# Icositetrachoric tetracomb

(Redirected from 24-cell honeycomb)
Icositetrachoric tetracomb
Rank5
TypeRegular
SpaceEuclidean
Notation
Bowers style acronymIcot
Coxeter diagramx3o4o3o3o ()
Schläfli symbol{3,4,3,3}
Elements
TeraN icositetrachora
Cells12N octahedra
Faces32N triangles
Edges24N
Vertices3N
Vertex figureTesseract, edge length 1
Measures (edge length 1)
Vertex density${\displaystyle {\frac {3}{2}}=1.5}$
Related polytopes
ArmyIcot
RegimentIcot
Petrie dualPetrial 24-cell honeycomb
ConjugateNone
Abstract & topological properties
OrientableYes
Properties
SymmetryU5
ConvexYes
NatureTame

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:

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

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

An alternate set of coordinates can be given by:

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