Icosidodecahedron
In geometry, an icosidodecahedron is a polyhedron with twenty (icosi) triangular faces and twelve (dodeca) 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.
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} 
t_{1}{5,3}  
Wythoff symbol  2  3 5 
Coxeter diagram  
Symmetry group  I_{h}, H_{3}, [5,3], (*532), order 120 
Rotation group  I, [5,3]^{+}, (532), order 60 
Dihedral angle  142.62° 
References  U_{24}, C_{28}, W_{12} 
Properties  Semiregular convex quasiregular 
Colored faces 
3.5.3.5 (Vertex figure) 
Rhombic triacontahedron (dual polyhedron) 
Net 
Geometry
An icosidodecahedron has icosahedral symmetry, and its first stellation is the compound of a dodecahedron and its dual icosahedron, with the vertices of the icosidodecahedron 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 wireframe figure of the icosidodecahedron consists of six flat regular decagons, meeting in pairs at each of the 30 vertices.
The icosidodecahedron has 6 central decagons. Projected into a sphere, they define 6 great circles. Buckminster Fuller used these 6 great circles, along with 15 and 10 others in two other polyhedra to define his 31 great circles of the spherical icosahedron.
Cartesian coordinates
Convenient Cartesian coordinates for the vertices of an icosidodecahedron with unit edges are given by the even permutations of:[1]
 (0, 0, ±φ)
 (±1/2, ±φ/2, ±φ^{2}/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 A_{2} and H_{2} Coxeter planes.
Centered by  Vertex  Edge  Face Triangle 
Face Pentagon 

Solid  
Wireframe  
Projective symmetry 
[2]  [2]  [6]  [10] 
Dual 
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.
Pentagoncentered 
Trianglecentered  
Orthographic projection  Stereographic projections 

Orthographic projections  

2fold, 3fold and 5fold symmetry axes 
Related polytopes
The icosidodecahedron is a rectified dodecahedron and also a rectified icosahedron, existing as the fulledge 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}  t{3,5}  {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 exists in a sequence of symmetries of quasiregular polyhedra and tilings with vertex configurations (3.n)^{2}, progressing from tilings of the sphere to the Euclidean plane and into the hyperbolic plane. With orbifold notation symmetry of *n32 all of these tilings are wythoff construction within a fundamental domain of symmetry, with generator points at the right angle corner of the domain.[2][3]
*n32 orbifold symmetries of quasiregular tilings: (3.n)^{2}  

Construction 
Spherical  Euclidean  Hyperbolic  
*332  *432  *532  *632  *732  *832...  *∞32  
Quasiregular figures 

Vertex  (3.3)^{2}  (3.4)^{2}  (3.5)^{2}  (3.6)^{2}  (3.7)^{2}  (3.8)^{2}  (3.∞)^{2} 
*5n2 symmetry mutations of quasiregular tilings: (5.n)^{2}  

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] 
[ni,5]  
Figures  
Config.  (5.3)^{2}  (5.4)^{2}  (5.5)^{2}  (5.6)^{2}  (5.7)^{2}  (5.8)^{2}  (5.∞)^{2}  (5.ni)^{2} 
Rhombic figures 

Config.  V(5.3)^{2}  V(5.4)^{2}  V(5.5)^{2}  V(5.6)^{2}  V(5.7)^{2}  V(5.8)^{2}  V(5.∞)^{2}  V(5.∞)^{2} 
Dissection
The icosidodecahedron 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) 

Related polyhedra
The truncated cube can be turned into an icosidodecahedron by dividing the octagons into two pentagons and two triangles. It has pyritohedral symmetry.
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.
Related polychora
In fourdimensional geometry the icosidodecahedron appears in the regular 600cell as the equatorial slice that belongs to the vertexfirst passage of the 600cell through 3D space. In other words: the 30 vertices of the 600cell 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 600cell 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  

5fold symmetry Schlegel diagram  
Vertices  30 
Edges  60 
Automorphisms  120 
Properties  Quartic graph, Hamiltonian, regular 
Table of graphs and parameters 
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.[4]
Trivia
In Star Trek Universe, the Vulcan game of logic KalToh has the goal to create a holographic icosidodecahedron.
In The Wrong Stars, book one of the Axiom series, by Tim Pratt, Elena has a icosidodecahedron machine on either side of her. [Paperback p 336]
See also
Notes
 Weisstein, Eric W. "Icosahedral group". MathWorld.
 Coxeter Regular Polytopes, Third edition, (1973), Dover edition, ISBN 0486614808 (Chapter V: The Kaleidoscope, Section: 5.7 Wythoff's construction)
 Two Dimensional symmetry Mutations by Daniel Huson
 Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269
References
 Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 048623729X. (Section 39)
 Cromwell, P. (1997). Polyhedra. United Kingdom: Cambridge. pp. 79–86 Archimedean solids. ISBN 0521554322.
External links
 Eric W. Weisstein, Icosidodecahedron (Archimedean solid) at MathWorld.
 Klitzing, Richard. "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