Gosset's figures

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

Ingeometry, byThorold Gosset's definition asemiregular polytope is usually taken to be apolytope that isvertex-transitive and has all itsfacets beingregular polytopes.E.L. Elte compiled alonger list in 1912 asThe Semiregular Polytopes of the Hyperspaces which included a wider definition.

Inthree-dimensional space and below, the termssemiregular polytope anduniform polytope have identical meanings, because all uniformpolygons must beregular. However, since not alluniform polyhedra areregular, 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 semiregular4-polytopes are therectified 5-cell,snub 24-cell andrectified 600-cell. The only semiregular polytopes in higher dimensions are thek₂₁ polytopes, where the rectified 5-cell is the special case ofk = 0. These were all listed by Gosset, but a proof of the completeness of this list was not published until the work ofMakarov (1988) for four dimensions, andBlind & 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), a5-polytope, ↔

2₂₁ polytope (6-ic semi-regular), a6-polytope, or

3₂₁ polytope (7-ic semi-regular), a7-polytope,

4₂₁ polytope (8-ic semi-regular), an8-polytope,

Semiregular polytopes can be extended to semiregularhoneycombs. The semiregular Euclidean honeycombs are thetetrahedral-octahedral honeycomb (3D),gyrated alternated cubic honeycomb (3D) and the5₂₁ honeycomb (8D).

Gossethoneycombs:

1. Tetrahedral-octahedral honeycomb oralternated cubic honeycomb (Simple tetroctahedric check), ↔ (Alsoquasiregular polytope)
2. Gyrated alternated cubic honeycomb (Complex tetroctahedric check),

Semiregular E-honeycomb:

• 5₂₁ honeycomb (9-ic check) (8D Euclidean honeycomb),

Gosset (1900) additionally allowed Euclidean honeycombs as facets of higher-dimensional Euclidean honeycombs, giving the following additional figures:

1. Hypercubic honeycomb prism, named by Gosset as the (n – 1)-ic semi-check (analogous to a single rank or file of a chessboard)
2. Alternated hexagonal slab honeycomb (tetroctahedric semi-check),

There are also hyperbolic uniform honeycombs composed of only regular cells (Coxeter & Whitrow 1950), including:

• Hyperbolic uniform honeycombs, 3D honeycombs:
  1. Alternated order-5 cubic honeycomb, ↔ (Alsoquasiregular 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. Rectified order-4 square tiling honeycomb, ↔
  4. Alternated order-6 cubic honeycomb, ↔ (Also quasiregular)
  5. Alternated hexagonal tiling honeycomb, ↔
  6. Alternated order-4 hexagonal tiling honeycomb, ↔
  7. Alternated order-5 hexagonal tiling honeycomb, ↔
  8. Alternated order-6 hexagonal tiling honeycomb, ↔
  9. Alternated square tiling honeycomb, ↔ (Also quasiregular)
  10. Cubic-square tiling honeycomb,
  11. Order-4 square tiling honeycomb, =
  12. Tetrahedral-triangular tiling honeycomb,
• 9D hyperbolic paracompact honeycomb:
  1. 6₂₁ honeycomb (10-ic check),

• Semiregular polyhedron

• Blind, G.;Blind, R. (1991). "The semiregular polytopes".Commentarii Mathematici Helvetici.66 (1):150–154.doi:10.1007/BF02566640.MR 1090169.S2CID 119695696.
• Coxeter, H. S. M. (1973).Regular Polytopes (3rd ed.). New York: Dover Publications.ISBN 0-486-61480-8.
• Coxeter, H. S. M.; Whitrow, G. J. (1950). "World-structure and non-Euclidean honeycombs".Proceedings of the Royal Society.201 (1066):417–437.Bibcode:1950RSPSA.201..417C.doi:10.1098/rspa.1950.0070.MR 0041576.S2CID 120322123.
• Elte, E. L. (1912).The Semiregular Polytopes of the Hyperspaces. Groningen: University of Groningen.ISBN 1-4181-7968-X.
• Gosset, Thorold (1900). "On the regular and semi-regular figures in space ofn dimensions".Messenger of Mathematics.29:43–48.
• Makarov, P. V. (1988). "On the derivation of four-dimensional semi-regular polytopes". Voprosy Diskret. Geom.Mat. Issled. Akad. Nauk. Mold.103:139–150, 177.MR 0958024.

• Uniform polytopes

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