Icositetrachoric tetracomb

Icositetrachoric tetracomb
Rank	5
Type	Regular
Space	Euclidean
Notation
Bowers style acronym	Icot
Coxeter diagram	x3o4o3o3o ()
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	${\frac {3}{2}}=1.5$
Related polytopes
Army	Icot
Regiment	Icot
Dual	Hexadecachoric tetracomb
Petrie dual	Petrial 24-cell honeycomb
Conjugate	None
Abstract & topological properties
Orientable	Yes
Properties
Symmetry	U5
Convex	Yes
Nature	Tame

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.

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:

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

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

An alternate set of coordinates can be given by:

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

where i , j , k , l  are integers.

Representations

An icositetrachoric tetracomb has the following Coxeter diagrams:

• x3o4o3o3o () (full symmetry)
• o3x3o4o3o () (U₅ symmetry, as rectified hexadecachoric tetracomb)
• o4o3x3o4o () (R₅ symmetry, birectified tesseractic tetracomb)
• o4o3x3o *c3o () (S₅ symmetry, rectified demitesseractic tetracomb)
• o3x3o *b3o *b3o () (Q₅ symmetry, cells of four different types)

External links

• Klitzing, Richard. "icot".
• Wikipedia contributors. "24-cell honeycomb".