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'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 polytopes.

E.L. Elte Emanuel Lodewijk Elte (16 March 1881 in Amsterdam – 9 April 1943 in Sobibór) Emanuël Lodewijk 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'' 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 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, snub 24-cell and rectified 600-cell. 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 (Octicosahedric), : Snub 24-cell (Tetricosahedric), , or ; Semiregular E-polytopes in higher dimensions: : 5-demicube (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, 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 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 (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 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

) *# 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 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, ↔ *#

Alternated order-6 hexagonal tiling honeycomb The triangular tiling honeycomb is one of 11 paracompact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space. It is called ''paracompact'' because it has infinite cells and vertex figures, with all vertices as ideal points at ...

, ↔ *#

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, = *#

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