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, an ...

'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 face in ...

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

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 inform ...

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 v ...

'' 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. 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-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), an ...

s are the rectified 5-cell,

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 ico ...

. 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 (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 ico ...

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

Euclidean honeycombs

Semiregular polytopes can be extended to semiregular honeycombs. The semiregular Euclidean honeycombs are the

tetrahedral-octahedral 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 inc ...

(3D), gyrated alternated cubic honeycomb (3D) and the 5₂₁ honeycomb (8D). Gosset honeycombs: #

Tetrahedral-octahedral 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 inc ...

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 in ...

(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) *#

, *# Tetrahedron-icosahedron honeycomb, * 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