Icositetrachoric tetracomb

The icositetrachoric tetracomb or icot, also known as the 24-cell tetracomb or 24-cell honeycomb, is one of three regular tetracombs or tessellations of 4D Euclidean space. 3 icositetrachora join at each face, and 8 join at each vertex of this honeycomb.

Icositetrachoric tetracomb
Rank	5
Type	Regular
Space	Euclidean
Notation
Bowers style acronym	Icot
Coxeter diagram
Schläfli symbol	{3,4,3,3}
Elements
Tera	N icositetrachora
Cells	12N octahedra
Faces	32N triangles
Edges	24N
Vertices	3N
Vertex figure	Tesseract, edge length 1
Measures (edge length 1)
Vertex density	$\frac32 = 1.5$
Related polytopes
Army	Icot
Regiment	Icot
Dual	Hexadecachoric tetracomb
Conjugate	None
Abstract & topological properties
Orientable	Yes
Properties
Symmetry	U5
Convex	Yes

It can be obtained from the tesseractic tetracomb by decomposing alternate tesseracts into 8 cubic pyramids and attaching those to the neighbouring tesseract, making it an icositetrachoron. It can also be obtained as a birectified tesseractic tetracomb, or a rectified hexadecachoric tetracomb.

Vertex coordinates

The vertices of an icositetrachoric tetracomb of edge length 1 are given by all permutations of:

• $\left(\sqrt2i,\,\sqrt2j,\,±\frac{\sqrt2}{2}+\sqrt2k,\,±\frac{\sqrt2}{2}+\sqrt2l\right),$

where i, j, k, and l range over the integers.

An alternate set of coordinates can be given by:

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

where i, j, k, l are integers.

Representations

An icositetrachoric tetracomb has the following Coxeter diagrams:

• (full symmetry)
• (U5 symmetry, as rectified hexadecachoric tetracomb)
• (R5 symmetry, birectified tesseractic tetracomb)
• (S5 symmetry, rectified demitesseractic tetracomb)
• (Q5 symmetry, cells of four different types)

External links

• Klitzing, Richard. "icot".

• Wikipedia Contributors. "24-cell honeycomb".