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.
| Icosidodecahedron | |
|---|---|
 | |
| Type | Archimedean solid Uniform polyhedron Quasiregular polyhedron |
| Faces | 32 |
| Edges | 60 |
| Vertices | 30 |
| Symmetry group | Icosahedral symmetry Ih |
| Dihedral angle (degrees) | 142.62° |
| Dual polyhedron | Rhombic triacontahedron |
| Properties | convex |
| Vertex figure | |
 | |
| Net | |
 | |
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.
There is another way to construct it, and that is rectification of an Icosahedron or a Dodecahedron.
Convenient Cartesian coordinates for the vertices of an icosidodecahedron with unit edges are given by the even permutations of: 
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:
The dihedral angle of an icosidodecahedron between pentagon-to-triangle is 
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. The polygonal faces that meet for every vertex are two equilateral triangles and two regular pentagons, and the vertex figure of an icosidodecahedron is 
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.
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.
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.
| *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.∞)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.
 Icosidodecahedron |  Small icosihemidodecahedron |  Small dodecahemidodecahedron |
 Great icosidodecahedron |  Great dodecahemidodecahedron |  Great icosihemidodecahedron |
 Dodecadodecahedron |  Small dodecahemicosahedron |  Great dodecahemicosahedron |
 Compound of five octahedra |  Compound 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.
Applications
The icosidodecahedron may appear in structures, as in the geodesic dome or the Hoberman sphere.
Icosidodecahedra can be found in all eukaryotic cells, including human cells, as Sec13/31 COPII coat-protein formations.
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.
See also
- Cuboctahedron
- Great truncated icosidodecahedron
- Icosahedron
- Rhombicosidodecahedron
- Truncated icosidodecahedron
References
- 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.
- de Graef, Marc; McHenry, Michael (2012). Structure of Materials: An Introduction to Crystallography, Diffraction and Symmetry (2nd ed.). Cambridge University Press. p. 500. ISBN 978-1-139-56047-4.
- 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, 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
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 IcosidodecahedronTypeArchimedean solid Uniform polyhedron Quasiregular polyhedronFaces32Edges60Vertices30Symmetry groupIcosahedral symmetry IhDihedral angle degrees 142 62 Dual polyhedronRhombic triacontahedronPropertiesconvexVertex figureNet3D model of an icosidodecahedronConstructionOne 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 The difference between icosidodecahedron and pentagonal orthobirotunda and its dissection There is another way to construct it and that is rectification of an Icosahedron or a Dodecahedron Convenient Cartesian coordinates for the vertices of an icosidodecahedron with unit edges are given by the even permutations of 1 0 0 12 f 1f f displaystyle pm 1 0 0 qquad tfrac 1 2 left pm varphi pm tfrac 1 varphi pm varphi right where f displaystyle varphi denotes the golden ratio PropertiesThe 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 A 53 325 105 a2 29 306a2V 45 1756a3 13 836a3 displaystyle begin aligned A amp left 5 sqrt 3 3 sqrt 25 10 sqrt 5 right a 2 amp approx 29 306a 2 V amp frac 45 17 sqrt 5 6 a 3 amp approx 13 836a 3 end aligned The dihedral angle of an icosidodecahedron between pentagon to triangle is arccos 5 2515 142 62 displaystyle arccos left sqrt frac 5 2 sqrt 5 15 right approx 142 62 circ determined by calculating the angle of a pentagonal rotunda 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 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 5 2 32 52 displaystyle 3 cdot 5 2 3 2 cdot 5 2 Its dual polyhedron is rhombic triacontahedron a Catalan solid 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 The long radius center to vertex of the icosidodecahedron is in the golden ratio to its edge length thus its radius is f if its edge length is 1 and its edge length is 1 f 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 Related polytopesThe 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 polyhedraSymmetry 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 polyhedraV5 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 n32 orbifold symmetries of quasiregular tilings 3 n 2Construction Spherical Euclidean Hyperbolic 332 432 532 632 732 832 32Quasiregular figuresVertex 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 2vteSymmetry 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 FiguresConfig 5 3 2 5 4 2 5 5 2 5 6 2 5 2 5 ni 2Rhombic figuresConfig 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 2Related polyhedra A topological icosidodecahedron in truncated cube inserting 6 vertices in center of octagons and dissecting them into 2 pentagons and 2 triangles 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 Icosidodecahedron Small icosihemidodecahedron Small dodecahemidodecahedronGreat icosidodecahedron Great dodecahemidodecahedron Great icosihemidodecahedronDodecadodecahedron Small dodecahemicosahedron Great dodecahemicosahedronCompound of five octahedra Compound of five tetrahemihexahedraRelated 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 GraphThe graph of an icosidodecahedron 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 ApplicationsThe icosidodecahedron may appear in structures as in the geodesic dome or the Hoberman sphere Icosidodecahedra can be found in all eukaryotic cells including human cells as Sec13 31 COPII coat protein formations 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 See alsoCuboctahedron Great truncated icosidodecahedron Icosahedron Rhombicosidodecahedron Truncated icosidodecahedronReferencesBerman 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 de Graef Marc McHenry Michael 2012 Structure of Materials An Introduction to Crystallography Diffraction and Symmetry 2nd ed Cambridge University Press p 500 ISBN 978 1 139 56047 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 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 linksWeisstein 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
