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
Rectified 600-cell schlegel halfsolid.png
Ortho solid 969-uniform polychoron 343-snub.png

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 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)

Beyond Gosset's list[edit]

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

See also[edit]