Semiregular polytope

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Gosset's figures
3D honeycombs
HC P1-P3.png
Simple tetroctahedric check
Gyrated alternated cubic honeycomb.png
Complex tetroctahedric check
4D polytopes
Schlegel half-solid rectified 5-cell.png
Tetroctahedric
Rectified 600-cell schlegel halfsolid.png
Octicosahedric
Ortho solid 969-uniform polychoron 343-snub.png
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[edit]

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 k21 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), CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Rectified 600-cell (Octicosahedric), CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
Snub 24-cell (Tetricosahedric), CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png, CDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png or CDel node h.pngCDel 3.pngCDel node h.pngCDel split1.pngCDel nodes hh.png
Semiregular E-polytopes in higher dimensions
5-demicube (5-ic semi-regular), a 5-polytope, CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
221 polytope (6-ic semi-regular), a 6-polytope, CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png or CDel nodes 10r.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
321 polytope (7-ic semi-regular), a 7-polytope, CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
421 polytope (8-ic semi-regular), an 8-polytope, CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png

Euclidean honeycombs[edit]

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:

  1. Tetrahedral-octahedral honeycomb or alternated cubic honeycomb (Simple tetroctahedric check), CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 4.pngCDel node.png (Also quasiregular polytope)
  2. Gyrated alternated cubic honeycomb (Complex tetroctahedric check), CDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel infin.pngCDel node.png

Semiregular E-honeycomb:

Hyperbolic honeycombs[edit]

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:

See also[edit]

References[edit]