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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.
A vertex configuration can also be represented as a polygonal vertex figure showing the faces around the vertex. This vertex figure has a 3-dimensional structure since the faces are not in the same plane for polyhedra, but for vertex-uniform polyhedra all the neighboring vertices are in the same plane and so this plane projection can be used to visually represent the vertex configuration.
1-uniform tilings include 3 regular tilings, and 8 semiregular ones, with 2 or more types of regular polygon faces. There are 20 2-uniform tilings, 61 3-uniform tilings, 151 4-uniform tilings, 332 5-uniform tilings and 673 6-uniform tilings. Each can be grouped by the number m of distinct vertex figures, which are also called m-Archimedean tilings.
Clusters of planigons which cannot tile the plane. Note the 8-cluster of V3.8.24 and the 10-cluster of V3.10.15 imply overlaps for the 24-gons and 15-gons, respectively. Also, V4.5.20 and V5 2.10 can generate lines and curves, but those cannot be completed without overlap.
10: 8{4} +2{8} Enneagonal prism: 4.4.9 ... The white polygon lines represent the "vertex figure" polygon. The colored faces are included on the vertex figure images ...
A regular digon has both angles equal and both sides equal and is represented by Schläfli symbol {2}. It may be constructed on a sphere as a pair of 180 degree arcs connecting antipodal points, when it forms a lune. The digon is the simplest abstract polytope of rank 2. A truncated digon, t{2} is a square, {4}. An alternated digon, h{2} is a ...
A regular polygon can also be represented by its Coxeter-Dynkin diagram, , and its uniform truncation , and its complete truncation . The graph represents Coxeter group I 2 (n), with each node representing a mirror, and the edge representing the angle π/ n between the mirrors, and a circle is given around one or both mirrors to show which ones ...
[9] [10] The square can be replaced by any parallelogram (the translation surfaces obtained are exactly those obtained as ramified covers of a flat torus). In fact the Veech group is arithmetic (which amounts to it being commensurable to the modular group) if and only if the surface is tiled by parallelograms. [10]