Compound of six pentagrammic crossed antiprisms

This uniform polyhedron compound 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 pentagrammic crossed antiprisms
Type	Uniform compound
Index	UC29
Polyhedra	6 pentagrammic crossed antiprisms
Faces	60 triangles, 12 pentagrams
Edges	120
Vertices	60
Symmetry group	icosahedral (Ih)
Subgroup restricting to one constituent	5-fold antiprismatic (D5d)

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 (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.