Small rhombated icositetrachoron

Small rhombated icositetrachoron
(OFF file)
Rank	4
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Srico
Coxeter diagram	x3o4x3o ()
Elements
Cells	96 triangular prisms, 24 cuboctahedra, 24 small rhombicuboctahedra
Faces	96+192 triangles, 144+288 squares
Edges	288+576
Vertices	288
Vertex figure	Rectangular wedge, edge lengths 1 (two edges of base rectangle and top edge) and √2 (other edges of base rectangle and side edges)
Measures (edge length 1)
Circumradius	$\sqrt{4+2\sqrt2} ≈ 2.61313$
Hypervolume	$85+64\sqrt2 ≈ 175.50967$
Dichoral angles	Co–3–trip: 150°
	Sirco–4–trip: $\arccos\left(-\frac{\sqrt6}{3}\right) ≈ 144.73561^\circ$
	Sirco–4–co: 135°
	Sirco–3–sirco: 120°
Central density	1
Number of external pieces	144
Level of complexity	9
Related polytopes
Army	Srico
Regiment	Srico
Dual	Notched diacosioctacontoctachoron
Conjugate	Quasirhombated icositetrachoron
Abstract & topological properties
Flag count	10368
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	F4, order 1152
Convex	Yes
Nature	Tame

The small rhombated icositetrachoron, or srico, also commonly called the cantellated 24-cell, is a convex uniform polychoron that consists of 96 triangular prisms, 24 cuboctahedra and 24 small rhombicuboctahedra. 2 triangular prisms, 1 cuboctahedron, and 2 small rhombicuboctahedra join at each vertex. As one of its names suggests, it can be obtained by cantellating the icositetrachoron.

Vertex coordinates

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

• $\left(±(1+\sqrt2),\,±\frac{\sqrt2}{2},\,±\frac{\sqrt2}{2},\,0\right),$
• $\left(±\frac{1+2\sqrt2}{2},\,±\frac{1+\sqrt2}{2},\,±\frac{1+\sqrt2}{2},\,±\frac12\right).$

The second set of vertices are identical to those of an inscribed prismatorhombated hexadecachoron.

The cantellation of the dual icositetrachoron has vertex coordinates given by all permutations of:

• $\left(±\frac{2+\sqrt2}{2},\,±\frac{2+\sqrt2}{2},\,±1,\,0\right),$
• $\left(±\frac{3+\sqrt2}{2},\,±\frac{1+\sqrt2}{2},\,±\frac12,\,±\frac12\right).$

Representations

A small rhombated icositetrachoron has the following Coxeter diagrams:

• x3o4x3o (full symmetry)
• s3s4x3o (half symmetry, as cantic snub icositetrachoron)
• oxxowqwoxxo4xxxxoooxxxx3ooxwxwxwxoo&#xt (B3 axial, cuboctahedron-first)
• xoxoxuxoxox4oqowxxxwoqo3xxwoqoqowxx&#xt (B3 axial, small rhombicuboctahedron-first)
• ox4xx3oo3wx&#zx (B4 symmetry)
• xo4oq3xx3qo&#zx (B4 symmetry, dual ico positioning)
• wxx3ooo3xwx *b3xxw&#zx (D4 symmetry)

Semi-uniform variant

The small rhombated icositetrachoron has a semi-uniform variant of the form x3o4y3o that maintains its full symmetry. This variant uses 24 cuboctahedra of size y, 24 semi-uniform small rhombicuboctahedra of form y4o3x, and 96 triangular prisms of form x y3o as cells, with 2 edge lengths.

With edges of length a (surrounded by two small rhombicuboctahedra) and b (of cuboctahedra), its circumradius is given by $\sqrt{a^2+3b^2+2ab\sqrt2}$ and its hypervolume is given by $2a^4+54a^2b^2+29b^4+(12a^3b+52ab^3)\sqrt2$.

Variations

Besides the semi-uniform variant, another isogonal variant known as the cantic snub icositetrachoron also exists, where the small rhombicuboctahedra have pyritohedral symmetry, the cuboctahedra have tetrahedral symmetry, and the triangular prisms have pyramidal symmetry only.

Related polychora

The small rhombated icositetrachoron is the colonel of a 7-member regiment. Its other members include the retrosphenoverted trisicositetrachoron, small rhombic disicositetrachoron, small pseudorhombic disicositetrachoron, grand rhombic disicositetrachoron, disicositetrintercepted disicositetrachoron, and icositetrintercepted prismatodisicositetrachoron.

The segmentochoron cuboctahedron atop truncated cube can be obtained as a cap of the small rhombated icositetrachoron. If 8 of these caps are removed, the result is the prismatorhombated hexadecachoron, with the small rhombicuboctahedral cells all cut down to their central octagonal prism segments only.

Uniform polychoron compounds coposed of small rhombated icositetrachora include:

• Small rhombated stellated tetracontoctachoron (2)
• Small rhombated dodecahedronary hexacosichoron (25)
• Small rhombated pentishecatonicosachoron (25)

External links

• Bowers, Jonathan. "Category 6: Sphenoverts" (#151).

• Klitzing, Richard. "srico".

• Quickfur. "The Cantellated 24-cell".

• Wikipedia Contributors. "Cantellated 24-cell".