Semiregular polytope

Convex semiregular polychora

Rectified 5-cell

Snub 24-cell

Rectified 600-cell

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.

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).

Gosset's list

Semiregular figures Gosset enumerated: (his names in parentheses)

• Convex uniform honeycombs, two 3D honeycombs:
  1. Tetrahedral-octahedral honeycomb (Simple tetroctahedric check)
  2. Gyrated alternated cubic honeycomb (Complex tetroctahedric check)
• Uniform polychora, three 4-polytopes:
  1. Rectified 5-cell (Tetroctahedric)
  2. Snub 24-cell (Tetricosahedric)
     • Snub 24-cell honeycomb
  3. Rectified 600-cell (Octicosahedric)
• 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 honeycomb)

See also

• Semiregular polyhedron

References

• Coxeter, H. S. M. (1973). Regular Polytopes ((3rd ed.) 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.