Icosidodecahedron

Icosidodecahedron
(Click here for rotating model)
Type	Archimedean solid Uniform polyhedron
Elements	F = 32, E = 60, V = 30 (χ = 2)
Faces by sides	20{3}+12{5}
Schläfli symbol	t1{5,3}
Wythoff symbol	2 | 3 5
Coxeter diagram
Symmetry group	Ih, H3, [5,3], (*532), order 120
Rotation group	I, [5,3]+, (532), order 60
Dihedral Angle	142.62° $\cos^{-1} \left(-\sqrt{\frac{1}{15}\left(5+2\sqrt{5}\right)}\right)$
References	U24, C28, W12
Properties	Semiregular convex quasiregular
Colored faces	3.5.3.5 (Vertex figure)
Rhombic triacontahedron (dual polyhedron)	Net

A Hoberman sphere as an icosidodecahedron

In geometry, an icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon. As such it is one of the Archimedean solids and more particularly, a quasiregular polyhedron.

An icosidodecahedron has icosahedral symmetry, and its first stellation is the compound of a dodecahedron and its dual icosahedron, with the vertices of the icosahedron located at the midpoints of the edges of either.

Its dual polyhedron is the rhombic triacontahedron. An icosidodecahedron can be split along any of six planes to form a pair of pentagonal rotundae, which belong among the Johnson solids.

The icosidodecahedron can be considered a pentagonal gyrobirotunda, as a combination of two rotundae (compare pentagonal orthobirotunda, one of the Johnson solids). In this form its symmetry is D_5d, [10,2⁺], (2*5), order 20.

The wire-frame figure of the icosidodecahedron consists of six flat regular decagons, meeting in pairs at each of the 30 vertices.

Cartesian coordinates

Convenient Cartesian coordinates for the vertices of an icosidodecahedron with unit edges are given by:

• (0,0,±τ)
• (0,±τ,0)
• (±τ,0,0)
• (±1/2, ±τ/2, ±(1+τ)/2)
• (±τ/2, ±(1+τ)/2, ±1/2)
• (±(1+τ)/2, ±1/2, ±τ/2)

where τ is the golden ratio, (1+√5)/2.

Orthogonal projections

The icosidodecahedron has four special orthogonal projections, centered on a vertex, an edge, a triangular face, and a pentagonal face. The last two correspond to the A₂ and H₂ Coxeter planes.

Orthogonal projections

Centered by	Vertex	Edge	Face Triangle	Face Pentagon
Image
Projective symmetry	[2]	[2]	[6]	[10]

Surface area and volume

The surface area A and the volume V of the icosidodecahedron of edge length a are:

$A = (5\sqrt{3}+3\sqrt{25+10\sqrt{5}}) a^2 \approx 29.3059828a^2$

$V = \frac{1}{6} (45+17\sqrt{5}) a^3 \approx 13.8355259a^3.$

Related polyhedra

The icosidodecahedron is a rectified dodecahedron and also a rectified icosahedron, existing as the full-edge truncation between these regular solids.

The icosidodecahedron contains 12 pentagons of the dodecahedron and 20 triangles of the icosahedron:

Family of uniform icosahedral polyhedra

Symmetry: [5,3], (*532)							[5,3]+, (532)
{5,3}	t0,1{5,3}	t1{5,3}	t0,1{3,5}	{3,5}	t0,2{5,3}	t0,1,2{5,3}	s{5,3}
Duals to uniform polyhedra
V5.5.5	V3.10.10	V3.5.3.5	V5.6.6	V3.3.3.3.3	V3.4.5.4	V4.6.10	V3.3.3.3.5

The icosidodecahedron can be seen in a sequence of quasiregular polyhedrons and tilings:

Dimensional family of quasiregular polyhedra and tilings: 3.n.3.n

Symmetry *n32 [n,3]	Spherical			Euclidean	Hyperbolic tiling
	*332 [3,3] Td	*432 [4,3] Oh	*532 [5,3] Ih	*632 [6,3] p6m	*732 [7,3]	*832 [8,3]	*∞32 [∞,3]
Quasiregular figures configuration	3.3.3.3	3.4.3.4	3.5.3.5	3.6.3.6	3.7.3.7	3.8.3.8	3.∞.3.∞
Coxeter diagram
Dual (rhombic) figures configuration	V3.3.3.3	V3.4.3.4	V3.5.3.5	V3.6.3.6	V3.7.3.7	V3.8.3.8	V3.∞.3.∞
Coxeter diagram

Pentagonal gyrobirotunda

It is also related to the Johnson solid called a pentagonal orthobirotunda created by two pentagonal rotunda connected as mirror images.

(Dissection)

Icosidodecahedron (pentagonal gyrobirotunda) Pentagonal orthobirotunda Pentagonal rotunda

Eight uniform star polyhedra share the same vertex arrangement. Of these, two also share the same edge arrangement: the small icosihemidodecahedron (having the triangular faces in common), and the small dodecahemidodecahedron (having the pentagonal faces in common). The vertex arrangement is also shared with the compounds of five octahedra and of five tetrahemihexahedra.

Icosidodecahedron	Small icosihemidodecahedron	Small dodecahemidodecahedron
Great icosidodecahedron	Great dodecahemidodecahedron	Great icosihemidodecahedron
Dodecadodecahedron	Small dodecahemicosahedron	Great dodecahemicosahedron
Compound of five octahedra	Compound of five tetrahemihexahedra

Related polytopes

In four-dimensional geometry the icosidodecahedron appears in the regular 600-cell as the equatorial slice that belongs to the vertex-first passage of the 600-cell through 3D space. In other words: the 30 vertices of the 600-cell which lie at arc distances of 90 degrees on its circumscribed hypersphere from a pair of opposite vertices, are the vertices of an icosidodecahedron. The wire frame figure of the 600-cell consists of 72 flat regular decagons. Six of these are the equatorial decagons to a pair of opposite vertices. They are precisely the six decagons which form the wire frame figure of the icosidodecahedron.

See also

• Cuboctahedron
• Great truncated icosidodecahedron
• Icosahedron
• Rhombicosidodecahedron
• Truncated icosidodecahedron

References

• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X.  (Section 3-9)

External links

• Eric W. Weisstein, Icosidodecahedron (Archimedean solid) at MathWorld
• Richard Klitzing, 3D convex uniform polyhedra, o3x5o - id
• Editable printable net of an icosidodecahedron with interactive 3D view
• The Uniform Polyhedra
• Virtual Reality Polyhedra The Encyclopedia of Polyhedra