Omnitruncated 4-simplex honeycomb
(No image)
Type	Uniform honeycomb
Family	Omnitruncated simplectic honeycomb
Schläfli symbol	t0,1,2,3,4{3[5]}
Coxeter–Dynkin diagrams
4-face types	t0,1,2,3{3,3,3}
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....   {6}x{} 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 (geometry) In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles... {6}
Vertex figure	Irr. 5-cell
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[5]]
Properties	vertex-transitive Vertex-transitive In geometry, a polytope is isogonal or vertex-transitive if, loosely speaking, all its vertices are the same... , cell-transitive

In four-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 4-simplex honeycomb or omnitruncated 5-cell 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...

 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-cell (omnitruncated 4-simplex) facets.

Coxeter calls this Hinton's honeycomb after C. H. Hinton

Charles Howard Hinton

Charles Howard Hinton was a British mathematician and writer of science fiction works titled Scientific Romances. He was interested in higher dimensions, particularly the fourth dimension, and is known for coining the word tesseract and for his work on methods of visualising the geometry of...

, who described it in his book The Fourth Dimension in 1906.

Related polytopes and honeycombs

This honeycomb is one of 7 unique uniform honycombs 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 six are: , , , , ,.

See also

• Regular and uniform honeycombs in 4-space:
  • tesseractic honeycomb
  • 16-cell honeycomb
  • 24-cell honeycomb
  • Truncated 24-cell honeycomb
    Truncated 24-cell honeycomb
    In four-dimensional Euclidean geometry, the truncated 24-cell honeycomb is a uniform space-filling honeycomb. It can be seen as a truncation of the regular 24-cell honeycomb, containing tesseract and truncated 24-cell cells....
    
    
  • Snub 24-cell honeycomb
    Snub 24-cell honeycomb
    In four-dimensional Euclidean geometry, the snub 24-cell honeycomb, or snub icositetrachoric honeycomb is a uniform space-filling tessellation by snub 24-cells, 16-cells, and 5-cells. It was discovered by Thorold Gosset with his 1900 paper of semiregular polytopes...
    
    
  • 5-cell honeycomb
    5-cell honeycomb
    In four-dimensional Euclidean geometry, the 4-simplex honeycomb, 5-cell honeycomb or pentachoric-dispentachoric honeycomb is a space-filling tessellation honeycomb...
    
    
  • Truncated 5-cell honeycomb