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Icosidodecahedron
 
 
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An icosidodecahedron is a polyhedron
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
 with twenty triangular faces and twelve 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.






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Hoberman Sphere
An icosidodecahedron is a polyhedron
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
 with twenty triangular faces and twelve 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 solid
Archimedean solid

In geometry an Archimedean solid is a highly symmetric, semi-regular convex set polyhedron composed of two or more types of regular polygons meeting in identical vertex ....
s and more particularly, a quasiregular polyhedron
Quasiregular polyhedron

A polyhedron which has regular faces and is transitive on its edges but not transitive on its faces is said to be quasiregular.A quasiregular polyhedron can have faces of only two kinds and these must alternate around each vertex....
.

An icosidodecahedron has icosahedral symmetry, and its first stellation
Stellation

Stellation is a process of constructing new polygons , new polyhedron in three dimensions, or, in general, new polytopes in n dimensions. The process consists of extending elements such as edges or face planes, usually in a symmetrical way, until they meet each other again....
 is the compound of a dodecahedron
Dodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
 and its dual icosahedron
Icosahedron

In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....
, with the vertices of the icosahedron located at the midpoints of the edges of either. Canonical coordinates for the vertices of an icosidodecahedron with unit edges are the cyclic permutation
Cyclic permutation

A cyclic permutation is built from one or more Set of elements in cyclic order.The notion cyclic permutation is used in different, but similar ways:...
s of (0,0,±τ), (±1/2, ±τ/2, ±(1+τ)/2), where τ is the golden ratio
Golden ratio

In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller....
, (1+√5)/2. Its dual polyhedron
Dual polyhedron

In geometry, polyhedron are associated into pairs called duals, where the wikt:vertex of one correspond to the face s of the other. The dual of the dual is the original polyhedron....
 is the rhombic triacontahedron
Rhombic triacontahedron

In geometry, the rhombic triacontahedron is a convex set polyhedron with 30 rhombus faces. It is an Archimedean solid solid, or a Catalan solid....
. An icosidodecahedron can be split along several planes to form pentagonal rotunda
Pentagonal rotunda

In geometry, the pentagonal rotunda is one of the Johnson solids . It can be seen as half an icosidodecahedron.The 92 Johnson solids were named and described by Norman Johnson in 1966....
e, which belong among the Johnson solid
Johnson solid

In geometry, a Johnson solid is a strictly convex set polyhedron, each face of which is a regular polygon, but which is not uniform polyhedron, i.e., not a Platonic solid, Archimedean solid, prism or antiprism....
s.

In the standard nomenclature used for the Johnson solid
Johnson solid

In geometry, a Johnson solid is a strictly convex set polyhedron, each face of which is a regular polygon, but which is not uniform polyhedron, i.e., not a Platonic solid, Archimedean solid, prism or antiprism....
s, an icosidodecahedron would be called a pentagonal gyrobirotunda.

Area and volume

The area A and the volume V of the icosidodecahedron of edge length a are:

Related polyhedra


The icosidodecahedron is a rectified
Rectification (geometry)

In Euclidean geometry, rectification is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points....
 dodecahedron
Dodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
 and also a rectified icosahedron
Icosahedron

In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....
, existing as the full-edge truncation between these regular solids.

The Icosidodecahedron contains 12 pentagons of the dodecahedron
Dodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
 and 20 triangles of the icosahedron
Icosahedron

In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....
:

Uniform Polyhedron 53 T0

Dodecahedron
Dodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
Uniform Polyhedron 53 T01

Truncated dodecahedron
Truncated dodecahedron

In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangle faces, 60 vertices and 90 edges....
Uniform Polyhedron 53 T1

Icosidodecahedron
Uniform Polyhedron 53 T12

Truncated icosahedron
Truncated icosahedron

The truncated icosahedron is an Archimedean solid. It comprises 12 regular pentagon faces, 20 regular hexagon faces, 60 vertices and 90 edges....
Uniform Polyhedron 53 T2

Icosahedron
Icosahedron

In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....


It is also related to the Johnson solid
Johnson solid

In geometry, a Johnson solid is a strictly convex set polyhedron, each face of which is a regular polygon, but which is not uniform polyhedron, i.e., not a Platonic solid, Archimedean solid, prism or antiprism....
 called a pentagonal orthobirotunda
Pentagonal orthobirotunda

In geometry, the pentagonal orthobirotunda is one of the Johnson solids . As the name suggests, it can be constructed by joining two pentagonal rotundae along their decagonal faces, matching like faces....
 created by two pentagonal rotunda
Pentagonal rotunda

In geometry, the pentagonal rotunda is one of the Johnson solids . It can be seen as half an icosidodecahedron.The 92 Johnson solids were named and described by Norman Johnson in 1966....
 connected as mirror images.

(Dissection)
Icosidodecahedron

Icosidodecahedron
(pentagonal gyrobirotunda)

Pentagonal orthobirotunda
Pentagonal Rotunda

Pentagonal rotunda


There are also 8 other uniform star polyhedra which share the same vertex arrangement
Vertex arrangement

In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes....
:
Icosidodecahedron

Icosidodecahedron
Icosidodecahedron

An icosidodecahedron is a polyhedron with twenty triangular faces and twelve 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....
Small Icosihemidodecahedron

Small icosihemidodecahedron
Small icosihemidodecahedron

In geometry, the Small icosihemidodecahedron is a nonconvex uniform polyhedron, indexed as U49.Its shares the same edge arrangement, along with its 20 triangular faces, with the icosidodecahedron....
Small Dodecahemidodecahedron

Small dodecahemidodecahedron
Small dodecahemidodecahedron

In geometry, the small dodecahemidodecahedron is a nonconvex uniform polyhedron, indexed as U51.Its edge arrangement, along with its 12 pentagonal faces, with the icosidodecahedron....
Great Icosidodecahedron

Great icosidodecahedron
Great icosidodecahedron

In geometry, the great icosidodecahedron is a nonconvex uniform polyhedron, indexed as U54.This polyhedron can be considered a Rectification great icosahedron....
Great Dodecahemidodecahedron

Great dodecahemidodecahedron
Great dodecahemidodecahedron

In geometry, the great dodecahemidodecahedron is a nonconvex uniform polyhedron, indexed as U70.It shared the same edge arrangement, along with its 12 pentagrammic faces, with the great icosidodecahedron....
Great Icosihemidodecahedron

Great icosihemidodecahedron
Great icosihemidodecahedron

In geometry, the great icosihemidodecahedron is a nonconvex uniform polyhedron, indexed as U71.It shares the same edge arrangement, along with its 20 triangular faces, with the great icosidodecahedron....
Dodecadodecahedron

Dodecadodecahedron
Dodecadodecahedron

In geometry, the dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U36.This polyhedron can be considered a Rectification great dodecahedron....
Small Dodecahemicosahedron

Small dodecahemicosahedron
Small dodecahemicosahedron

In geometry, the small dodecahemicosahedron is a nonconvex uniform polyhedron, indexed as U62.It shares the same edge arrangement, along with its 12 pentagrammic faces, with the convex dodecadodecahedron....
Great Dodecahemicosahedron

Great dodecahemicosahedron
Great dodecahemicosahedron

In geometry, the great dodecahemicosahedron is a nonconvex uniform polyhedron, indexed as U65.It shares the same edge arrangement, along with its 12 pentagonal faces, with the convex dodecadodecahedron....


See also

  • Cuboctahedron
    Cuboctahedron

    In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....
  • Dodecahedron
    Dodecahedron

    A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
  • Great truncated icosidodecahedron
    Great truncated icosidodecahedron

    In geometry, the great truncated icosidodecahedron is a nonconvex uniform polyhedron, indexed as U68....
  • Icosahedron
    Icosahedron

    In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....
  • Rhombicosidodecahedron
    Rhombicosidodecahedron

    The rhombicosidodecahedron, or small rhombicosidodecahedron, is an Archimedean solid. It has 20 regular triangle faces, 30 square faces, 12 regular pentagonal faces, 60 vertices and 120 edges....
  • Truncated icosidodecahedron
    Truncated icosidodecahedron

    The truncated icosidodecahedron is an Archimedean solid. It has 30 square faces, 20 regular hexagonal faces, 12 regular decagonal faces, 120 vertices and 180 edges....

External links

  • The Encyclopedia of Polyhedra