Icosidodecahedron: Wikis

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Encyclopedia

Icosidodecahedron 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 Ih
or (*532)
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

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: 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

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 Small icosihemidodecahedron Small dodecahemidodecahedron Great icosidodecahedron Great dodecahemidodecahedron Great icosihemidodecahedron Dodecadodecahedron Small dodecahemicosahedron Great dodecahemicosahedron Compound of five octahedra Compound of five tetrahemihexahedra

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)