Icosidodecahedron

(Click here for rotating model)
TypeArchimedean solid
Uniform polyhedron
ElementsF = 32, E = 60, V = 30 (χ = 2)
Faces by sides20{3}+12{5}
Conway notationaD
Schläfli symbolsr{5,3}
t1{5,3}
Wythoff symbol2 | 3 5
Coxeter diagram
Symmetry groupIh, H3, [5,3], (*532), order 120
Rotation groupI, [5,3]+, (532), order 60
Dihedral angle
ReferencesU24, C28, W12
PropertiesSemiregular convex quasiregular

Colored faces
3.5.3.5
(Vertex figure)

Rhombic triacontahedron
(dual polyhedron)
Net
3D model of an icosidodecahedron

In geometry, an icosidodecahedron 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.

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 pentagonal gyrobirotunda, as a combination of two rotundae (compare pentagonal orthobirotunda, one of the Johnson solids). In this form its symmetry is D5d, [10,2+], (2*5), order 20.

The wire-frame figure of the icosidodecahedron consists of six flat regular decagons, meeting in pairs at each of the 30 vertices.

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 Cartesian coordinates for the vertices of an icosidodecahedron with unit edges are given by the even permutations of:[1]

  • (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 orthogonal projections, centered on a vertex, an edge, a triangular face, and a pentagonal face. The last two correspond to the A2 and H2 Coxeter planes.

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

The icosidodecahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.


Pentagon-centered
Triangle-centered
Orthographic projection Stereographic projections

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.[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)

Icosidodecahedron
(pentagonal gyrobirotunda)

Pentagonal orthobirotunda

Pentagonal rotunda
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 dodecahemidodecahedron

Great icosidodecahedron
Great dodecahemidodecahedron
Great icosihemidodecahedron

Dodecadodecahedron
Small dodecahemicosahedron
Great dodecahemicosahedron

Compound of five octahedra
Compound of five tetrahemihexahedra

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 wire frame figure of the 600-cell consists of 72 flat regular decagons. Six of these are the equatorial decagons to a pair of opposite vertices. They are precisely the six decagons which form the wire frame figure of the 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.

Icosidodecahedral graph

Icosidodecahedral graph
5-fold symmetry Schlegel diagram
Vertices30
Edges60
Automorphisms120
PropertiesQuartic graph, Hamiltonian, regular
Table of graphs and parameters

In the mathematical field of graph theory, a icosidodecahedral graph is the graph of vertices and edges of the icosidodecahedron, one of the Archimedean solids. It has 30 vertices and 60 edges, and is a quartic graph Archimedean graph.[4]

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 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

Notes

  1. ↑ Weisstein, Eric W. "Icosahedral group". MathWorld.
  2. ↑ Coxeter Regular Polytopes, Third edition, (1973), Dover edition, ISBN 0-486-61480-8 (Chapter V: The Kaleidoscope, Section: 5.7 Wythoff's construction)
  3. ↑ Two Dimensional symmetry Mutations by Daniel Huson
  4. ↑ Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269
  5. ↑ 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.

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)
  • Cromwell, P. (1997). Polyhedra. United Kingdom: Cambridge. pp. 79–86 Archimedean solids. ISBN 0-521-55432-2.
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