# Small icosicosidodecahedron

Small icosicosidodecahedron Rank3
TypeUniform
SpaceSpherical
Notation
Bowers style acronymSiid
Coxeter diagramx5/2o3x3*a (    )
Elements
Faces20 triangles, 12 pentagrams, 20 hexagons
Edges60+60
Vertices60
Vertex figureIsosceles trapezoid, edge lengths 1, 3, (5–1)/2, 3 Measures (edge length 1)
Circumradius$\sqrt{\frac{17+3\sqrt5}{8}} \approx 1.72149$ Volume$\frac{30+29\sqrt5}{3} \approx 31.61533$ Dihedral angles5/2–6: $\arccos\left(-\sqrt{\frac{5+2\sqrt5}{15}}\right) \approx 142.62263^\circ$ 3–6: $\arccos\left(-\frac{\sqrt5}{3}\right) \approx 138.18969^\circ$ Central density2
Number of external pieces92
Level of complexity7
Related polytopes
ArmySemi-uniform Srid, edge lengths $\frac{\sqrt5-1}{2}$ (pentagons), 1 (triangles)
RegimentSiid
DualSmall icosacronic hexecontahedron
ConjugateGreat icosicosidodecahedron
Convex coreTruncated icosahedron
Abstract & topological properties
Flag count480
Euler characteristic–8
OrientableYes
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

The small icosicosidodecahedron, or siid, is a uniform polyhedron. It consists of 20 triangles, 12 pentagrams, and 20 hexagons. One triangle, one pentagram, and two hexagons join at each vertex.

A semi-uniform variant of this polyhedron can be constructed as a rectified small ditrigonary icosidodecahedron.

## Vertex coordinates

A small icosicosidodecahedron of edge length 1 has vertex coordinates given by all even permutations of:

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

A small icosicosidodecahedron has the following Coxeter diagrams:

• x5/2o3x3*a
• ß5o3x (as holosnub)

## Related polyhedra

The small icosicosidodecahedron is the colonel of a three-member regiment that also includes the small ditrigonal dodecicosidodecahedron and the small dodecicosahedron.

o5/2o3o3*a truncations
Name OBSA CD diagram Picture
Small ditrigonary icosidodecahedron sidtid x5/2o3o3*a (   )
(degenerate, double cover of id) x5/2x3o3*a (   )
(degenerate, double cover of ike) o5/2o3x3*a (   )
Small icosicosidodecahedron siid x5/2o3x3*a (   )
(degenerate, double cover of ti) x5/2x3x3*a (   )
Small snub icosicosidodecahedron seside s5/2s3s3*a (   )