Icosidodecahedron

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Icosidodecahedron
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Icosidodecahedron
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 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 Icosidodecahedron
3.5.3.5
(Vertex figure)
Rhombictriacontahedron.svg
Rhombic triacontahedron
(dual polyhedron) Icosidodecahedron Net
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 any of six planes to form a pair of 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.
Contents
[hide]

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

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
Dodecahedron Uniform polyhedron-53-t01.png
Truncated dodecahedron Uniform polyhedron-53-t1.png
Icosidodecahedron Uniform polyhedron-53-t12.png
Truncated icosahedron Uniform polyhedron-53-t2.png
Icosahedron

It is also related to the Johnson solid called a pentagonal orthobirotunda created by two pentagonal rotunda connected as mirror images.
Dissected icosidodecahedron.png
(Dissection)
Icosidodecahedron.png
Icosidodecahedron
(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.
Icosidodecahedron.png
Icosidodecahedron 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
Dodecadodecahedron.png
Dodecadodecahedron 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

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

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)

External links

* Eric W. Weisstein, Icosidodecahedron (Archimedean solid) at MathWorld.
* The Uniform Polyhedra
* Virtual Reality Polyhedra The Encyclopedia of Polyhedra

[show]Archimedean solids
Truncated tetrahedron.png
Truncated tetrahedron Truncated hexahedron.png
Truncated
cube Truncated octahedron.png
Truncated octahedron Truncated dodecahedron.png
Truncated dodecahedron Truncated icosahedron.png
Truncated icosahedron
Cuboctahedron.png
Cuboctahedron Icosidodecahedron.png
Icosidodecahedron
Snub hexahedron.png
Snub
cube Small rhombicuboctahedron.png
Rhombi-
cuboctahedron Great rhombicuboctahedron.png
Truncated cuboctahedron Great rhombicosidodecahedron.png
Truncated icosidodecahedron Small rhombicosidodecahedron.png
Rhomb-
icosidodecahedron Snub dodecahedron ccw.png
Snub
dodecahedron
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Polyhedron navigator
Platonic solids (regular)
tetrahedron · cube · octahedron · dodecahedron · icosahedron
Archimedean solids
(Semiregular/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
(Dual semiregular)
triakis tetrahedron · rhombic dodecahedron · triakis octahedron · tetrakis cube · 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 of dihedral uniform
bipyramids · trapezohedra
Dihedral others
pyramids · truncated trapezohedra · gyroelongated bipyramid · cupola · bicupola · pyramidal frusta
Degenerate polyhedra are in italics.
Retrieved from "http://en.wikipedia.org/wiki/Icosidodecahedron"
Categories: Archimedean solids | Quasiregular polyhedra
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