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}
Conway notation	aD
Schläfli symbols	r{5,3}
	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°
References	U24, C28, W12
Properties	Semiregular convex quasiregular
Colored faces	3.5.3.5 (Vertex figure)
Rhombic triacontahedron (dual polyhedron)	Net

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:^[1]

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

where φ 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]
Dual image

Surface area and volume

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

Spherical tiling

The icosidodecahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.

	Pentagon-centered	Triangle-centered
Orthographic projection	Stereographic projections

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}	t{5,3}	r{5,3}	2t{5,3}=t{3,5}	2r{5,3}={3,5}	rr{5,3}	tr{5,3}	sr{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 spherical polyhedra and tilings: (3.n)²

Sym. *n32 [n,3]	Spherical			Euclid.	Compact hyperb.		Paraco.	Noncompact hyperbolic
	*332 [3,3] Td	*432 [4,3] Oh	*532 [5,3] Ih	*632 [6,3] p6m	*732 [7,3]	*832 [8,3]...	*∞32 [∞,3]	[12i,3]	[9i,3]	[6i,3]	[3i,3]
Figure
Config.	r{3,3}	r{4,3}	r{5,3}	r{6,3}	r{7,3}	r{8,3}	r{∞,3}	r{12i,3}	r{9i,3}	r{6i,3}	r{3i,3}
Coxeter
Dual uniform figures
Dual conf.	V(3.3)2	V(3.4)2	V(3.5)2	V(3.6)2	V(3.7)2	V(3.8)2	V(3.∞)2
Coxeter

Dimensional family of quasiregular polyhedra and tilings: 5.n.5.n

Symmetry *5n2 [n,5]	Spherical	Hyperbolic...					Paracompact	Noncompact
	*352 [3,5]	*452 [4,5]	*552 [5,5]	*652 [6,5]	*752 [7,5]	*852 [8,5]...	*∞52 [∞,5]	[iπ/λ,5]
Coxeter
Quasiregular figures configuration	5.3.5.3	5.4.5.4	5.5.5.5	5.6.5.6	5.7.5.7	5.8.5.8	5.∞.5.∞	5.∞.5.∞
Dual figures
Coxeter
Dual (rhombic) figures configuration	V5.3.5.3	V5.4.5.4	V5.5.5.5	V5.6.5.6	V5.7.5.7	V5.8.5.8	V5.∞.5.∞	V5.∞.5.∞

Dissection

The icosidodecahedron and is related to the Johnson solid called a pentagonal orthobirotunda created by two pentagonal rotunda connected as mirror images. The icosidodecahedron can therefore be called a pentagonal gyrobirotunda with the gyration between top and bottom halves.

(Dissection)

Icosidodecahedron (pentagonal gyrobirotunda) Pentagonal orthobirotunda Pentagonal rotunda

Related polyhedra

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.

Icosidodecahedral graph

Icosidodecahedral graph
5-fold symmetry Schlegel diagram
Vertices	30
Edges	60
Automorphisms	120
Properties	Quartic graph, Hamiltonian, regular

In the mathematical field of graph theory, a icosidodecahedral graph is the graph of vertices and edges of the icosidodecahedron, one of the Archimedean solids. It has 30 vertices and 60 edges, and is a quartic graph Archimedean graph.^[2]

Orthographic projections

6-fold symmetry

10-fold symmetry

See also

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

Notes

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)
• Cromwell, P. (1997). Polyhedra. United Kingdom: Cambridge. pp. 79–86 Archimedean solids. ISBN 0-521-55432-2. 

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

Archimedean solids Truncated tetrahedron Truncated cube Truncated octahedron Truncated dodecahedron Truncated icosahedron Cuboctahedron Icosidodecahedron Truncated cuboctahedron Rhombi- cuboctahedron Snub cube Truncated icosidodecahedron Rhombicosi- dodecahedron Snub dodecahedron

Convex polyhedra Platonic solids (regular) tetrahedron cube octahedron dodecahedron icosahedron Archimedean solids (semiregular or uniform) truncated tetrahedron cuboctahedron truncated cube truncated octahedron rhombicuboctahedron truncated cuboctahedron snub cube icosidodecahedron truncated dodecahedron truncated icosahedron rhombicosidodecahedron truncated icosidodecahedron snub dodecahedron Catalan solids (duals of Archimedean) triakis tetrahedron rhombic dodecahedron triakis octahedron tetrakis hexahedron deltoidal icositetrahedron disdyakis dodecahedron pentagonal icositetrahedron rhombic triacontahedron triakis icosahedron pentakis dodecahedron deltoidal hexecontahedron disdyakis triacontahedron pentagonal hexecontahedron Dihedral regular dihedron hosohedron Dihedral uniform prisms antiprisms duals: bipyramids trapezohedra Dihedral others pyramids truncated trapezohedra gyroelongated bipyramid cupola bicupola pyramidal frusta Degenerate polyhedra are in italics.