Icosidodecahedron

Icosidodecahedron
	(3D model); (OFF file)
Rank	3
Type	Uniform
Space	Spherical
Notation
Bowers style acronym	Id
Coxeter diagram	o5x3o ()
Elements
Faces	20 triangles, 12 pentagons
Edges	60
Vertices	30
Vertex figure	Rectangle, edge lengths 1 and (1+√5)/2 ;
Measures (edge length 1)
Circumradius
Volume
Dihedral angle
Central density	1
Number of external pieces	32
Level of complexity	2
Related polytopes
Army	Id
Regiment	Id
Dual	Rhombic triacontahedron
Conjugate	Great icosidodecahedron
Abstract & topological properties
Flag count	240
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	H3, order 120
Convex	Yes
Nature	Tame

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[edit | edit source]

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

$\left(±\frac{1+\sqrt5}{2},\,0,\,0\right),$

and even permutations of

$\left(±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{4},\,±\frac12\right).$

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

Representations[edit | edit source]

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 (A1×A1×A1 symmetry)
oxFf(oV)fFxo ofxF(Vo)Fxfo&#xt (A1×A1 axial)

Related polyhedra[edit | edit source]

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.

o5o3o truncations
Name	OBSA	Schläfli symbol	CD diagram
Dodecahedron	doe	{5,3}	x5o3o
Truncated dodecahedron	tid	t{5,3}	x5x3o
Icosidodecahedron	id	r{5,3}	o5x3o
Truncated icosahedron	ti	t{3,5}	o5x3x
Icosahedron	ike	{3,5}	o5o3x
Small rhombicosidodecahedron	srid	rr{5,3}	x5o3x
Great rhombicosidodecahedron	grid	tr{5,3}	x5x3x
Snub dodecahedron	snid	sr{5,3}	s5s3s