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
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Rico
Coxeter diagram	o3x4o3o ()
Elements
Cells	24 cubes, 24 cuboctahedra
Faces	96 triangles, 144 squares
Edges	288
Vertices	96
Vertex figure	Semi-uniform triangular prism, edge lengths √2 (base) and 1 (side)
Edge figure	cube 4 co 3 co 4
Measures (edge length 1)
Circumradius	$\sqrt3 ≈ 1.73205$
Hypervolume	29
Dichoral angles	Co–4–cube: 135°
	Co–3–co: 120°
Central density	1
Number of external pieces	48
Level of complexity	3
Related polytopes
Army	Rico
Regiment	Rico
Dual	Joined icositetrachoron
Conjugate	None
Abstract & topological properties
Flag count	3456
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	F4, order 1152
Convex	Yes
Nature	Tame

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

Vertex coordinates

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

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

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

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

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:

• Small rhombated hexadecachoron - 8 cuboctahedra, 16 rhombitetratetrahedra, and 24 square prisms with a wedge verf
• Prismatorhombated demitesseract - 3 sets of 8 rhombitetratetrahedra and 24 cuboids, with a skewed wedge verf

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:

• Rectified stellated tetracontoctachoron (2)
• Rectified chirohecatonicosachoron (5)
• Rectified dodecahedronary hexacosichoron (25)
• Rectified pentishecatonicosachoron (25)

Isogonal derivatives

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

• Cuboctahedron (24): Icositetrachoron
• Cube (24): Icositetrachoron
• Square (144): Semi-uniform small prismatotetracontoctachoron
• Triangle (96): Rectified icositetrachoron
• Edge (288): Small rhombated icositetrachoron

External links

• Bowers, Jonathan. "Category 3: Triangular Rectates" (#45).

• Bowers, Jonathan. "Category 6: Sphenoverts" (#45 - recount under tesseract symmetry).

• Bowers, Jonathan. "Tessic Isogonals".

• Klitzing, Richard. "rico".

• Quickfur. "The Rectified Icositetrachoron".

• Wikipedia Contributors. "Rectified 24-cell".