Icosidodecahedron

Icosidodecahedron
Icosidodecahedron
Type	Archimedean solid; Uniform polyhedron; Quasiregular polyhedron
Faces	32
Edges	60
Vertices	30
Symmetry group	Icosahedral symmetry
Dihedral angle (degrees)	142.62°
Dual polyhedron	Rhombic triacontahedron
Properties	convex
Vertex figure
Net

In geometry, an icosidodecahedron or pentagonal gyrobirotunda is a polyhedron with twenty (icosi) triangular faces and twelve (dodeca) 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.

Construction

One way to construct the icosidodecahedron is to start with two pentagonal rotunda by attaching them to their bases. These rotundas cover their decagonal base so that the resulting polyhedron has 32 faces, 30 vertices, and 60 edges. This construction is similar to one of the Johnson solids, the pentagonal orthobirotunda. The difference is that the icosidodecahedron is constructed by twisting its rotundas by 36°, a process known as gyration, resulting in the pentagonal face connecting to the triangular one. The icosidodecahedron has an alternative name, pentagonal gyrobirotunda.^[1]^[2]

The difference between icosidodecahedron and pentagonal orthobirotunda, and its dissection.

Convenient Cartesian coordinates for the vertices of an icosidodecahedron with unit edges are given by the even permutations of: $(0,0,\pm \varphi ),\qquad \left(\pm {\frac {1}{2}},\pm {\frac {\varphi }{2}},\pm {\frac {\varphi ^{2}}{2}}\right),$ where $\varphi$ denotes the golden ratio.^[3]

Properties

The surface area of an icosidodecahedron $A$ can be determined by calculating the area of all pentagonal faces. The volume of an icosidodecahedron $V$ can be determined by slicing it off into two pentagonal rotunda, after which summing up their volumes. Therefore, its surface area and volume can be formulated as:^[1] ${\begin{aligned}A&=\left(5{\sqrt {3}}+3{\sqrt {25+10{\sqrt {5}}}}\right)a^{2}&\approx 29.306a^{2}\\V&={\frac {45+17{\sqrt {5}}}{6}}a^{3}&\approx 13.836a^{3}.\end{aligned}}$

The dihedral angle of an icosidodecahedron between pentagon-to-triangle is $\arccos \left(-{\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 142.62^{\circ },$ determined by calculating the angle of a pentagonal rotunda.^[4]

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.

The icosidodecahedron is an Archimedean solid, meaning it is a highly symmetric and semi-regular polyhedron, and two or more different regular polygonal faces meet in a vertex.^[5] The polygonal faces that meet for every vertex are two equilateral triangles and two regular pentagons, and the vertex figure of an icosidodecahedron is $(3\cdot 5)^{2}=3^{2}\cdot 5^{2}$ . Its dual polyhedron is rhombic triacontahedron, a Catalan solid.^[4]

The 60 edges form 6 decagons corresponding to great circles in the spherical tiling.

The icosidodecahedron has 6 central decagons. Projected into a sphere, they define 6 great circles. Fuller (1975) 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.^[6]

The long radius (center to vertex) of the icosidodecahedron is in the golden ratio to its edge length; thus its radius is $\varphi$ if its edge length is 1, and its edge length is $1/\varphi$ if its radius is 1.^[4] 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.

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
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.^[7]^[8]

*n32 orbifold symmetries of quasiregular tilings: (3.n)²
Construction	Spherical			Euclidean	Hyperbolic
Construction	*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* v t e
Symmetry *5n2 [n,5]	Spherical	Hyperbolic					Paracompact	Noncompact
Symmetry *5n2 [n,5]	*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

Related polyhedra

The truncated cube can be turned into an icosidodecahedron by dividing the octagons into two pentagons and two triangles. It has pyritohedral symmetry.

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.

Related polychora

In four-dimensional geometry, the icosidodecahedron appears in the regular 600-cell as the equatorial slice that belongs to the vertex-first passage of the 600-cell through 3D space. In other words: the 30 vertices of the 600-cell which lie at arc distances of 90 degrees on its circumscribed hypersphere from a pair of opposite vertices, are the vertices of an icosidodecahedron. The wireframe figure of the 600-cell consists of 72 flat regular decagons. Six of these are the equatorial decagons to a pair of opposite vertices, and these six form the wireframe figure of an icosidodecahedron.

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.

Graph

The skeleton of an icosidodecahedron can be represented as the graph with 30 vertices and 60 edges, one of the Archimedean graphs. It is quartic, meaning that each of its vertex is connected by four other vertices.^[9]

Appearance

The icosidodecahedron may appears in structural, as in the geodesic dome of the Hoberman sphere.

Icosidodecahedra can be found in all eukaryotic cells, including human cells, as Sec13/31 COPII coat-protein formations.^[10]

The icosidodecahedron may also found in popular culture. In Star Trek universe, the Vulcan game of logic Kal-Toh has the goal of creating a shape with two nested holographic icosidodecahedra joined at the midpoints of their segments.

References

^ ^a ^b Berman, Martin (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute. 291 (5): 329–352. doi:10.1016/0016-0032(71)90071-8. MR 0290245.
^ Ogievetsky, O.; Shlosman, S. (2021). "Platonic compounds and cylinders". In Novikov, S.; Krichever, I.; Ogievetsky, O.; Shlosman, S. (eds.). Integrability, Quantization, and Geometry: II. Quantum Theories and Algebraic Geometry. American Mathematical Society. p. 477. ISBN 978-1-4704-5592-7.
^ Weisstein, Eric W. "Icosahedral group". MathWorld.
^ ^a ^b ^c Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. p. 86. ISBN 978-0-486-23729-9.
^ Diudea, M. V. (2018). Multi-shell Polyhedral Clusters. Carbon Materials: Chemistry and Physics. Vol. 10. Springer. p. 39. doi:10.1007/978-3-319-64123-2. ISBN 978-3-319-64123-2.
^ Fuller, R. B. (1975). Synergetics: Explorations in the Geometry of Thinking. MacMillan. p. 183–185. ISBN 978-0-02-065320-2.
^ Coxeter Regular Polytopes, Third edition, (1973), Dover edition, 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
^ Russell, Christopher; Stagg, Scott (11 February 2010). "New Insights into the Structural Mechanisms of the COPII Coat". Traffic. 11 (3): 303–310. doi:10.1111/j.1600-0854.2009.01026.x. PMID 20070605.

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

[berman-1] Berman, Martin (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute. 291 (5): 329–352. doi:10.1016/0016-0032(71)90071-8. MR 0290245.

[os-2] Ogievetsky, O.; Shlosman, S. (2021). "Platonic compounds and cylinders". In Novikov, S.; Krichever, I.; Ogievetsky, O.; Shlosman, S. (eds.). Integrability, Quantization, and Geometry: II. Quantum Theories and Algebraic Geometry. American Mathematical Society. p. 477. ISBN 978-1-4704-5592-7.

[3] Weisstein, Eric W. "Icosahedral group". MathWorld.

[williams-4] Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. p. 86. ISBN 978-0-486-23729-9.

[diudea-5] Diudea, M. V. (2018). Multi-shell Polyhedral Clusters. Carbon Materials: Chemistry and Physics. Vol. 10. Springer. p. 39. doi:10.1007/978-3-319-64123-2. ISBN 978-3-319-64123-2.

[fuller-6] Fuller, R. B. (1975). Synergetics: Explorations in the Geometry of Thinking. MacMillan. p. 183–185. ISBN 978-0-02-065320-2.

[7] Coxeter Regular Polytopes, Third edition, (1973), Dover edition, ISBN 0-486-61480-8 (Chapter V: The Kaleidoscope, Section: 5.7 Wythoff's construction)

[8] Two Dimensional symmetry Mutations by Daniel Huson

[9] Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269

[rs-10] Russell, Christopher; Stagg, Scott (11 February 2010). "New Insights into the Structural Mechanisms of the COPII Coat". Traffic. 11 (3): 303–310. doi:10.1111/j.1600-0854.2009.01026.x. PMID 20070605.

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