geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, by

Thorold Gosset John Herbert de Paz Thorold Gosset (16 October 1869 – December 1962) was an English lawyer and an amateur mathematician. In mathematics, he is noted for discovering and classifying the semiregular polytopes in dimensions four and higher, a ...

's definition a semiregular polytope is usually taken to be a

polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...

that is

vertex-transitive In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of fa ...

and has all its facets being

regular polytope In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...

s. E.L. Elte compiled a longer list in 1912 as ''The Semiregular Polytopes of the Hyperspaces'' which included a wider definition.

Gosset's list

three-dimensional space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informal ...

and below, the terms ''semiregular polytope'' and ''

uniform polytope In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons (the definition is different in 2 dimensions to exclude ver ...

'' have identical meanings, because all uniform

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two t ...

s must be

regular The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * "Regular" (Badfinger song) * Regular tunings of stringed instrum ...

. However, since not all

uniform polyhedra In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent. Uniform polyhedra may be regular (if also f ...

are

, 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-polytope In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces ( polygons), ...

s are the

rectified 5-cell In four-dimensional geometry, the rectified 5-cell is a uniform 4-polytope composed of 5 regular tetrahedral and 5 regular octahedral cells. Each edge has one tetrahedron and two octahedra. Each vertex has two tetrahedra and three octahedra. In ...

snub 24-cell In geometry, the snub 24-cell or snub disicositetrachoron is a convex uniform 4-polytope composed of 120 regular tetrahedral and 24 icosahedral cells. Five tetrahedra and three icosahedra meet at each vertex. In total it has 480 triangular faces, ...

and

rectified 600-cell In geometry, the rectified 600-cell or rectified hexacosichoron is a convex uniform 4-polytope composed of 600 regular octahedra and 120 icosahedra cells. Each edge has two octahedra and one icosahedron. Each vertex has five octahedra and two ic ...

. The only semiregular polytopes in higher dimensions are the ''k''₂₁ 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 for four dimensions, and for higher dimensions. ;Gosset's 4-polytopes (with his names in parentheses): :

Rectified 5-cell In four-dimensional geometry, the rectified 5-cell is a uniform 4-polytope composed of 5 regular tetrahedral and 5 regular octahedral cells. Each edge has one tetrahedron and two octahedra. Each vertex has two tetrahedra and three octahedra. In ...

(Tetroctahedric), :

Rectified 600-cell In geometry, the rectified 600-cell or rectified hexacosichoron is a convex uniform 4-polytope composed of 600 regular octahedra and 120 icosahedra cells. Each edge has two octahedra and one icosahedron. Each vertex has five octahedra and two ic ...

(Octicosahedric), :

Snub 24-cell In geometry, the snub 24-cell or snub disicositetrachoron is a convex uniform 4-polytope composed of 120 regular tetrahedral and 24 icosahedral cells. Five tetrahedra and three icosahedra meet at each vertex. In total it has 480 triangular faces, ...

(Tetricosahedric), , or ; Semiregular E-polytopes in higher dimensions: :

5-demicube In five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a ''5-hypercube'' ( penteract) with alternated vertices removed. It was discovered by Thorold Gosset. Since it was the only semiregular 5- ...

(5-ic semi-regular), a 5-polytope, ↔ : 2₂₁ polytope (6-ic semi-regular), a

6-polytope In six-dimensional geometry, a six-dimensional polytope or 6-polytope is a polytope, bounded by 5-polytope facets. Definition A 6-polytope is a closed six-dimensional figure with vertices, edges, faces, cells (3-faces), 4-faces, and 5-faces. ...

, or : 3₂₁ polytope (7-ic semi-regular), a

7-polytope In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets. A uniform 7-polytope is one whose symmetry group is transitive on vertices and whose f ...

, : 4₂₁ polytope (8-ic semi-regular), an 8-polytope,

Euclidean honeycombs

Semiregular polytopes can be extended to semiregular

honeycombs A honeycomb is a mass of hexagonal prismatic wax cells built by honey bees in their nests to contain their larvae and stores of honey and pollen. Beekeepers may remove the entire honeycomb to harvest honey. Honey bees consume about of hone ...

. The semiregular Euclidean honeycombs are the tetrahedral-octahedral honeycomb (3D), gyrated alternated cubic honeycomb (3D) and the 5₂₁ honeycomb (8D). Gosset

: # Tetrahedral-octahedral honeycomb or

alternated cubic honeycomb The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2. Other names incl ...

(Simple tetroctahedric check), ↔ (Also quasiregular polytope) # Gyrated alternated cubic honeycomb (Complex tetroctahedric check), Semiregular E-honeycomb: * 5₂₁ honeycomb (9-ic check) (8D Euclidean honeycomb), additionally allowed Euclidean honeycombs as facets of higher-dimensional Euclidean honeycombs, giving the following additional figures: #Hypercubic honeycomb prism, named by Gosset as the (''n'' – 1)-ic semi-check (analogous to a single rank or file of a chessboard) #Alternated hexagonal slab honeycomb (tetroctahedric semi-check),

Hyperbolic honeycombs

There are also hyperbolic uniform honeycombs composed of only regular cells , including: * Hyperbolic uniform honeycombs, 3D honeycombs: *# Alternated order-5 cubic honeycomb, ↔ (Also quasiregular polytope) *# Tetrahedral-octahedral honeycomb, *#

Tetrahedron-icosahedron honeycomb In the geometry of hyperbolic 3-space, the tetrahedral-icosahedral honeycomb is a compact uniform honeycomb, constructed from icosahedron, tetrahedron, and octahedron cells, in an icosidodecahedron vertex figure. It has a single-ring Coxeter diagra ...

, * Paracompact uniform honeycombs, 3D honeycombs, which include uniform tilings as cells: *# Rectified order-6 tetrahedral honeycomb, *# Rectified square tiling honeycomb, *# Rectified order-4 square tiling honeycomb, ↔ *# Alternated order-6 cubic honeycomb, ↔ (Also quasiregular) *# Alternated hexagonal tiling honeycomb, ↔ *# Alternated order-4 hexagonal tiling honeycomb, ↔ *# Alternated order-5 hexagonal tiling honeycomb, ↔ *# Alternated order-6 hexagonal tiling honeycomb, ↔ *# Alternated square tiling honeycomb, ↔ (Also quasiregular) *# Cubic-square tiling honeycomb, *# Order-4 square tiling honeycomb, = *# Tetrahedral-triangular tiling honeycomb, *9D hyperbolic paracompact honeycomb: *# 6₂₁ honeycomb (10-ic check),

References

* * * * * * {{cite journal , last = Makarov , first = P. V. , department = Voprosy Diskret. Geom. , journal = Mat. Issled. Akad. Nauk. Mold. , mr = 958024 , pages = 139–150, 177 , title = On the derivation of four-dimensional semi-regular polytopes , volume = 103 , year = 1988 Polytopes

Gosset's list

Euclidean honeycombs

Hyperbolic honeycombs

See also

References