Rectified icositetrachoron

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Rectified icositetrachoron
Schlegel half-solid cantellated 16-cell.png
Rank4
TypeUniform
SpaceSpherical
Notation
Bowers style acronymRico
Coxeter diagramo3x4o3o (CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png)
Elements
Cells24 cubes, 24 cuboctahedra
Faces96 triangles, 144 squares
Edges288
Vertices96
Vertex figureSemi-uniform triangular prism, edge lengths 2 (base) and 1 (side)
Edge figurecube 4 co 3 co 4
Measures (edge length 1)
Circumradius
Hypervolume29
Dichoral anglesCo–4–cube: 135°
 Co–3–co: 120°
Central density1
Number of pieces48
Level of complexity3
Related polytopes
ArmyRico
RegimentRico
DualJoined icositetrachoron
ConjugateNone
Abstract properties
Euler characteristic0
Topological properties
OrientableYes
Properties
SymmetryF4, order 1152
ConvexYes
NatureTame

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.

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.

Cross-sections[edit | edit source]

Rico sections Bowers.png Rico-slices.gif

Vertex coordinates[edit | edit source]

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:

Representations[edit | edit source]

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)

Variations[edit | edit source]

The rectified icositetrachoron has two semi-uniform subsymmetrical variations:

Related polychora[edit | edit source]

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.

The three full-symmetric members form uniform compounds of 25 in the rissidtixhi regiment.

o3o4o3o truncations
Name OBSA CD diagram Picture
Icositetrachoron ico CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Schlegel wireframe 24-cell.png
Truncated icositetrachoron tico CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Schlegel half-solid truncated 24-cell.png
Rectified icositetrachoron rico CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Schlegel half-solid cantellated 16-cell.png
Tetracontoctachoron cont CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
Bitruncated 24-cell Schlegel halfsolid.png
Rectified icositetrachoron rico CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
Schlegel half-solid cantellated 16-cell.png
Truncated icositetrachoron tico CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Schlegel half-solid truncated 24-cell.png
Icositetrachoron ico CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
Schlegel wireframe 24-cell.png
Small rhombated icositetrachoron srico CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
Cantel 24cell1.png
Great rhombated icositetrachoron grico CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
Cantitruncated 24-cell schlegel halfsolid.png
Small rhombated icositetrachoron srico CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
Cantel 24cell1.png
Great rhombated icositetrachoron grico CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Cantitruncated 24-cell schlegel halfsolid.png
Small prismatotetracontoctachoron spic CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
Runcinated 24-cell Schlegel halfsolid.png
Prismatorhombated icositetrachoron prico CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
Prico.png
Prismatorhombated icositetrachoron prico CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Prico.png
Great prismatotetracontoctachoron gippic CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Gippic.png
Snub disicositetrachoron sadi CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Schlegel half-solid alternated cantitruncated 16-cell.png
o4o3o3o truncations
Name OBSA CD diagram Picture
Tesseract tes x4o3o3o
Schlegel wireframe 8-cell.png
Truncated tesseract tat x4x3o3o
Schlegel half-solid truncated tesseract.png
Rectified tesseract rit o4x3o3o
Schlegel half-solid rectified 8-cell.png
Tesseractihexadecachoron tah o4x3x3o
Schlegel half-solid bitruncated 8-cell.png
Rectified hexadecachoron = Icositetrachoron ico o4o3x3o
Schlegel half-solid rectified 16-cell.png
Truncated hexadecachoron thex o4o3x3x
Schlegel half-solid truncated 16-cell.png
Hexadecachoron hex o4o3o3x
Schlegel wireframe 16-cell.png
Small rhombated tesseract srit x4o3x3o
Schlegel half-solid cantellated 8-cell.png
Great rhombated tesseract grit x4x3x3o
Schlegel half-solid cantitruncated 8-cell.png
Small rhombated hexadecachoron = Rectified icositetrachoron rico o4x3o3x
Schlegel half-solid cantellated 16-cell.png
Great rhombated hexadecachoron = Truncated icositetrachoron tico o4x3x3x
Schlegel half-solid cantitruncated 16-cell.png
Small disprismatotesseractihexadecachoron sidpith x4o3o3x
Schlegel half-solid runcinated 8-cell.png
Prismatorhombated hexadecachoron proh x4x3o3x
Schlegel half-solid runcitruncated 8-cell.png
Prismatorhombated tesseract prit x4o3x3x
Schlegel half-solid runcitruncated 16-cell.png
Great disprismatotesseractihexadecachoron gidpith x4x3x3x
Schlegel half-solid omnitruncated 8-cell.png

Isogonal derivatives[edit | edit source]

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

External links[edit | edit source]