Hexadecachoric tetracomb

Hexadecachoric tetracomb
Rank	5
Type	Regular
Space	Euclidean
Notation
Bowers style acronym	Hext
Coxeter diagram	o3o4o3o3x ()
Schläfli symbol	{3,3,4,3}
Elements
Tera	3N hexadecachora
Cells	24N tetrahedra
Faces	32N triangles
Edges	12N
Vertices	N
Vertex figure	Icositetrachoron, edge length 1
Measures (edge length 1)
Vertex density	$2$
Related polytopes
Army	Hext
Regiment	Hext
Dual	Icositetrachoric tetracomb
Conjugate	None
Abstract & topological properties
Orientable	Yes
Properties
Symmetry	U5
Convex	Yes

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:

• $\frac{\sqrt2}{2}\left(i,\,j,\,k,\,l\right),$

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:

• $\left(i,\,j,\,k,\,l\right),$
• $\left(i+\frac12,\,j+\frac12,\,k+\frac12,\,l+\frac12\right).$

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

Representations

A hexadecachoric tetracomb has the following Coxeter diagrams:

• o3o4o3o3x (full symmetry)
• x3o3o *b3o4o (S5 symmetry, demitesseractic tetracomb, facets of two types)
• x3o3o *b3o *b3o (Q5 symmetry, facets of three types)

External links

• Klitzing, Richard. "hext".

• Wikipedia Contributors. "16-cell honeycomb".