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(Click here for rotating model)
Type Archimedean solid
Elements F = 32, E = 60, V = 30 (χ = 2)
Faces by sides 20{3}+12{5}
Schläfli symbol \begin{Bmatrix} 3 \\ 5 \end{Bmatrix}
Wythoff symbol 2 | 3 5
Coxeter-Dynkin CDW dot.pngCDW 5.pngCDW ring.pngCDW 3.pngCDW dot.png
Symmetry Ih
or (*532)
References U24, C28, W12
Properties Semiregular convex quasiregular
Icosidodecahedron color
Colored faces
(Vertex figure)
Rhombic triacontahedron
(dual polyhedron)
Icosidodecahedron 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. Convenient Cartesian coordinates for the vertices of an icosidodecahedron with unit edges are given by the cyclic permutations of (0,0,±τ), (±1/2, ±τ/2, ±(1+τ)/2), where τ is the golden ratio, (1+√5)/2. Its dual polyhedron is the rhombic triacontahedron. An icosidodecahedron can be split along several planes to form pentagonal rotundae, which belong among the Johnson solids.

In the standard nomenclature used for the Johnson solids, an icosidodecahedron would be called a pentagonal gyrobirotunda.


Area and volume

The 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:

Uniform polyhedron-53-t0.png
Uniform polyhedron-53-t01.png
Truncated dodecahedron
Uniform polyhedron-53-t1.png
Uniform polyhedron-53-t12.png
Truncated icosahedron
Uniform polyhedron-53-t2.png

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

Dissected icosidodecahedron.png
(pentagonal gyrobirotunda)
Pentagonal orthobirotunda solid.png
Pentagonal orthobirotunda
Pentagonal rotunda.png
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.

Small icosihemidodecahedron.png
Small icosihemidodecahedron
Small dodecahemidodecahedron.png
Small dodecahemidodecahedron
Great icosidodecahedron.png
Great icosidodecahedron
Great dodecahemidodecahedron.png
Great dodecahemidodecahedron
Great icosihemidodecahedron.png
Great icosihemidodecahedron
Small dodecahemicosahedron.png
Small dodecahemicosahedron
Great dodecahemicosahedron.png
Great dodecahemicosahedron
Compound of five octahedra.png
Compound of five octahedra
UC18-5 tetrahemihexahedron.png
Compound of five tetrahemihexahedra

See also


  • 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


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