Compound of six pentagonal antiprisms

The compound of six pentagonal antiprisms is a uniform polyhedron compound. It's composed of a symmetric arrangement of 6 pentagonal antiprisms. It can be constructed by inscribing one pentagonal antiprism within an icosahedron 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 pentagonal faces).

Compound of six pentagonal antiprisms
Type	Uniform compound
Index	UC27
Polyhedra	6 pentagonal antiprisms
Faces	60 triangles, 12 pentagons
Edges	120
Vertices	60
Symmetry group	icosahedral (Ih)
Subgroup restricting to one constituent	5-fold antiprismatic (D5d)

3D model of a compound of six pentagonal antiprisms

It shares its vertex arrangement with the compound of six pentagrammic crossed antiprisms.

Cartesian coordinates

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

(±(3 + 4τ), 0, ±(4 − 3τ))

(±(2 − 4τ), ±5τ, ±(1 − 2τ))

(±(2 + τ), ±5, ±(4 + 2τ))

where τ = (1 + √5)/2 is the golden ratio (sometimes written φ).

References

• Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society, 79: 447–457, doi:10.1017/S0305004100052440, MR 0397554.