(Redirected from 16-cell honeycomb)
Rank5
TypeRegular
SpaceEuclidean
Notation
Bowers style acronymHext
Coxeter diagramo3o4o3o3x ()
Schläfli symbol{3,3,4,3}
Elements
Cells24N tetrahedra
Faces32N triangles
Edges12N
VerticesN
Vertex figureIcositetrachoron, edge length 1
Measures (edge length 1)
Vertex density${\displaystyle 2}$
Related polytopes
ArmyHext
RegimentHext
DualIcositetrachoric tetracomb
ConjugateNone
Abstract & topological properties
OrientableYes
Properties
SymmetryU5
ConvexYes
NatureTame

The hexadecachoric tetracomb or hext, also called the demitesseractic tetracomb, 16-cell tetracomb, also -honeycomb, is one of three regular tetracombs or tessellations of 4D Euclidean space. 3 hexadecachora join at each face, and 24 join at each vertex of this honeycomb. It is the 4D demihypercubic honeycomb, obtained by alternation of the tesseractic tetracomb. It can also be formed as the hull of two dual tesseractic tetracombs, with the hexadecachoric facets being seen as square duotegums.

## Vertex coordinates

The vertices of a hexadecachoric tetracomb of edge length 1 are given by:

• ${\displaystyle {\frac {\sqrt {2}}{2}}\left(i,\,j,\,k,\,l\right)}$,

where i, j, k, and l are integers, and i+j+k+l is even.

These coordinates are due to the hexadecachoric tetracomb's construction as an alternated tesseractic tetracomb. Another set of coordinates, formed from two dual tesseractic tetracombs, are given by:

• ${\displaystyle \left(i,\,j,\,k,\,l\right)}$,
• ${\displaystyle \left(i+{\frac {1}{2}},\,j+{\frac {1}{2}},\,k+{\frac {1}{2}},\,l+{\frac {1}{2}}\right)}$.

where i, j, k, and l are integers.

## Representations

A hexadecachoric tetracomb has the following Coxeter diagrams:

• o3o4o3o3x () (full symmetry)
• x3o3o4o *b3o () (S5 symmetry, demitesseractic tetracomb, facets of two types)
• x3o3o *b3o *b3o () (Q5 symmetry, facets of three types)