# Small rhombated icositetrachoron

Small rhombated icositetrachoron
Rank4
TypeUniform
Notation
Bowers style acronymSrico
Coxeter diagramx3o4x3o ()
Elements
Cells96 triangular prisms, 24 cuboctahedra, 24 small rhombicuboctahedra
Faces96+192 triangles, 144+288 squares
Edges288+576
Vertices288
Vertex figureRectangular 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${\displaystyle {\sqrt {4+2{\sqrt {2}}}}\approx 2.61313}$
Hypervolume${\displaystyle 85+64{\sqrt {2}}\approx 175.50967}$
Dichoral anglesCo–3–trip: 150°
Sirco–4–trip: ${\displaystyle \arccos \left(-{\frac {\sqrt {6}}{3}}\right)\approx 144.73561^{\circ }}$
Sirco–4–co: 135°
Sirco–3–sirco: 120°
Central density1
Number of external pieces144
Level of complexity9
Related polytopes
ArmySrico
RegimentSrico
DualNotched diacosioctacontoctachoron
ConjugateQuasirhombated icositetrachoron
Abstract & topological properties
Flag count10368
Euler characteristic0
OrientableYes
Properties
SymmetryF4, order 1152
Flag orbits9
ConvexYes
NatureTame

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:

• ${\displaystyle \left(\pm (1+{\sqrt {2}}),\,\pm {\frac {\sqrt {2}}{2}},\,\pm {\frac {\sqrt {2}}{2}},\,0\right)}$,
• ${\displaystyle \left(\pm {\frac {1+2{\sqrt {2}}}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}}\right)}$.

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:

• ${\displaystyle \left(\pm {\frac {2+{\sqrt {2}}}{2}},\,\pm {\frac {2+{\sqrt {2}}}{2}},\,\pm 1,\,0\right)}$,
• ${\displaystyle \left(\pm {\frac {3+{\sqrt {2}}}{2}},\,\pm {\frac {1+{\sqrt {2}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right)}$.

## 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 ${\displaystyle {\sqrt {a^{2}+3b^{2}+2ab{\sqrt {2}}}}}$ and its hypervolume is given by ${\displaystyle 2a^{4}+54a^{2}b^{2}+29b^{4}+(12a^{3}b+52ab^{3}){\sqrt {2}}}$.

## 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: