# Small rhombicosidodecahedral prism

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Small rhombicosidodecahedral prism
Rank4
TypeUniform
SpaceSpherical
Notation
Bowers style acronymSriddip
Coxeter diagramx x5o3x ()
Elements
Cells20 triangular prisms, 30 cubes, 12 pentagonal prisms, 2 small rhombicosidodecahedra
Faces40 triangles, 60+60+60 squares, 24 pentagons
Edges60+120+120
Vertices120
Vertex figureIsosceles trapezoidal pyramid, edge lengths 1, 2, (1+5)/2, 2 (base), 2 (legs)
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt2+\sqrt{10}}{2} ≈ 2.28825}$
Hypervolume${\displaystyle \frac{60+29\sqrt5}{3} ≈ 41.61532}$
Dichoral anglesTrip–4–cube: ${\displaystyle \arccos\left(-\frac{\sqrt3+\sqrt{15}}{6}\right) ≈ 159.09484°}$
Cube–4–pip: ${\displaystyle \arccos\left(-\sqrt{\frac{5+\sqrt5}{10}}\right) ≈ 148.28253°}$
Srid–5–pip: 90°
Srid–3–trip: 90°
Srid–4–cube: 90°
Height1
Central density1
Number of external pieces64
Level of complexity16
Related polytopes
ArmySriddip
RegimentSriddip
DualDeltoidal hexecontahedral tegum
ConjugateQuasirhombicosidodecahedral prism
Abstract & topological properties
Flag count3840
Euler characteristic0
OrientableYes
Properties
SymmetryH3×A1, order 240
ConvexYes
NatureTame

The small rhombicosidodecahedral prism or sriddip is a prismatic uniform polychoron that consists of 2 small rhombicosidodecahedra, 12 pentagonal prisms, 20 triangular prisms, and 30 cubes. Each vertex joins 1 small rhombicosidodecahedron, 1 pentagonal prism, 1 triangular prism, and 2 cubes. As the name suggests, it is a prism based on the small rhombicosidodecahedron. As such it is also a convex segmentochoron (designated K-4.111 on Richard Klitzing's list).

The small rhombicosidodecahedral prism can be vertex-inscribed into the small ditetrahedronary hexacosihecatonicosachoron.

## Vertex coordinates

The vertices of a small rhombicosidodecahedral prism of edge length 1 are given by all permutations of the first three coordinates of:

• ${\displaystyle \left(±\frac12,\,±\frac12,\,±\frac{2+\sqrt5}{2},\,±\frac12\right),}$

along with all even permutations of the first three coordinates of:

• ${\displaystyle \left(0,\,±\frac{3+\sqrt5}{4},\,±\frac{5+\sqrt5}{4},\,±\frac12\right),}$
• ${\displaystyle \left(±\frac{1+\sqrt5}{4},\,±\frac{1+\sqrt5}{2},\,±\frac{3+\sqrt5}{4},\,±\frac12\right).}$

## Representations

The small rhombicosidodecahedral prism has the following Coxeter diagrams:

• x x5o3x (full symmetry)
• xx5oo3xx&#x (bases considered separately)

## Related polychora

The pentagonal cupolic prism is a segmentochoron that can be obtained as a cap of the small rhombicosidodecahedral prism.

The regiment of the small rhombicosidodecahedral prism also includes the small dodecicosidodecahedral prism and the small rhombidodecahedral prism.