# Icosidodecahedron

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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,±&tau;), (±1/2, ±&tau;/2, ±(1+&tau;)/2), where &tau; is the golden ratio, (1+&radic;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::$A = \left(5sqrt\left\{3\right\}+3sqrt\left\{25+10sqrt\left\{5\right) a^2 approx 29.3059828a^2$:$V = frac\left\{1\right\}\left\{6\right\} \left(45+17sqrt\left\{5\right\}\right) a^3 approx 13.8355259a^3.$

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:

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

* Cuboctahedron
* Dodecahedron
* Great truncated icosidodecahedron
* Icosahedron
* Rhombicosidodecahedron
* Truncated icosidodecahedron

References

* (Section 3-9)

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* [http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
* [http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] The Encyclopedia of Polyhedra
* [http://www.lifeisastoryproblem.org/explore/net_icosidodecahedron.pdf Printable Geometric Net of an Icosidodecahedron] [http://www.lifeisastoryproblem.org Life is a Story Problem.org]

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