Compound of six pentagrammic crossed antiprisms
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	UC29
Polyhedra	6 pentagrammic crossed antiprisms
Faces	60 triangles, 12 pentagrams
Edges	120
Vertices	60
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	5-fold antiprismatic Dihedral symmetry in three dimensions This article deals with three infinite sequences of point groups in three dimensions which have a symmetry group that as abstract group is a dihedral group Dihn .See also point groups in two dimensions.Chiral:...  (D5d)

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 6 pentagrammic crossed antiprisms. It can be constructed by inscribing within a great icosahedron one pentagrammic crossed antiprism in each of the six possible ways, and then rotating each by 36 degrees about its axis (that passes through the centres of the two opposite pentagrammic faces). It shares its vertices with the compound of 6 pentagonal antiprisms

Compound of six pentagonal antiprisms

This uniform polyhedron compound is a symmetric arrangement of 6 pentagonal antiprisms. It can be constructed by inscribing within an icosahedron one pentagonal antiprism in each of the six possible ways, and then rotating each by 36 degrees about its axis .It shares its vertex arrangement with the...

.

Cartesian coordinates

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

(±(3−4τ⁻¹), 0, ±(4+3τ⁻¹))

(±(2+4τ⁻¹), ±τ⁻¹, ±(1+2τ⁻¹))

(±(2−τ⁻¹), ±1, ±(4−2τ⁻¹))

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).