# Small icosicosidodecahedron

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.

Small icosicosidodecahedron
Rank3
TypeUniform
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${\displaystyle {\sqrt {\frac {17+3{\sqrt {5}}}{8}}}\approx 1.72149}$
Volume${\displaystyle {\frac {30+29{\sqrt {5}}}{3}}\approx 31.61533}$
Dihedral angles5/2–6: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }}$
3–6: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18969^{\circ }}$
Central density2
Number of external pieces92
Level of complexity7
Related polytopes
ArmySemi-uniform Srid, edge lengths ${\displaystyle {\frac {{\sqrt {5}}-1}{2}}}$ (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
Flag orbits4
ConvexNo
NatureTame

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:

• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {{\sqrt {5}}-1}{4}}\right),}$
• ${\displaystyle \left(0,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {\sqrt {5}}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm 1,\,\pm {\frac {3+{\sqrt {5}}}{4}}\right).}$

## Representations

A small icosicosidodecahedron has the following Coxeter diagrams:

• x3x5/2o3*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 (   )