Icosidodecahedron

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Icosidodecahedron
(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
Symmetry	I_h
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

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. Canonical coordinates for the vertices of an icosidodecahedron with unit edges are 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.

1 Area and volume
2 Related polyhedra
3 See also
4 References
5 External links

[edit] Area and volume

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

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

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

[edit] 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:

Dodecahedron

Truncated dodecahedron

Icosidodecahedron

Truncated icosahedron

Icosahedron

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

There are also 9 uniform star polyhedra which share the same vertex arrangement:

Great icosicosidodecahedron	Small icosihemidodecahedron	Small dodecahemidodecahedron
Great icosidodecahedron	Great dodecahemidodecahedron	Great icosihemidodecahedron
Dodecadodecahedron	Small dodecahemicosahedron	Great dodecahemicosahedron

[edit] See also

[edit] 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)