Rectified icositetrachoron

The rectified icositetrachoron, or rico, also commonly called the rectified 24-cell, is a convex uniform polychoron that consists of 24 cubes and 24 cuboctahedra. Two cubes and three cuboctahedra join at each triangular prismatic vertex. As the name suggests, it can be obtained by rectifying the icositetrachoron.

Rectified icositetrachoron
Schlegel half-solid cantellated 16-cell.png
Bowers style acronymRico
Coxeter diagramo3x4o3o (CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png)
Cells24 cubes, 24 cuboctahedra
Faces96 triangles, 144 squares
Vertex figureSemi-uniform triangular prism, edge lengths 2 (base) and 1 (side)
Edge figurecube 4 co 3 co 4
Measures (edge length 1)
Dichoral anglesCo–4–cube: 135°
 Co–3–co: 120°
Central density1
Number of external pieces48
Level of complexity3
Related polytopes
DualJoined icositetrachoron
Abstract & topological properties
Flag count3456
Euler characteristic0
SymmetryF4, order 1152

It can also be constructed as the cantellated hexadecachoron, under B4 symmetry. This is due to the fact that the regular icositetrachoron coincides with the rectified hexadecachoron.



Vertex coordinatesEdit

The vertices of a rectified icositetrachoron of edge length 1 are given by all permutations of:


The rectification of the dual icositetrachoron has vertices given by all permutations of:



A rectified icositetrachoron has the following Coxeter diagrams:

  • o3x4o3o (full symmetry)
  • o4x3o3x (B4 symetry, small rhombated hexadecachoron)
  • x3o3x *b3x (D4 symetry, as prismatorhombated demitesseract)
  • s4x3o3x (as snub)
  • ooqoo4xxoxx3oxxxo&#xt *B3 axial, cuboctahedron-first)
  • xoxuxox4oqoooqo3ooqoqoo&#xt (B3 axial, cube-first)
  • xo4oq3oo3qo&#zx (B4 symmetry, rectified dual ico)
  • qqo3ooo3qoq *b3oqq&#zx (D4 symmetry)
  • Qqo oqq4xxo3oxx&#zx (B3×A1 symmetry)


The rectified icositetrachoron has two semi-uniform subsymmetrical variations:

Related polychoraEdit

The rectified icositetrachoron is the colonel of a regiment of 7 members. Of these, 2 more (the facetorectified icositetrachoron and icositetrintercepted icositetrachoron) have full F4 symmetry, while the other 4 (the retrosphenoverted hexadecatesseractioctachoron, small rhombic tesseractihexadecachoron, grand rhombic dishexadecachoron, and hexadecintercepted prismatotesseractihexadecachoron) have only BC4 symmetry. It is also uniform under D4 symmetry, acting as a skewvert, but there are no members with D4 symmetry only.

Uniform polychoron compounds composed of rectified icositetrachora include:

o3o4o3o truncations
Name OBSA CD diagram Picture
Icositetrachoron ico        
Truncated icositetrachoron tico        
Rectified icositetrachoron rico        
Tetracontoctachoron cont        
Rectified icositetrachoron rico        
Truncated icositetrachoron tico        
Icositetrachoron ico        
Small rhombated icositetrachoron srico        
Great rhombated icositetrachoron grico        
Small rhombated icositetrachoron srico        
Great rhombated icositetrachoron grico        
Small prismatotetracontoctachoron spic        
Prismatorhombated icositetrachoron prico        
Prismatorhombated icositetrachoron prico        
Great prismatotetracontoctachoron gippic        
Snub disicositetrachoron sadi        
o4o3o3o truncations
Name OBSA CD diagram Picture
Tesseract tes x4o3o3o
Truncated tesseract tat x4x3o3o
Rectified tesseract rit o4x3o3o
Tesseractihexadecachoron tah o4x3x3o
Rectified hexadecachoron = Icositetrachoron ico o4o3x3o
Truncated hexadecachoron thex o4o3x3x
Hexadecachoron hex o4o3o3x
Small rhombated tesseract srit x4o3x3o
Great rhombated tesseract grit x4x3x3o
Small rhombated hexadecachoron = Rectified icositetrachoron rico o4x3o3x
Great rhombated hexadecachoron = Truncated icositetrachoron tico o4x3x3x
Small disprismatotesseractihexadecachoron sidpith x4o3o3x
Prismatorhombated hexadecachoron proh x4x3o3x
Prismatorhombated tesseract prit x4o3x3x
Great disprismatotesseractihexadecachoron gidpith x4x3x3x

Isogonal derivativesEdit

Substitution by vertices of these following elements will produce these convex isogonal polychora:

External linksEdit