Compound of six decagonal prisms

Compound of six decagonal prisms
Type Uniform compound
Index UC40
Polyhedra 6 decagonal prisms
Faces 12 decagons,
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 decagonal 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−1/√5), ±2τ, ±√(τ/√5))
(±(√(τ−1/√5)−τ2), ±1, ±(√(τ/√5)+τ))
(±(√(τ−1/√5)−τ), ±τ2, ±(√(τ/√5)+1))
(±(√(τ−1/√5)+τ), ±τ2, ±(√(τ/√5)−1))
(±(√(τ−1/√5)+τ2), ±1, ±(√(τ/√5)−τ))

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 (3): 447–457, Bibcode:1976MPCPS..79..447S, doi:10.1017/S0305004100052440, MR 0397554, S2CID 123279687.