Compound of six decagrammic prisms

Compound of six decagrammic prisms
Type Uniform compound
Index UC41
Polyhedra 6 decagrammic prisms
Faces 12 decagrams, 60 squares
Edges 180
Vertices 120
Symmetry group icosahedral (Ih)
Subgroup restricting to one constituent 5-fold antiprismatic (D5d)

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

Cartesian coordinates

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

(±√(τ/√5), ±2τ−1, ±√(τ−1/√5))
(±(√(τ/√5)+τ−2), ±1, ±(√(τ−1/√5)−τ−1))
(±(√(τ/√5)−τ−1), ±τ−2, ±(√(τ−1/√5)+1))
(±(√(τ/√5)+τ−1), ±τ−2, ±(√(τ−1/√5)−1))
(±(√(τ/√5)−τ−2), ±1, ±(√(τ−1/√5)+τ−1))

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

References

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  • Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society, 79 (03): 447–457, doi:10.1017/S0305004100052440, MR 0397554.