# Icosidodecahedron

Icosidodecahedron
Rank3
TypeUniform
Notation
Bowers style acronymId
Coxeter diagramo5x3o ()
Stewart notationB5
Elements
Faces20 triangles, 12 pentagons
Edges60
Vertices30
Vertex figureRectangle, edge lengths 1 and (1+5)/2
Measures (edge length 1)
Circumradius${\displaystyle {\frac {1+{\sqrt {5}}}{2}}\approx 1.61803}$
Volume${\displaystyle {\frac {45+17{\sqrt {5}}}{6}}\approx 13.83552}$
Dihedral angle${\displaystyle \arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62263^{\circ }}$
Central density1
Number of external pieces32
Level of complexity2
Related polytopes
ArmyId
RegimentId
DualRhombic triacontahedron
ConjugateGreat icosidodecahedron
Abstract & topological properties
Flag count240
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryH3, order 120
Flag orbits2
ConvexYes
NatureTame

The icosidodecahedron, or id, is a quasiregular polyhedron and one of the 13 Archimedean solids. It consists of 20 equilateral triangles and 12 pentagons, with two of each joining at a vertex. It can be derived as a rectified dodecahedron or icosahedron.

## Vertex coordinates

An icosidodecahedron of side length 1 has vertex coordinates given by all permutations of

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

and even permutations of

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

The first set of vertices corresponds to a scaled octahedron which can be inscribed into the icosidodecahedron.

## Representations

An icosidodecahedron can be represented by the following Coxeter diagrams:

• o5x3o () (full symmetry)
• xoxfo5ofxox&#xt (H2 axial, as pentagonal gyrobirotunda, triangle-first)
• oxFfofx3xfofFxo&#xt (A2 axial)
• VooFxf oVofFx ooVxfF&#zx (K3 symmetry)
• oxFf(oV)fFxo ofxF(Vo)Fxfo&#xt (K2 axial)

## Related polyhedra

The icosidodecahedron is the colonel of a three-member regiment that also includes the small icosihemidodecahedron and the small dodecahemidodecahedron.

The icosidodecahedron can be split along an equatorial decagonal section to produce two pentagonal rotundas. Since the bases of these rotundas are in opposite orientations, an icosidodecahedron can be called the pentagonal gyrobirotunda. If one rotunda is rotated 36°, so that triangles connect to triangles and pentagons connect to pentagons, the result is the pentagonal orthobirotunda. If a decagonal prism is inserted between the two halves of the icosidodecahedron, the result is the elongated pentagonal gyrobirotunda.