Icosidodecahedron

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Icosidodecahedron
Icosidodecahedron.jpg
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
Polyhedron 12-20 vertfig.svg
Net
Polyhedron 12-20 net.svg
3D model of an icosidodecahedron Icosidodecahedron.stl
3D model of an icosidodecahedron

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.

Contents

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]

Icosidodecahedron.png
Dissected icosidodecahedron.png
Pentagonal orthobirotunda solid.png
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: where denotes the golden ratio. [3]

Properties

The surface area of an icosidodecahedron can be determined by calculating the area of all pentagonal faces. The volume of an icosidodecahedron 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]

The dihedral angle of an icosidodecahedron between pentagon-to-triangle is 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 . Its dual polyhedron is rhombic triacontahedron, a Catalan solid. [4]

Polyhedron 12-20, davinci.png
Spherical icosidodecahedron with colored cicles.png
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 if its edge length is 1, and its edge length is 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.

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)
Uniform polyhedron-53-t0.svg Uniform polyhedron-53-t01.svg Uniform polyhedron-53-t1.svg Uniform polyhedron-53-t12.svg Uniform polyhedron-53-t2.svg Uniform polyhedron-53-t02.png Uniform polyhedron-53-t012.png Uniform polyhedron-53-s012.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.png
{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
Icosahedron.svg Triakisicosahedron.jpg Rhombictriacontahedron.svg Pentakisdodecahedron.jpg Dodecahedron.svg Deltoidalhexecontahedron.jpg Disdyakistriacontahedron.jpg Pentagonalhexecontahedronccw.jpg
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
Quasiregular fundamental domain.png
Construction
Spherical EuclideanHyperbolic
*332*432*532*632*732*832...*32
Quasiregular
figures
Uniform tiling 332-t1-1-.png Uniform tiling 432-t1.png Uniform tiling 532-t1.png Uniform tiling 63-t1.svg Triheptagonal tiling.svg H2-8-3-rectified.svg H2 tiling 23i-2.png
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 HyperbolicParacompactNoncompact
*352
[3,5]
*452
[4,5]
*552
[5,5]
*652
[6,5]
*752
[7,5]
*852
[8,5]...
*52
[,5]
 
[ni,5]
Figures Uniform tiling 532-t1.png H2-5-4-rectified.svg H2 tiling 255-2.png H2 tiling 256-2.png H2 tiling 257-2.png H2 tiling 258-2.png H2 tiling 25i-2.png
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
Rhombictriacontahedron.svg H2-5-4-rhombic.svg H2-5-4-primal.svg Order-6-5 quasiregular rhombic tiling.png
Config. V(5.3)2 V(5.4)2 V(5.5)2 V(5.6)2V(5.7)2V(5.8)2V(5.)2V(5.)2
A topological icosidodecahedron in truncated cube, inserting 6 vertices in center of octagons, and dissecting them into 2 pentagons and 2 triangles. Icosidecahedron in truncated cube.png
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.png
Icosidodecahedron
Small icosihemidodecahedron.png
Small icosihemidodecahedron
Small dodecahemidodecahedron.png
Small dodecahemidodecahedron
Great icosidodecahedron.png
Great icosidodecahedron
Great dodecahemidodecahedron.png
Great dodecahemidodecahedron
Great icosihemidodecahedron.png
Great icosihemidodecahedron
Dodecadodecahedron.png
Dodecadodecahedron
Small dodecahemicosahedron.png
Small dodecahemicosahedron
Great dodecahemicosahedron.png
Great dodecahemicosahedron
Compound of five octahedra.png
Compound of five octahedra
UC18-5 tetrahemihexahedron.png
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 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 graph of an icosidodecahedron Icosidodecahedral graph.png
The 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. [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.

See also

Related Research Articles

In geometry, a dodecahedron or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120.

<span class="mw-page-title-main">Regular icosahedron</span> Convex polyhedron with 20 triangular faces

In geometry, the regular icosahedron is a convex polyhedron that can be constructed from pentagonal antiprism by attaching two pentagonal pyramids with regular faces to each of its pentagonal faces, or by putting points onto the cube. The resulting polyhedron has 20 equilateral triangles as its faces, 30 edges, and 12 vertices. It is an example of a Platonic solid and of a deltahedron. The icosahedral graph represents the skeleton of a regular icosahedron.

<span class="mw-page-title-main">Kepler–Poinsot polyhedron</span> Any of 4 regular star polyhedra

In geometry, a Kepler–Poinsot polyhedron is any of four regular star polyhedra.

In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent regular polygons, and the same number of faces meet at each vertex. There are only five such polyhedra:

<span class="mw-page-title-main">Truncated icosahedron</span> A polyhedron resembling a soccerball

In geometry, the truncated icosahedron is a polyhedron that can be constructed by truncating all of the regular icosahedron's vertices. Intuitively, it may be regarded as footballs that are typically patterned with white hexagons and black pentagons. It can be found in the application of geodesic dome structures such as those whose architecture Buckminster Fuller pioneered are often based on this structure. It is an example of an Archimedean solid, as well as a Goldberg polyhedron.

<span class="mw-page-title-main">Rhombicosidodecahedron</span> Archimedean solid

In geometry, the rhombicosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces.

<span class="mw-page-title-main">Truncated icosidodecahedron</span> Archimedean solid

In geometry, a truncated icosidodecahedron, rhombitruncated icosidodecahedron, great rhombicosidodecahedron, omnitruncated dodecahedron or omnitruncated icosahedron is an Archimedean solid, one of thirteen convex, isogonal, non-prismatic solids constructed by two or more types of regular polygon faces.

<span class="mw-page-title-main">Snub dodecahedron</span> Archimedean solid with 92 faces

In geometry, the snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces.

<span class="mw-page-title-main">Truncated dodecahedron</span> Archimedean solid with 32 faces

In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges.

<span class="mw-page-title-main">Rhombic triacontahedron</span> Catalan solid with 30 faces

The rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types. It is a Catalan solid, and the dual polyhedron of the icosidodecahedron. It is a zonohedron.

<span class="mw-page-title-main">Triakis icosahedron</span> Catalan solid with 60 faces

In geometry, the triakis icosahedron is an Archimedean dual solid, or a Catalan solid, with 60 isosceles triangle faces. Its dual is the truncated dodecahedron. It has also been called the kisicosahedron. It was first depicted, in a non-convex form with equilateral triangle faces, by Leonardo da Vinci in Luca Pacioli's Divina proportione, where it was named the icosahedron elevatum. The capsid of the Hepatitis A virus has the shape of a triakis icosahedron.

<span class="mw-page-title-main">Disdyakis triacontahedron</span> Catalan solid with 120 faces

In geometry, a disdyakis triacontahedron, hexakis icosahedron, decakis dodecahedron or kisrhombic triacontahedron is a Catalan solid with 120 faces and the dual to the Archimedean truncated icosidodecahedron. As such it is face-uniform but with irregular face polygons. It slightly resembles an inflated rhombic triacontahedron: if one replaces each face of the rhombic triacontahedron with a single vertex and four triangles in a regular fashion, one ends up with a disdyakis triacontahedron. That is, the disdyakis triacontahedron is the Kleetope of the rhombic triacontahedron. It is also the barycentric subdivision of the regular dodecahedron and icosahedron. It has the most faces among the Archimedean and Catalan solids, with the snub dodecahedron, with 92 faces, in second place.

<span class="mw-page-title-main">Pentagonal hexecontahedron</span>

In geometry, a pentagonal hexecontahedron is a Catalan solid, dual of the snub dodecahedron. It has two distinct forms, which are mirror images of each other. It has 92 vertices that span 60 pentagonal faces. It is the Catalan solid with the most vertices. Among the Catalan and Archimedean solids, it has the second largest number of vertices, after the truncated icosidodecahedron, which has 120 vertices.

<span class="mw-page-title-main">Great icosahedron</span> Kepler-Poinsot polyhedron with 20 faces

In geometry, the great icosahedron is one of four Kepler–Poinsot polyhedra, with Schläfli symbol {3,52} and Coxeter-Dynkin diagram of . It is composed of 20 intersecting triangular faces, having five triangles meeting at each vertex in a pentagrammic sequence.

<span class="mw-page-title-main">Great truncated icosidodecahedron</span> Polyhedron with 62 faces

In geometry, the great truncated icosidodecahedron (or great quasitruncated icosidodecahedron or stellatruncated icosidodecahedron) is a nonconvex uniform polyhedron, indexed as U68. It has 62 faces (30 squares, 20 hexagons, and 12 decagrams), 180 edges, and 120 vertices. It is given a Schläfli symbol t0,1,2{5/3,3}, and Coxeter-Dynkin diagram, .

<span class="mw-page-title-main">Great snub icosidodecahedron</span> Polyhedron with 92 faces

In geometry, the great snub icosidodecahedron is a nonconvex uniform polyhedron, indexed as U57. It has 92 faces (80 triangles and 12 pentagrams), 150 edges, and 60 vertices. It can be represented by a Schläfli symbol sr{52,3}, and Coxeter-Dynkin diagram .

<span class="mw-page-title-main">Chamfered dodecahedron</span> Goldberg polyhedron with 42 faces

In geometry, the chamfered dodecahedron is a convex polyhedron with 80 vertices, 120 edges, and 42 faces: 30 hexagons and 12 pentagons. It is constructed as a chamfer (edge-truncation) of a regular dodecahedron. The pentagons are reduced in size and new hexagonal faces are added in place of all the original edges. Its dual is the pentakis icosidodecahedron.

<span class="mw-page-title-main">Regular dodecahedron</span> Convex polyhedron with 12 regular pentagonal faces

A regular dodecahedron or pentagonal dodecahedron is a dodecahedron composed of regular pentagonal faces, three meeting at each vertex. It is an example of Platonic solids, described as cosmic stellation by Plato in his dialogues, and it was used as part of Solar System proposed by Johannes Kepler. However, the regular dodecahedron, including the other Platonic solids, has already been described by other philosophers since antiquity.

<span class="mw-page-title-main">Medial rhombic triacontahedron</span> Polyhedron with 30 faces

In geometry, the medial rhombic triacontahedron is a nonconvex isohedral polyhedron. It is a stellation of the rhombic triacontahedron, and can also be called small stellated triacontahedron. Its dual is the dodecadodecahedron.

<span class="mw-page-title-main">Pentakis icosidodecahedron</span> Geodesic polyhedron with 80 faces

In geometry, the pentakis icosidodecahedron or subdivided icosahedron is a convex polyhedron with 80 triangular faces, 120 edges, and 42 vertices. It is a dual of the truncated rhombic triacontahedron.

References

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  8. Two Dimensional symmetry Mutations by Daniel Huson
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