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 (I_h)
Subgroup restricting to one constituent	5-fold antiprismatic (D_5d)

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

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).

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 .

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