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Isogonal polytope with regular facets
In
geometry , by
Thorold Gosset 's definition a semiregular polytope is usually taken to be a
polytope that is
vertex-transitive and has all its
facets being
regular polytopes .
E.L. Elte compiled a
longer list in 1912 as The Semiregular Polytopes of the Hyperspaces which included a wider definition.
In
three-dimensional space and below, the terms semiregular polytope and
uniform polytope have identical meanings, because all uniform
polygons must be
regular . However, since not all
uniform polyhedra are
regular , the number of semiregular polytopes in dimensions higher than three is much smaller than the number of uniform polytopes in the same number of dimensions.
The three convex semiregular
4-polytopes are the
rectified 5-cell ,
snub 24-cell and
rectified 600-cell . The only semiregular polytopes in higher dimensions are the
k 21 polytopes , where the rectified 5-cell is the special case of k = 0. These were all listed by Gosset, but a proof of the completeness of this list was not published until the work of
Makarov (1988) for four dimensions, and
Blind & Blind (1991) for higher dimensions.
Gosset's 4-polytopes (with his names in parentheses)
Rectified 5-cell (Tetroctahedric),
Rectified 600-cell (Octicosahedric),
Snub 24-cell (Tetricosahedric), , or
Semiregular E-polytopes in higher dimensions
5-demicube (5-ic semi-regular), a
5-polytope , ↔
221 polytope (6-ic semi-regular), a
6-polytope , or
321 polytope (7-ic semi-regular), a
7-polytope ,
421 polytope (8-ic semi-regular), an
8-polytope ,
The
tetrahedral-octahedral honeycomb in Euclidean 3-space has alternating tetrahedral and octahedral cells.
Semiregular polytopes can be extended to semiregular
honeycombs . The semiregular Euclidean honeycombs are the
tetrahedral-octahedral honeycomb (3D),
gyrated alternated cubic honeycomb (3D) and the
521 honeycomb (8D).
Gosset
honeycombs :
Tetrahedral-octahedral honeycomb or
alternated cubic honeycomb (Simple tetroctahedric check), ↔ (Also
quasiregular polytope )
Gyrated alternated cubic honeycomb (Complex tetroctahedric check),
Semiregular E-honeycomb:
Gosset (1900) additionally allowed Euclidean honeycombs as facets of higher-dimensional Euclidean honeycombs, giving the following additional figures:
Hypercubic honeycomb prism, named by Gosset as the (n – 1)-ic semi-check (analogous to a single rank or file of a chessboard)
Alternated hexagonal slab honeycomb (tetroctahedric semi-check),
The
hyperbolic tetrahedral-octahedral honeycomb has tetrahedral and two types of octahedral cells.
There are also hyperbolic uniform honeycombs composed of only regular cells (
Coxeter & Whitrow 1950 ), including:
Hyperbolic uniform honeycombs , 3D honeycombs:
Alternated order-5 cubic honeycomb , ↔ (Also
quasiregular polytope )
Tetrahedral-octahedral honeycomb ,
Tetrahedron-icosahedron honeycomb ,
Paracompact uniform honeycombs , 3D honeycombs, which include uniform tilings as cells:
Rectified order-6 tetrahedral honeycomb ,
Rectified square tiling honeycomb ,
Rectified order-4 square tiling honeycomb , ↔
Alternated order-6 cubic honeycomb , ↔ (Also quasiregular)
Alternated hexagonal tiling honeycomb , ↔
Alternated order-4 hexagonal tiling honeycomb , ↔
Alternated order-5 hexagonal tiling honeycomb , ↔
Alternated order-6 hexagonal tiling honeycomb , ↔
Alternated square tiling honeycomb , ↔ (Also quasiregular)
Cubic-square tiling honeycomb ,
Order-4 square tiling honeycomb , =
Tetrahedral-triangular tiling honeycomb ,
9D hyperbolic paracompact honeycomb:
621 honeycomb (10-ic check),