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Spherical tilings with dihedral symmetry exist for all =,,, … many with digon faces which become degenerate polyhedra. Two of the eight forms (Rectified and cantillated) are replications and are skipped in the table.
In 1981, Grünbaum, Miller, and Shephard, in their paper Uniform Tilings with Hollow Tiles, list 25 tilings using the first two expansions and 28 more when the third is added (making 53 using Coxeter et al.'s definition). When the fourth is added, they list an additional 23 uniform tilings and 10 families (8 depending on continuous parameters ...
Each Convex regular 4-polytope has a rectified form as a uniform 4-polytope. A regular 4-polytope {p,q,r} has cells {p,q}. Its rectification will have two cell types, a rectified {p,q} polyhedron left from the original cells and {q,r} polyhedron as new cells formed by each truncated vertex.
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 ...
It can be seen as constructed as a rectified tetraoctagonal tiling, r ... Related polyhedra and tiling. n42 symmetry mutation of expanded tilings: n.4.4.4 .
It can be seen as constructed as a rectified tetraheptagonal tiling, r ... Related polyhedra and tiling. n42 symmetry mutation of expanded tilings: n.4.4.4 .
The following list of polygons, polyhedra and polytopes gives the names of various classes of polytopes and lists some specific examples. ... Rectified cubic honeycomb;
In geometry, the triheptagonal tiling is a semiregular tiling of the hyperbolic plane, representing a rectified Order-3 heptagonal tiling. There are two triangles and two heptagons alternating on each vertex. It has Schläfli symbol of r{7,3}. Compare to trihexagonal tiling with vertex configuration 3.6.3.6.
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