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, and ...

'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 -d ...

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 A facet is a flat surface of a geometric shape, e.g., of a cut gemstone. Facet may also refer to: Arts, entertainment, and media * ''Facets'' (album), an album by Jim Croce * ''Facets'', a 1980 album by jazz pianist Monty Alexander and his tri ...

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, f ...

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 (geometry), position of an element (i.e., Point (m ...

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 vert ...

'' 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 toge ...

s must be regular. 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 fa ...

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 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 t ...

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 face ...

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 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 t ...

(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 face ...

(Tetricosahedric), , or ;

Semiregular E-polytope In geometry, a uniform ''k''21 polytope is a polytope in ''k'' + 4 dimensions constructed from the ''E'n'' Coxeter group, and having only regular polytope facets. The family was named by their Coxeter symbol ''k''21 by its bifurcatin ...

s 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 In geometry, a five-dimensional polytope (or 5-polytope) is a polytope in five-dimensional space, bounded by (4-polytope) facets, pairs of which share a polyhedral cell. Definition A 5-polytope is a closed five-dimensional figure with vertices ...

, ↔ : 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. A ...

, 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 In Eight-dimensional space, eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets. Each 6-polytope Ridge (geometry), ridge being shared by exactly two 7-polytope Facet (mathematics), fa ...

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 incl ...

(3D),

gyrated 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 ...

(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 incl ...

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 In geometry, a quasiregular polyhedron is a uniform polyhedron that has exactly two kinds of regular polygon, regular faces, which alternate around each vertex (geometry), vertex. They are vertex-transitive and edge-transitive, hence a step closer ...

) #

Gyrated 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 ...

(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 honeycomb In hyperbolic geometry, a uniform honeycomb in hyperbolic space is a uniform tessellation of uniform polyhedron, uniform polyhedral Cell (geometry), cells. In 3-dimensional hyperbolic space there are nine Coxeter group families of compact convex u ...

s, 3D honeycombs: *#

Alternated order-5 cubic honeycomb In hyperbolic geometry, the order-5 cubic honeycomb is one of four compact regular polytope, regular space-filling tessellations (or honeycomb (geometry), honeycombs) in Hyperbolic space, hyperbolic 3-space. With Schläfli symbol it has five cub ...

, ↔ (Also

) *#

, *#

Tetrahedron-icosahedron honeycomb In the geometry of Hyperbolic space, hyperbolic 3-space, the tetrahedral-icosahedral honeycomb is a compact uniform honeycomb (geometry), honeycomb, constructed from icosahedron, tetrahedron, and octahedron cells, in an icosidodecahedron vertex fig ...

, *

Paracompact uniform honeycomb In geometry, uniform honeycombs in hyperbolic space are tessellations of convex uniform polyhedron Cell (geometry), cells. In 3-dimensional hyperbolic space there are 23 Coxeter group families of Coxeter diagram#Paracompact (Koszul simplex groups), ...

s, 3D honeycombs, which include uniform tilings as cells: *#

Rectified order-6 tetrahedral honeycomb In Hyperbolic space, hyperbolic 3-space, the order-6 tetrahedral honeycomb is a paracompact regular space-filling tessellation (or honeycomb (geometry), honeycomb). It is ''paracompact'' because it has vertex figures composed of an infinite number ...

, *#

Rectified square tiling honeycomb In the geometry of Hyperbolic space, hyperbolic 3-space, the square tiling honeycomb is one of 11 paracompact regular honeycombs. It is called ''paracompact'' because it has infinite Cell (geometry), cells, whose vertices exist on horospheres and ...

, *#

Rectified order-4 square tiling honeycomb In the geometry of Hyperbolic space, hyperbolic 3-space, the order-4 square tiling honeycomb is one of 11 paracompact regular honeycombs. It is ''paracompact'' because it has infinite Cell (geometry), cells and vertex figures, with all vertices as ...

, ↔ *#

Alternated order-6 cubic honeycomb The order-6 cubic honeycomb is a paracompact regular polytope, regular space-filling tessellation (or honeycomb (geometry), honeycomb) in Hyperbolic space, hyperbolic 3-space. It is ''paracompact'' because it has vertex figures composed of an infin ...

, ↔ (Also quasiregular) *#

Alternated hexagonal tiling honeycomb In three-dimensional hyperbolic geometry, the alternated hexagonal tiling honeycomb, h, or , is a semiregular polytope, semiregular tessellation with tetrahedron and triangular tiling cells arranged in an octahedron vertex figure. It is named afte ...

, ↔ *#

Alternated order-4 hexagonal tiling honeycomb In the field of Hyperbolic space, hyperbolic geometry, the order-4 hexagonal tiling honeycomb arises as one of 11 Paracompact uniform honeycombs#Regular paracompact honeycombs, regular paracompact honeycombs in 3-dimensional hyperbolic space. It is ...

, ↔ *#

Alternated order-5 hexagonal tiling honeycomb In the field of Hyperbolic space, hyperbolic geometry, the order-5 hexagonal tiling honeycomb arises as one of 11 Paracompact uniform honeycombs#Regular paracompact honeycombs, regular paracompact honeycombs in 3-dimensional hyperbolic space. It is ...

, ↔ *# Alternated order-6 hexagonal tiling honeycomb, ↔ *#

Alternated square tiling honeycomb In the geometry of Hyperbolic space, hyperbolic 3-space, the square tiling honeycomb is one of 11 paracompact regular honeycombs. It is called ''paracompact'' because it has infinite Cell (geometry), cells, whose vertices exist on horospheres and ...

, ↔ (Also quasiregular) *#

Cubic-square tiling honeycomb In the geometry of hyperbolic 3-space, the cubic-square tiling honeycomb is a paracompact uniform honeycomb, constructed from cube and square tiling cells, in a rhombicuboctahedron vertex figure. It has a single-ring Coxeter diagram, , and is name ...

, *#

Order-4 square tiling honeycomb In the geometry of hyperbolic 3-space, the order-4 square tiling honeycomb is one of 11 paracompact regular honeycombs. It is ''paracompact'' because it has infinite cells and vertex figures, with all vertices as ideal points at infinity. Given b ...

, = *#

Tetrahedral-triangular tiling honeycomb In the geometry of Hyperbolic space, hyperbolic 3-space, the tetrahedral-triangular tiling honeycomb is a paracompact uniform honeycomb, constructed from triangular tiling, tetrahedron, and octahedron cells, in an icosidodecahedron vertex figure. I ...

, *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