An example of uniform tiling in the Archeological Museum of Seville, Sevilla, Spain: rhombitrihexagonal tiling
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.
John Conway called these uniform duals Catalan tilings, in parallel to the Catalan solid polyhedra.
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 known nonconvex tilings, using star polygons, and reverse orientation vertex configurations. A further 28 uniform tilings are known using apeirogons. If zigzags are also allowed, there are 23 more known uniform tilings and 10 more known families depending on a parameter: in 8 cases the parameter is continuous, and in the other 2 it is discrete. The set is not known to be complete.
and 26 Related for: List of Euclidean uniform tilings information
convex uniformtilings (regular and semiregular) of the Euclidean plane, and their dual tilings. There are three regular and eight semiregular tilings in...
Euclidean plane tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler...
polygons Tilings and patterns, from list of 107 isohedral tilings, pp. 473–481 Tilings and patterns, uniformtilings that are not edge-to-edge Order in Space:...
Commons has media related to Uniformtiling 3-3-3-3-6 (snub trihexagonal tiling). Tilingsof regular polygons Listofuniformtilings Order in Space: A design...
geometry, the trihexagonal tiling is one of 11 uniformtilingsof the Euclidean plane by regular polygons. It consists of equilateral triangles and regular...
In geometry, the truncated trihexagonal tiling is one of eight semiregular tilingsof the Euclidean plane. There are one square, one hexagon, and one...
3 regular and 8 semiregular tilings in the plane. There are two distinct uniform colorings of a truncated square tiling. (Naming the colors by indices...
related to Uniformtiling 3-12-12 (truncated hexagonal tiling). Tilingsof regular polygons Listofuniformtilings Chavey, D. (1989). "Tilings by Regular...
the triangular tiling or triangular tessellation is one of the three regular tilingsof the Euclidean plane, and is the only such tiling where the constituent...
Commons has media related to Uniformtiling 3-4-6-4 (rhombitrihexagonal tiling). Tilingsof regular polygons Listofuniformtilings Conway, 2008, p288 table...
geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells...
non-edge-to-edge tilingsof the Euclidean plane are possible, including the family of Pythagorean tilings, tessellations that use two (parameterised) sizes of square...
square tiling, t{4,8}. Like the hexagonal tilingof the Euclidean plane, there are 3 uniform colorings of this hyperbolic tiling. The dual tiling V8.8.8...
symbol can be applied to all uniform polyhedra. However, the construction methods do not lead to all uniformtilings in Euclidean or hyperbolic space. The...
heptagon in Euclidean space is unable to generate uniformtilings alongside other polygons, like the regular pentagon. However, it is one of fourteen polygons...
Commons has media related to Uniformtiling 3-4-i-4. Listofuniform planar tilingsTilingsof regular polygons Uniformtilings in hyperbolic plane John H...