Compound of two icosahedra
Type	Uniform compound Uniform polyhedron compound A uniform polyhedron compound is a polyhedral compound whose constituents are identical uniform polyhedra, in an arrangement that is also uniform: the symmetry group of the compound acts transitively on the compound's vertices.The uniform polyhedron compounds were first enumerated by John Skilling...
Index	UC46
Polyhedra	2 icosahedra Icosahedron In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....
Faces	16+24 triangles
Edges	60
Vertices	24
Symmetry group Symmetry group The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...	octahedral Octahedral symmetry 150px|thumb|right|The [[cube]] is the most common shape with octahedral symmetryA regular octahedron has 24 rotational symmetries, and a symmetry order of 48 including transformations that combine a reflection and a rotation...  (Oh)
Subgroup Subgroup In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...  restricting to one constituent	pyritohedral Tetrahedral symmetry 150px|right|thumb|A regular [[tetrahedron]], an example of a solid with full tetrahedral symmetryA regular tetrahedron has 12 rotational symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation.The group of all symmetries is isomorphic to the group...  (Th)

This uniform polyhedron compound

Uniform polyhedron compound

A uniform polyhedron compound is a polyhedral compound whose constituents are identical uniform polyhedra, in an arrangement that is also uniform: the symmetry group of the compound acts transitively on the compound's vertices.The uniform polyhedron compounds were first enumerated by John Skilling...

 is a composition of 2 icosahedra

Icosahedron

In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....

. It has octahedral symmetry

Octahedral symmetry

150px|thumb|right|The [[cube]] is the most common shape with octahedral symmetryA regular octahedron has 24 rotational symmetries, and a symmetry order of 48 including transformations that combine a reflection and a rotation...

 O_h.

The triangles in this compound decompose into two orbits under action of the symmetry group: 16 of the triangles lie in coplanar pairs in octahedral

Octahedron

In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex....

 planes, while the other 24 lie in unique planes.

It shares 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....

 as a nonuniform

Uniform polyhedron

A uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive...

 truncated octahedron

Truncated octahedron

In geometry, the truncated octahedron is an Archimedean solid. It has 14 faces , 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a zonohedron....

, having irregular hexagons alternating with long and short edges.

Nonuniform and uniform Uniform polyhedron A uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive...  truncated octahedra. The first shares its vertex arrangement with this compound.

The icosahedron, as a uniform snub tetrahedron, is similar to these snub-pair compounds: compound of two snub cubes

Compound of two snub cubes

This uniform polyhedron compound is a composition of the 2 enantiomers of the snub cube.The vertex arrangement of this compound is shared by a convex nonuniform truncated cuboctahedron, having rectangular faces, alongside irregular hexagons and octagons, each alternating with two edge lengths....

 and compound of two snub dodecahedra

Compound of two snub dodecahedra

This uniform polyhedron compound is a composition of the 2 enantiomers of the snub dodecahedron.The vertex arrangement of this compound is shared by a convex nonuniform truncated icosidodecahedron, with rectangular faces, alongside irregular hexagons and decagons, each alternating two different...

.

Cartesian coordinates

Cartesian coordinates for the vertices of this compound are all the permutations of

(±1, 0, ±τ)

where τ = (1+√5)/2 is the golden ratio

Golden ratio

In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874989...

 (sometimes written φ).

External links

• VRML
  VRML
  VRML is a standard file format for representing 3-dimensional interactive vector graphics, designed particularly with the World Wide Web in mind...
  
  model: http://www.interocitors.com/polyhedra/UCs/46%20-%202%20Icosahedra.wrl