In
geometry, by
Thorold Gosset's definition a semiregular polytope is usually taken to be a
polytope that is
vertex-transitive and has all its
facets being
regular polytopes.
compiled a
longer list in 1912 as ''The Semiregular Polytopes of the Hyperspaces'' which included a wider definition.
Gosset's list
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
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 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
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,
:
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
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 ...
:
#
Tetrahedral-octahedral honeycomb or
alternated cubic honeycomb (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
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 ...
)
*#
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),
See also
*
Semiregular polyhedron
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