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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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