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
Icositetrachoronic tetracomb.png
Bowers style acronymIcot
Coxeter diagramCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Schläfli symbol{3,4,3,3}
TeraN icositetrachora
Cells12N octahedra
Faces32N triangles
Vertex figureTesseract, edge length 1
Measures (edge length 1)
Vertex density
Related polytopes
DualHexadecachoric tetracomb
Abstract & topological properties

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 coordinatesEdit

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


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

An alternate set of coordinates can be given by:

  •   for i+j+k+l even

where i, j, k, l are integers.


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 polytopesEdit

o3o4o3o3o truncations
Name OBSA CD diagram Picture
Icositetrachoric tetracomb          
Rectified icositetrachoric tetracomb          
Birectified icositetrachoric tetracomb          
Icositetrachoric tetracomb          
Hexadecachoric tetracomb          
Truncated icositetrachoric tetracomb          
Bitruncated icositetrachoric tetracomb          
Bitruncated hexadecachoric tetracomb          
Truncated hexadecachoric 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          
Small prismated hexadecachoric tetracomb          
Prismatotruncated icositetrachoric tetracomb          
Prismatorhombated hexadecachoric tetracomb          
Prismatorhombated icositetrachoric tetracomb          
Prismatotruncated hexadecachoric tetracomb          
Great prismated icositetrachoric tetracomb          
Great prismated hexadecachoric tetracomb          
Small cellated icositetrachoric tetracomb          
Celliprismated hexadecachoric tetracomb          
Cellirhombated icositetrachoric tetracomb          
Celligreatorhombated icositetrachoric tetracomb          
Celliprismated icositetrachoric tetracomb          
Celliprismatotruncated icositetrachoric tetracomb          
Celligreatorhombated hexadecachoric 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          

External linksEdit