Icosidodecahedron
Icosidodecahedron | |
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(Click here for rotating model) | |
Type | Archimedean solid Uniform polyhedron |
Elements | F = 32, E = 60, V = 30 (χ = 2) |
Faces by sides | 20{3}+12{5} |
Conway notation | aD |
Schläfli symbols | r{5,3} |
t1{5,3} | |
Wythoff symbol | 2 | 3 5 |
Coxeter diagram |
|
Symmetry group | Ih, H3, [5,3], (*532), order 120 |
Rotation group | I, [5,3]+, (532), order 60 |
Dihedral angle | |
References | U24, C28, W12 |
Properties | Semiregular convex quasiregular
|
Colored faces |
3.5.3.5 (Vertex figure) |
Rhombic triacontahedron (dual polyhedron) |
Net |
In
Geometry
An icosidodecahedron has icosahedral symmetry, and its first stellation is the compound of a dodecahedron and its dual icosahedron, with the vertices of the icosidodecahedron located at the midpoints of the edges of either.
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.
The icosidodecahedron can be considered a
The
The icosidodecahedron has 6 central decagons. Projected into a sphere, they define 6 great circles. Buckminster Fuller used these 6 great circles, along with 15 and 10 others in two other polyhedra to define his 31 great circles of the spherical icosahedron.
Cartesian coordinates
Convenient
- (0, 0, ±φ)
- (±1/2, ±φ/2, ±φ2/2)
where φ is the golden ratio, 1 + √5/2.
The long radius (center to vertex) of the icosidodecahedron is in the golden ratio to its edge length; thus its radius is φ if its edge length is 1, and its edge length is 1/φ if its radius is 1. Only a few uniform polytopes have this property, including the four-dimensional 600-cell, the three-dimensional icosidodecahedron, and the two-dimensional decagon. (The icosidodecahedron is the equatorial cross section of the 600-cell, and the decagon is the equatorial cross section of the icosidodecahedron.) These radially golden polytopes can be constructed, with their radii, from golden triangles which meet at the center, each contributing two radii and an edge.
Orthogonal projections
The icosidodecahedron has four special
Centered by | Vertex | Edge | Face Triangle |
Face Pentagon |
---|---|---|---|---|
Solid | ||||
Wireframe | ||||
Projective symmetry |
[2] | [2] | [6] | [10] |
Dual |
Surface area and volume
The surface area A and the volume V of the icosidodecahedron of edge length a are:
Spherical tiling
The icosidodecahedron can also be represented as a
Pentagon-centered |
Triangle-centered | |
Orthographic projection | Stereographic projections |
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Orthographic projections | ||||
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2-fold, 3-fold and 5-fold symmetry axes |
Related polytopes
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:
Family of uniform icosahedral polyhedra | |||||||
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Symmetry: [5,3], (*532) | [5,3]+, (532) | ||||||
{5,3} | t{5,3} | r{5,3} | t{3,5} | {3,5} | rr{5,3} | tr{5,3} | sr{5,3} |
Duals to uniform polyhedra | |||||||
V5.5.5 | V3.10.10 | V3.5.3.5 | V5.6.6 | V3.3.3.3.3 | V3.4.5.4 | V4.6.10 | V3.3.3.3.5 |
The icosidodecahedron exists in a sequence of symmetries of quasiregular polyhedra and tilings with vertex configurations (3.n)2, progressing from tilings of the sphere to the Euclidean plane and into the hyperbolic plane. With orbifold notation symmetry of *n32 all of these tilings are wythoff construction within a fundamental domain of symmetry, with generator points at the right angle corner of the domain.[2][3]
*n32 orbifold symmetries of quasiregular tilings: (3.n)2 | |||||||
---|---|---|---|---|---|---|---|
Construction |
Spherical | Euclidean | Hyperbolic | ||||
*332 | *432 | *532 | *632 | *732 | *832... | *∞32 | |
Quasiregular figures |
|||||||
Vertex | (3.3)2 | (3.4)2 | (3.5)2 | (3.6)2 | (3.7)2 | (3.8)2 | (3.∞)2 |
*5n2 symmetry mutations of quasiregular tilings: (5.n)2 | ||||||||
---|---|---|---|---|---|---|---|---|
Symmetry *5n2 [n,5] |
Spherical | Hyperbolic | Paracompact | Noncompact | ||||
*352 [3,5] |
*452 [4,5] |
*552 [5,5] |
*652 [6,5] |
*752 [7,5] |
*852 [8,5]... |
*∞52 [∞,5] |
[ni,5] | |
Figures | ||||||||
Config. | (5.3)2 | (5.4)2 | (5.5)2
|
(5.6)2 | (5.7)2 | (5.8)2 | (5.∞)2 | (5.ni)2 |
Rhombic figures |
||||||||
Config.
|
V(5.3)2 | V(5.4)2 | V(5.5)2 | V(5.6)2 | V(5.7)2 | V(5.8)2 | V(5.∞)2 | V(5.∞)2 |
Dissection
The icosidodecahedron is related to the Johnson solid called a pentagonal orthobirotunda created by two pentagonal rotundae connected as mirror images. The icosidodecahedron can therefore be called a pentagonal gyrobirotunda with the gyration between top and bottom halves.
(Dissection) |
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Related polyhedra
The
Eight
Related polychora
In four-dimensional geometry the icosidodecahedron appears in the
If a 600-cell is stereographically projected to 3-space about any vertex and all points are normalised, the geodesics upon which edges fall comprise the icosidodecahedron's barycentric subdivision.
Icosidodecahedral graph
Icosidodecahedral graph | |
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Hamiltonian, regular | |
Table of graphs and parameters |
In the
Icosidodecahedra in nature
The Hoberman sphere is an icosidodecahedron.
Icosidodecahedra can be found in all eukaryotic cells, including human cells, as Sec13/31 COPII coat-protein formations. [5]
Trivia
In
See also
- Cuboctahedron
- Great truncated icosidodecahedron
- Icosahedron
- Rhombicosidodecahedron
- Truncated icosidodecahedron
Notes
- ^ Weisstein, Eric W. "Icosahedral group". MathWorld.
- ISBN 0-486-61480-8(Chapter V: The Kaleidoscope, Section: 5.7 Wythoff's construction)
- ^ Two Dimensional symmetry Mutations by Daniel Huson
- ^ Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269
- PMID 20070605.
References
- ISBN 0-486-23729-X. (Section 3-9)
- Cromwell, P. (1997). Polyhedra. United Kingdom: Cambridge. pp. 79–86 Archimedean solids. ISBN 0-521-55432-2.
External links
- Weisstein, Eric W., "Icosidodecahedron" ("Archimedean solid") at MathWorld.
- Klitzing, Richard. "3D convex uniform polyhedra o3x5o - id".
- Editable printable net of an icosidodecahedron with interactive 3D view
- The Uniform Polyhedra
- Virtual Reality Polyhedra The Encyclopedia of Polyhedra