Omnitruncated 5-simplex honeycomb
(No image)
Type	Uniform honeycomb
Family	Omnitruncated simplectic honeycomb
Schläfli symbol	t0,1,2,3,4,5{3[6]}
Coxeter–Dynkin diagrams
5-face types	t0,1,2,3,4{3,3,3,3}
4-face types	t0,1,2,3{3,3,3} {}×t0,1,2{3,3} Truncated octahedral prism In 4-dimensional geometry, a truncated octahedral prism is a convex uniform polychoron . This polychoron has 16 cells It has 64 faces , and 96 edges and 48 vertices.It has two symmetry constructions, one from the truncated octahedron, and one as an omnitruncation of... {6}×{6} Duoprism In geometry of 4 dimensions or higher, a duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher...
Cell types	t0,1,2{3,3} Truncated octahedron In geometry, the truncated octahedron is an Archimedean solid. It has 14 faces , 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a zonohedron.... {4,3} Cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and... {}x{6} Hexagonal prism In geometry, the hexagonal prism is a prism with hexagonal base. The shape has 8 faces, 18 edges, and 12 vertices.Since it has eight faces, it is an octahedron. However, the term octahedron is primarily used to refer to the regular octahedron, which has eight triangular faces...
Face types	{4} Square -Mathematics and science:* Square , a cryptographic block cipher* Square , a regular quadrilateral with four equal sides and four right angles* Square number, an integer that is the square of another integer-Construction:... {6}
Vertex figure	Irr. 5-simplex
Coxeter group Coxeter group In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example... s	, [3[6]]
Properties	vertex-transitive Vertex-transitive In geometry, a polytope is isogonal or vertex-transitive if, loosely speaking, all its vertices are the same...

In five-dimensional Euclidean geometry

Euclidean geometry

Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...

, the omnitruncated 5-simplex honeycomb or omnitruncated hexateric honeycomb is a space-filling tessellation

Tessellation

A tessellation or tiling of the plane is a pattern of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible. Tessellations frequently appeared in the art...

 (or honeycomb

Honeycomb (geometry)

In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions....

). It is composed entirely of omnitruncated 5-simplex facets.

Related polytopes and honeycombs

This honeycomb is one of 12 unique uniform honeycombs constructed by the Coxeter group

Coxeter group

In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...

. The Coxeter–Dynkin diagrams of the other 11 are:
,
,
,
,
,
,
,
,
,
,
, the first two being the 5-simplex honeycomb

5-simplex honeycomb

In five-dimensional Euclidean geometry, the 5-simplex honeycomb or hexateric honeycomb is a space-filling tessellation . Each vertex is shared by 12 5-simplexes, 30 rectified 5-simplexes, and 20 birectified 5-simplexes...

, and truncated 5-simplex honeycomb

Truncated 5-simplex honeycomb

In five-dimensional Euclidean geometry, the truncated 5-simplex honeycomb or truncated hexateric honeycomb is a space-filling tessellation...

 respectively.

Projection by folding

The omnitruncated 5-simplex honeycomb can be projected into the 3-dimensional omnitruncated cubic honeycomb

Omnitruncated cubic honeycomb

The omnitruncated cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space. It is composed of truncated cuboctahedra and octagonal prisms in a ratio of 1:3.A wireframe rendering- References :...

 by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same 3-space vertex arrangement

Vertex arrangement

In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes....

:

See also

• Regular and uniform honeycombs in 5-space:
  • Penteractic honeycomb
  • Demipenteractic honeycomb