# Icositetrachoric tetracomb

Icositetrachoric tetracomb
Rank5
TypeRegular
SpaceEuclidean
Notation
Bowers style acronymIcot
Coxeter diagram
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${\displaystyle \frac32 = 1.5}$
Related polytopes
ArmyIcot
RegimentIcot
ConjugateNone
Abstract & topological properties
OrientableYes
Properties
SymmetryU5
ConvexYes

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:

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

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

An alternate set of coordinates can be given by:

• ${\displaystyle \left(i,\,j,\,k,\,l\right),}$
• ${\displaystyle \left(i+\frac12,\,j+\frac12,\,k+\frac12,\,l+\frac12\right)}$ 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)

## Related polytopes

o3o4o3o3o truncations
Name OBSA CD diagram Picture
Icositetrachoric tetracomb
Rectified icositetrachoric tetracomb
Birectified icositetrachoric tetracomb
Icositetrachoric tetracomb
Truncated icositetrachoric tetracomb
Bitruncated icositetrachoric tetracomb
Small rhombated icositetrachoric tetracomb
Small birhombated icositetrachoric tetracomb
Rectified icositetrachoric tetracomb
Great rhombated icositetrachoric tetracomb
Great birhombated icositetrachoric tetracomb
Truncated icositetrachoric tetracomb
Small prismated icositetrachoric tetracomb
Prismatotruncated icositetrachoric tetracomb
Prismatorhombated icositetrachoric tetracomb
Great prismated icositetrachoric tetracomb
Small cellated icositetrachoric tetracomb
Cellirhombated icositetrachoric tetracomb
Celligreatorhombated icositetrachoric tetracomb
Celliprismated icositetrachoric tetracomb
Celliprismatotruncated icositetrachoric tetracomb
Great cellated icositetrachoric tetracomb
o4o3o3o4o truncations
Name OBSA CD diagram Picture
Tesseractic tetracomb test
Rectified tesseractic tetracomb rittit
Icositetrachoric tetracomb icot
Truncated tesseractic tetracomb tattit
Bitruncated tesseractic tetracomb batitit
Small rhombated tesseractic tetracomb srittit
Rectified icositetrachoric tetracomb ricot
Great rhombated tesseractic tetracomb grittit
Truncated icositetrachoric tetracomb ticot
Small prismated tesseractic tetracomb sidpitit
Prismatotruncated tesseractic tetracomb potatit
Prismatorhombated tesseractic tetracomb prittit
Great prismated tesseractic tetracomb gippittit
Tesseractic tetracomb test
Celliprismated tesseractic tetracomb capotat
Cellirhombated tesseractic tetracomb scartit
Celligreatorhombated tesseractic tetracomb gicartit
Celliprismatotruncated tesseractic tetracomb captatit
Great cellated tesseractic tetracomb otatit