Icositetrachoric tetracomb

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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
Related polytopes
ArmyIcot
RegimentIcot
DualHexadecachoric tetracomb
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[edit | edit source]

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.

Representations[edit | edit source]

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[edit | edit source]