular polygons List of uniform planar tilings List of regular polytopes John
Tetrahexagonal_tiling
40 potential uniform polyhedra with degenerate vertex figures which have overlapping edges (not counted by Coxeter); The uniform tilings (infinite polyhedra)
List_of_uniform_polyhedra
Classification of a two-dimensional repetitive pattern
the colorings of the snub square tiling (see also at pg) 4 co-uniform tiling (fractalization of snub square tiling) Orbifold signature: 333 Coxeter notation:
Wallpaper_group
Geometric tiling
related to Uniform tiling 3-4-7-4. Rhombitrihexagonal tiling Order-3 heptagonal tiling Tilings of regular polygons List of uniform tilings Kagome lattice
Rhombitriheptagonal_tiling
Polytope or tiling whose vertices are identical
G. C. (1987). Tilings and Patterns. W. H. Freeman and Company. ISBN 0-7167-1193-1. (p. 33 k-isogonal tiling, p. 65 k-uniform tilings) Weisstein, Eric
Isogonal_figure
tiling. There are different notations for expressing these uniform solutions, Wythoff symbol, Coxeter diagram, and Coxeter's t-notation. Simple tiles
Lists of uniform tilings on the sphere, plane, and hyperbolic plane
Lists_of_uniform_tilings_on_the_sphere,_plane,_and_hyperbolic_plane
Semiregular tiling of a plane
Like the uniform polyhedra there are eight uniform tilings that can be based from the regular hexagonal tiling (or the dual triangular tiling). Drawing
Truncated_hexagonal_tiling
Regular paracompact honeycomb
the hexagonal tiling honeycomb is {6,3,3}. Since that of the hexagonal tiling is {6,3}, this honeycomb has three such hexagonal tilings meeting at each
Hexagonal_tiling_honeycomb
Tiling of the plane with 60° rhombi
is the dual tiling of the trihexagonal tiling or kagome lattice. As the dual to a uniform tiling, it is one of eleven possible Laves tilings, and in the
Rhombille_tiling
the hexagonal tiling honeycomb, . It contains hexagonal tiling facets with a tetrahedral vertex figure. The bitruncated triangular tiling honeycomb, ,
Triangular_tiling_honeycomb
infinite-order triangular tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of t{3,∞}. The dual of this tiling represents the fundamental
Truncated infinite-order triangular tiling
Truncated_infinite-order_triangular_tiling
Semiregular tiling of the hyperbolic plane
to Uniform tiling 3-7-3-7. Trihexagonal tiling - 3.6.3.6 tiling Rhombille tiling - dual V3.6.3.6 tiling Tilings of regular polygons List of uniform tilings
Triheptagonal_tiling
Uniform tiling of the hyperbolic plane
In geometry, the tritetragonal tiling or alternated octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbols of {(4,3,3)}
Alternated_octagonal_tiling
Uniform tiling of the hyperbolic plane
to Uniform tiling 4-5-4-5. Binary tiling, an aperiodic tiling of the hyperbolic plane by pentagons Uniform tilings in hyperbolic plane List of regular
Tetrapentagonal_tiling
Isogonal honeycomb of uniform polytope facets
convex uniform polytopes used in uniform polyhedron, uniform 4-polytope, uniform 5-polytope, uniform 6-polytope, uniform tiling, and convex uniform honeycomb
Uniform_honeycomb
Semiregular tiling of the hyperbolic plane
media related to Uniform tiling 3-14-14. Truncated hexagonal tiling Heptagonal tiling Tilings of regular polygons List of uniform tilings John H. Conway
Truncated_heptagonal_tiling
Prism with an infinite-sided polygon base
Similarly to the uniform polyhedra and the uniform tilings, eight uniform tilings may be based from the regular apeirogonal tiling. The rectified and
Apeirogonal_prism
Semiregular tiling of the hyperbolic plane
geometry, the order-7 truncated triangular tiling, sometimes called the hyperbolic soccerball, is a semiregular tiling of the hyperbolic plane. There are two
Truncated order-7 triangular tiling
Truncated_order-7_triangular_tiling
Tiling of euclidean or hyperbolic space of three or more dimensions
that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions. Its dimension can be clarified
Honeycomb_(geometry)
Semiregular tiling of the plane
square tiling (quadrille). There are 3 regular and 8 semiregular tilings in the plane. There are two distinct uniform colorings of a snub square tiling. (Naming
Snub_square_tiling
Semiregular tiling of the hyperbolic plane
one uniform coloring of a truncated triheptagonal tiling. (Naming the colors by indices around a vertex: 123.) Each triangle in this dual tiling, order
Truncated triheptagonal tiling
Truncated_triheptagonal_tiling
Regular tiling of the hyperbolic plane
coloring represents the uniform tiling t1{5,5} and as a quasiregular tiling is called a pentapentagonal tiling. This tiling is topologically related
Order-4_pentagonal_tiling
(not just uniform) tiling of the genus 3 surface by 56 equilateral triangles, meeting at 24 vertices, each with degree 7. This regular tiling is significant
Small_cubicuboctahedron
Semiregular tiling
In geometry, the truncated square tiling is a semiregular tiling by regular polygons of the Euclidean plane with one square and two octagons on each vertex
Truncated_square_tiling
order-5 hexagonal tiling honeycomb is {6,3,5}. Since that of the hexagonal tiling is {6,3}, this honeycomb has five such hexagonal tilings meeting at each
Order-5 hexagonal tiling honeycomb
Order-5_hexagonal_tiling_honeycomb
order-4 hexagonal tiling honeycomb is {6,3,4}. Since that of the hexagonal tiling is {6,3}, this honeycomb has four such hexagonal tilings meeting at each
Order-4 hexagonal tiling honeycomb
Order-4_hexagonal_tiling_honeycomb
Isogonal polyhedron with regular faces
polyhedron Semiregular polyhedron Polyhedron model Pseudo-uniform polyhedron Uniform tiling Uniform tilings in hyperbolic plane Diudea (2018), p. 40. Coxeter
Uniform_polyhedron
similar to the 2D hyperbolic uniform triapeirogonal tiling, r{∞,3}, with triangle and apeirogonal faces. The truncated square tiling honeycomb, t{4,4,3}, has
Square_tiling_honeycomb
Plane tiling with two infinite-sided polygons
Similarly to the uniform polyhedra and the uniform tilings, eight uniform tilings may be based from the regular apeirogonal tiling. The rectified and
Order-2_apeirogonal_tiling
hexagonal tiling honeycomb is {6,3,6}. Since that of the hexagonal tiling of the plane is {6,3}, this honeycomb has six such hexagonal tilings meeting at
Order-6 hexagonal tiling honeycomb
Order-6_hexagonal_tiling_honeycomb
Natural number
generate uniform tilings alongside other polygons, like the regular pentagon. However, it is one of fourteen polygons that can fill a plane-vertex tiling, in
7
Uniform tiling of the hyperbolic plane
Wikimedia Commons has media related to Uniform tiling 3-8-4-8. Hyperbolic space Square tiling Uniform tilings in hyperbolic plane List of regular polytopes
Cantic order-4 hexagonal tiling
Cantic_order-4_hexagonal_tiling
In geometry, a uniform coloring is a property of a uniform figure (uniform tiling or uniform polyhedron) that is colored to be vertex-transitive. Different
Uniform_coloring
order-4 square tiling honeycomb. The quarter order-4 square tiling honeycomb, q{4,4,4}, , or , has truncated square tiling and square tiling facets, with
Order-4 square tiling honeycomb
Order-4_square_tiling_honeycomb
Uniform tiling of the hyperbolic plane
In geometry, the truncated order-7 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{4,7}. John H. Conway, Heidi
Truncated order-7 square tiling
Truncated_order-7_square_tiling
Uniform tiling of the hyperbolic plane
heptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{7,4}. There are two uniform constructions of this tiling, first
Truncated order-4 heptagonal tiling
Truncated_order-4_heptagonal_tiling
Semiregular tiling of the hyperbolic plane
In geometry, the order-3 snub heptagonal tiling is a semiregular tiling of the hyperbolic plane. There are four triangles and one heptagon on each vertex
Snub_triheptagonal_tiling
Uniform tiling of the hyperbolic plane
tiling, r{6,4}, as well as an expanded order-4 hexagonal tiling or expanded order-6 square tiling. There are two uniform constructions of this tiling
Rhombitetrahexagonal_tiling
Pattern in hyperbolic geometry
Commons has media related to Uniform tiling 4-12-12. Square tiling Tilings of regular polygons List of uniform planar tilings List of regular polytopes Weisstein
Truncated order-4 hexagonal tiling
Truncated_order-4_hexagonal_tiling
Semiregular tiling of the hyperbolic plane
In geometry, the truncated tetrahexagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one dodecagon
Truncated tetrahexagonal tiling
Truncated_tetrahexagonal_tiling
also be divided between compact, paracompact and divergent cases. The uniform tilings are the simplest application of these mutations, although more complex
Uniform tiling symmetry mutations
Uniform_tiling_symmetry_mutations
truncated order-8 octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{8,8}. This tiling can also be constructed in
Truncated order-8 octagonal tiling
Truncated_order-8_octagonal_tiling
Uniform tiling of the hyperbolic plane
rhombitetrapentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,2{4,5}. The dual is called the deltoidal tetrapentagonal tiling with
Rhombitetrapentagonal_tiling
itself in the process. The number of times the tiling winds round the sphere is the density of the tiling, and is denoted μ. Jonathan Bowers' short names
List of uniform polyhedra by Schwarz triangle
List_of_uniform_polyhedra_by_Schwarz_triangle
Concept in geometry
tiling. Infinite-order tetrahedral honeycomb List of regular polytopes List of uniform planar tilings Tilings of regular polygons Triangular tiling Uniform
Infinite-order triangular tiling
Infinite-order_triangular_tiling
Semi regularly Tiling
In geometry, the truncated hexaoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one dodecagon, and one hexakaidecagon
Truncated hexaoctagonal tiling
Truncated_hexaoctagonal_tiling
Generalisation of dice with identical faces
tiling (m = 1) has congruent faces, either directly or reflectively, which occur in one or more symmetry positions. An m-hedral polyhedron or tiling has
Isohedral_figure
Euclidean 3-space) 1 p + 1 q = 1 2 : Euclidean plane tiling 1 p + 1 q < 1 2 : Hyperbolic plane tiling {\displaystyle {\begin{aligned}&{\frac {1}{p}}+{\frac
List_of_regular_polytopes
of uniform tilings Uniform tilings in hyperbolic plane Archimedean tiling Square tiling Triangular tiling Hexagonal tiling Truncated square tiling Snub
List_of_mathematical_shapes
order-6 square tiling, h2{4,6} By *663 symmetry, this tiling can be constructed as an omnitruncation, t{(6,6,3)}: The dual to this tiling represent the
Truncated order-6 hexagonal tiling
Truncated_order-6_hexagonal_tiling
Commons has media related to Uniform tiling 3-3-5-3-6. Square tiling Tilings of regular polygons List of uniform planar tilings List of regular polytopes
Snub_pentahexagonal_tiling
"Hyperbolic tiling". MathWorld. Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld. Hyperbolic and Spherical Tiling Gallery KaleidoTile 3: Educational
Quarter_order-6_square_tiling
to Uniform tiling 3-3-3-3-3-i. Square tiling Uniform tilings in hyperbolic plane List of regular polytopes Weisstein, Eric W. "Hyperbolic tiling". MathWorld
Snub infinite-order triangular tiling
Snub_infinite-order_triangular_tiling
Notation for tesselations
three numbers and a vertical bar. It represents one uniform polyhedron or tiling, although the same tiling/polyhedron can have different Wythoff symbols from
Wythoff_symbol
Mathematics book
topics in tiling theory: colored patterns and tilings, polygonal tilings, aperiodic tilings, Wang tiles, and tilings with unusual kinds of tiles. Each chapter
Tilings_and_patterns
Regular tiling of the hyperbolic plane
coloring represents the uniform tiling t1{7,7} and as a quasiregular tiling is called a heptaheptagonal tiling. This tiling is topologically related
Order-4_heptagonal_tiling
Regular tiling of the hyperbolic plane
tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {8,4}. Its checkerboard coloring can be called a octaoctagonal tiling,
Order-4_octagonal_tiling
Semiregular tiling of the hyperbolic plane
to Uniform tiling 6-6-8. Triangular tiling Order-3 octagonal tiling Order-8 triangular tiling Tilings of regular polygons List of uniform tilings John
Truncated order-8 triangular tiling
Truncated_order-8_triangular_tiling
Pictorial representation of symmetry
(called branches) representing a Coxeter group or sometimes a uniform polytope or uniform tiling constructed from the group. A class of closely related objects
Coxeter–Dynkin_diagram
Manufactured pieces for covering surfaces
The techniques and tools for tiling is advanced, evidenced by the fine workmanship and close fit of the tiles. Such tiling can be seen in Ruwanwelisaya
Tile
Semi-regular arrangement of squares, decagons, and dodecagons
In geometry, the truncated tetrahexagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one decagon, and one dodecagon
Truncated pentahexagonal tiling
Truncated_pentahexagonal_tiling
Tiling of hyperbolic 3-space by uniform polyhedra
trihexagonal tiling as the vertex figure), and hypercompact for p>6. Again, the truncated and rectified versions of these honeycombs are still uniform. The cases
Uniform honeycombs in hyperbolic space
Uniform_honeycombs_in_hyperbolic_space
Uniform tiling of the hyperbolic plane in geometry
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Rhombipentahexagonal_tiling
tetrahedral-square tiling honeycomb is a paracompact uniform honeycomb, constructed from tetrahedron, cuboctahedron and square tiling cells, in a rhombicuboctahedron
Tetrahedral-square tiling honeycomb
Tetrahedral-square_tiling_honeycomb
Semiregular tiling of the hyperbolic plane
Commons has media related to Uniform dual tiling V 4-6-14. In geometry, the 3-7 kisrhombille tiling is a semiregular dual tiling of the hyperbolic plane.
3-7_kisrhombille
In geometry, the tetrakis square tiling is a tiling of the Euclidean plane. It is a square tiling with each square divided into four isosceles right triangles
Tetrakis_square_tiling
Periodic tiling of the hyperbolic disk
In geometry, the order-3 apeirogonal tiling is a regular tiling of the hyperbolic plane. It is represented by the Schläfli symbol {∞,3}, having three regular
Order-3_apeirogonal_tiling
"Hyperbolic tiling". MathWorld. Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld. Hyperbolic and Spherical Tiling Gallery KaleidoTile 3: Educational
Truncated order-5 square tiling
Truncated_order-5_square_tiling
Antiprism with an infinite-sided polygon base
Similarly to the uniform polyhedra and the uniform tilings, eight uniform tilings may be based from the regular apeirogonal tiling. The rectified and
Apeirogonal_antiprism
Regular pattern for geometric tiling
LCCN 99035678. Wikimedia Commons has media related to Uniform tiling 7-14-14. Weisstein, Eric W. "Hyperbolic tiling". MathWorld. Weisstein, Eric W. "Poincaré hyperbolic
Truncated order-7 heptagonal tiling
Truncated_order-7_heptagonal_tiling
heptagon. Wikimedia Commons has media related to Uniform tiling 3-3-7-3-7. Square tiling Uniform tilings in hyperbolic plane List of regular polytopes John
Snub_heptaheptagonal_tiling
Uniform tiling of the hyperbolic plane
media related to Uniform tiling 4-10-10. Uniform tilings in hyperbolic plane List of regular polytopes Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
Truncated order-4 pentagonal tiling
Truncated_order-4_pentagonal_tiling
tiling-triangular tiling honeycomb is a paracompact uniform honeycomb, constructed from triangular tiling, hexagonal tiling, and trihexagonal tiling cells
Hexagonal tiling-triangular tiling honeycomb
Hexagonal_tiling-triangular_tiling_honeycomb
Concept in mathematics
In geometry, the snub tritetratrigonal tiling or snub order-8 triangular tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbols of
Snub order-8 triangular tiling
Snub_order-8_triangular_tiling
Type of geometrical tiling
Commons has media related to Uniform tiling 5-6-5-6. Square tiling Tilings of regular polygons List of uniform planar tilings List of regular polytopes Weisstein
Pentahexagonal_tiling
Semiregular tiling of the hyperbolic plane
truncated order-8 hexagonal tiling is a semiregular tiling of the hyperbolic plane. It has Schläfli symbol of t{6,8}. This tiling can also be constructed
Truncated order-8 hexagonal tiling
Truncated_order-8_hexagonal_tiling
infinite-order square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{4,∞}. In (*∞44) symmetry this tiling has 3 colors. Bisecting
Truncated infinite-order square tiling
Truncated_infinite-order_square_tiling
Concept in mathematics
a cantic octagonal tiling, h2{8,3}. Wikimedia Commons has media related to Uniform tiling 3-6-4-6. Square tiling Uniform tilings in hyperbolic plane
Cantic_octagonal_tiling
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