Compound of six pentagrammic antiprisms

This uniform polyhedron compound is a chiral symmetric arrangement of 6 pentagrammic antiprisms, aligned with the axes of fivefold rotational symmetry of a dodecahedron.

Compound of six pentagrammic antiprisms
Type	Uniform compound
Index	UC44
Polyhedra	6 pentagrammic antiprisms
Faces	60 triangles, 12 pentagrams
Edges	120
Vertices	60
Symmetry group	chiral icosahedral (I)
Subgroup restricting to one constituent	5-fold dihedral (D5)

Cartesian coordinates

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

(±(τ+√τ⁻¹), ±τ⁻¹, ±(−1+√τ))

(±√τ⁻¹, ±2, ±√τ)

(±(−τ+√τ⁻¹), ±τ⁻¹, ±(1+√τ))

(±(−1+√τ⁻¹), ±(−τ), ±(τ⁻¹+√τ))

(±(1+√τ⁻¹), ±(−τ), ±(−τ⁻¹+√τ))

with an even number of minuses in the '±' choices, 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 (3): 447–457, doi:10.1017/S0305004100052440, MR 0397554.