Compound of twenty octahedra with rotational freedom
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	UC13
Polyhedra	20 octahedra 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....
Faces	40+120 triangles
Edges	240
Vertices	120
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...	icosahedral Icosahedral symmetry A regular icosahedron has 60 rotational symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation...  (Ih)
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	6-fold improper rotation Cyclic symmetries This article deals with the four infinite series of point groups in three dimensions with n-fold rotational symmetry about one axis , and no other rotational symmetry :Chiral:*Cn of order n - n-fold rotational symmetry...  (S6)

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 symmetric arrangement of 20 octahedra

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....

, considered as triangular antiprisms. It can be constructed by superimposing two copies of the compound of 10 octahedra UC₁₆, and for each resulting pair of octahedra, rotating each octahedron in the pair by an equal and opposite angle θ.

When θ is zero or 60 degrees, the octahedra coincide in pairs yielding (two superimposed copies of) the compounds of ten octahedra UC₁₆ and UC₁₅ respectively. At a certain intermediate angle, octahedra (from distinct rotational axes) coincide in sets four, yielding the compound of five octahedra

Compound of five octahedra

This polyhedron can be seen as either a polyhedral stellation or a compound. This compound was first described by Edmund Hess in 1876.- As a stellation :It is the second stellation of the icosahedron, and given as Wenninger model index 23....

. At another intermediate angle the vertices coincide in pairs, yielding the compound of twenty octahedra

Compound of twenty octahedra

This uniform polyhedron compound is a symmetric arrangement of 20 octahedra . It is a special case of the compound of 20 octahedra with rotational freedom, in which pairs of octahedral vertices coincide.- Related polyhedra :...

 (without rotational freedom).

Cartesian coordinates

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

(±2(√3)sinθ, ±(τ⁻¹√2+2τcosθ), ±(τ√2−2τ⁻¹cosθ))

(±(√2−τ²cosθ+τ⁻¹(√3)sinθ), ±(√2+(2τ−1)cosθ+(√3)sinθ), ±(√2+τ⁻²cosθ−τ(√3)sinθ))

(±(τ⁻¹√2−τcosθ−τ(√3)sinθ), ±(τ√2+τ⁻¹cosθ+τ⁻¹(√3)sinθ), ±(3cosθ−(√3)sinθ))

(±(−τ⁻¹√2+τcosθ−τ(√3)sinθ), ±(τ√2+τ⁻¹cosθ−τ⁻¹(√3)sinθ), ±(3cosθ+(√3)sinθ))

(±(−√2+τ²cosθ+τ⁻¹(√3)sinθ), ±(√2+(2τ−1)cosθ−(√3)sinθ), ±(√2+τ⁻²cosθ+τ(√3)sinθ))

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).