Semiregular polytope

Gosset's figures

3D honeycombs
Simple tetroctahedric check	Complex tetroctahedric check
4D polytopes
Tetroctahedric	Octicosahedric	Tetricosahedric

In geometry, by Thorold Gosset's definition a semiregular polytope is usually taken to be a polytope that is vertex-uniform 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.

Gosset's list

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 polychora (4-polytopes) are the rectified 5-cell, snub 24-cell and rectified 600-cell. The only semiregular polytopes in higher dimensions are the k₂₁ polytopes, where the rectified 5-cell is the special case of k = 0.

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).

Semiregular figures Gosset enumerated: (his names in parentheses)

• Convex uniform honeycombs, two 3D honeycombs:
  1. Tetrahedral-octahedral honeycomb (Simple tetroctahedric check), ↔ (Also quasiregular polytope)
  2. Gyrated alternated cubic honeycomb (Complex tetroctahedric check),
• Uniform polychora, three 4-polytopes:
  1. Rectified 5-cell (Tetroctahedric),
  2. Rectified 600-cell (Octicosahedric),
  3. Snub 24-cell (Tetricosahedric),
     • Snub 24-cell honeycomb,
• Semiregular E-polytopes, four polytopes, and one honeycomb:
  1. 5-demicube (5-ic semi-regular), a 5-polytope, ↔
  2. 2₂₁ polytope (6-ic semi-regular), a 6-polytope,
  3. 3₂₁ polytope (7-ic semi-regular), a 7-polytope,
  4. 4₂₁ polytope (8-ic semi-regular), an 8-polytope,
  5. 5₂₁ honeycomb (9-ic check) (8D Euclidean honeycomb),

Beyond Gosset's list

There are also hyperbolic uniform honeycombs composed of only regular cells, including:

• Hyperbolic uniform honeycombs, 3D honeycombs:
  1. Alternated order-5 cubic honeycomb, ↔ (Also quasiregular polytope)
  2. Tetrahedral-octahedral honeycomb,
  3. Tetrahedron-icosahedron honeycomb,
• Paracompact uniform honeycombs, 3D honeycombs, which include uniform tilings as cells:
  1. Rectified order-6 tetrahedral honeycomb,
  2. Rectified square tiling honeycomb,
  3. Alternated order-6 cubic honeycomb, ↔ (Also quasiregular)
  4. Alternated hexagonal tiling honeycomb, ↔
  5. Alternated square tiling honeycomb, ↔ (Also quasiregular)
  6. Cubic-square tiling honeycomb,
  7. Tetrahedral-triangular tiling honeycomb,
• 9D hyperbolic paracompact honeycomb:
  1. 6₂₁ honeycomb (10-ic check),

See also

• Semiregular polyhedron

References

• Coxeter, H. S. M. (1973). Regular Polytopes (3rd ed.). New York: Dover Publications. ISBN 0-486-61480-8.
• Gosset, Thorold (1900). "On the regular and semi-regular figures in space of n dimensions". Messenger of Mathematics. 29: 43–48.
• Elte, E. L. (1912). The Semiregular Polytopes of the Hyperspaces. Groningen: University of Groningen. ISBN 1-4181-7968-X.