Compound of ten octahedra

Polyhedral compound

Compounds of ten octahedra

Type | Uniform compound
Index | UC15 and UC16
Polyhedra | 10 octahedra
Faces | 20+60 triangles
Edges | 120
Vertices | 60
Symmetry group | icosahedral (Ih)
Subgroup restricting to one constituent | 3-fold antiprismatic (D3d)
3D model of the compound of ten octahedra UC15

3D model of the compound of ten octahedra UC16

The compounds of ten octahedra UC15 and UC16 are two uniform polyhedron compounds. They are composed of a symmetric arrangement of 10 octahedra, considered as triangular antiprisms, aligned with the axes of three-fold rotational symmetry of an icosahedron. The two compounds differ in the orientation of their octahedra: each compound may be transformed into the other by rotating each octahedron by 60 degrees.

## Cartesian coordinates

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

(0, ±(τ−1√2 \+ 2sτ), ±(τ√2 − 2sτ−1))
(±(√2 − sτ2), ±(√2 \+ s(2τ − 1)), ±(√2 \+ sτ−2))
(±(τ−1√2 − sτ), ±(τ√2 \+ sτ−1), ±3s)

where τ = (1 + √5)/2 is the golden ratio (sometimes written φ) and s is either +1 or −1\. Setting s = −1 gives UC15, while s = +1 gives UC16.

## See also

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* Compound of three octahedra
* Compound of four octahedra
* Compound of five octahedra
* Compound of twenty octahedra

## References

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* Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society, 79 (3): 447–457, doi:10.1017/S0305004100052440, MR 0397554447-457&rft.date=1976&rft_id=info:doi/10.1017/S0305004100052440&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=0397554#id-name=MR&rft.aulast=Skilling&rft.aufirst=John&rfr_id=info:sid/en.wikipedia.org:Compound+of+ten+octahedra" class="Z3988">.

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