# Small icosicosidodecahedron

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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

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:

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

## 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 ()