# Rectified icositetrachoron

Rectified icositetrachoron
Rank4
TypeUniform
SpaceSpherical
Notation
Bowers style acronymRico
Coxeter diagramo3x4o3o ()
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${\displaystyle \sqrt3 ≈ 1.73205}$
Hypervolume29
Dichoral anglesCo–4–cube: 135°
Co–3–co: 120°
Central density1
Number of external pieces48
Level of complexity3
Related polytopes
ArmyRico
RegimentRico
DualJoined icositetrachoron
ConjugateNone
Abstract & topological properties
Flag count3456
Euler characteristic0
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.

## Vertex coordinates

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

• ${\displaystyle \left(±\sqrt2,\,±\frac{\sqrt2}{2},\,±\frac{\sqrt2}{2},\,0\right).}$

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

• ${\displaystyle \left(±\frac32,\,±\frac12,\,±\frac12,\,±\frac12\right),}$
• ${\displaystyle \left(±1,\,±1,\,±1,\,0\right).}$

## Representations

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

The rectified icositetrachoron has two semi-uniform subsymmetrical variations:

## Related polychora

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
o4o3o3o truncations
Name OBSA CD diagram Picture
Tesseract tes x4o3o3o
Truncated tesseract tat x4x3o3o
Rectified tesseract rit o4x3o3o
Rectified hexadecachoron = Icositetrachoron ico o4o3x3o