Truncated 4-simplex honeycomb
(No image)
Type	Uniform honeycomb
Family	Truncated simplectic honeycomb
Schläfli symbol	t0,1{3[5]}
Coxeter–Dynkin diagrams
4-face types	{3,3,3} t0,1{3,3,3} Truncated 5-cell In geometry, a truncated 5-cell is a uniform polychoron formed as the truncation of the regular 5-cell.There are two degrees of trunctions, including a bitruncation.- Truncated 5-cell:...
Cell types	{3,3} Tetrahedron In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids... t0,1{3,3} Truncated tetrahedron In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 regular triangular faces, 12 vertices and 18 edges.- Area and volume :...
Face types	Triangle Triangle A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....  {3} Hexagon {6}
Vertex figure	Elongated tetrahedral antiprism
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...

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 truncated 4-simplex honeycomb, truncated 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 of 5-cells, truncated 5-cell

Truncated 5-cell

In geometry, a truncated 5-cell is a uniform polychoron formed as the truncation of the regular 5-cell.There are two degrees of trunctions, including a bitruncation.- Truncated 5-cell:...

s, and bitruncated 5-cells facets in a ratio of 2:2:1.

Parallel cells of this honeycomb can be grouped into 5 sets of parallel hyperplane

Hyperplane

A hyperplane is a concept in geometry. It is a generalization of the plane into a different number of dimensions.A hyperplane of an n-dimensional space is a flat subset with dimension n − 1...

s, each filled by a quarter cubic honeycomb

Quarter cubic honeycomb

The quarter cubic honeycomb is a space-filling tessellation in Euclidean 3-space. It is composed of tetrahedra and truncated tetrahedra in a ratio of 1:1...

.

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 other 6 have Coxeter–Dynkin diagrams as: , , , ,, .

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...
    
    
  • Omnitruncated 5-cell honeycomb