Compound of ten octahedra
{{Short description|Polyhedral compound}}
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!bgcolor=#e7dcc3 colspan=2|Compounds of ten octahedra | |
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| bgcolor=#e7dcc3|Type | Uniform compound |
| bgcolor=#e7dcc3|Index | UC15 and UC16 |
| bgcolor=#e7dcc3|Polyhedra | 10 octahedra |
| bgcolor=#e7dcc3|Faces | 20+60 triangles |
| bgcolor=#e7dcc3|Edges | 120 |
| bgcolor=#e7dcc3|Vertices | 60 |
| bgcolor=#e7dcc3|Symmetry group | icosahedral (Ih) |
| bgcolor=#e7dcc3|Subgroup restricting to one constituent | 3-fold antiprismatic (D3d) |
File:First compound of ten octahedra.stl
{{stack|File:Second compound of ten octahedra.stl}}
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.
For UC15, the convex hull of this compound is a nonuniform rhombicosidodecahedron. For UC16, the convex hull would be a nonuniform truncated icosahedron.
Cartesian coordinates
Cartesian coordinates for the vertices of this compound are all the cyclic permutations of
: (0, ±(φ−1{{radic|2}} + 2sφ), ±(φ{{radic|2}} − 2sφ−1))
: (±({{radic|2}} − sτφ2), ±({{radic|2}} + s(2τφ − 1)), ±({{radic|2}} + sφ−2))
: (±(φ−1{{radic|2}} − sφ), ±(φ{{radic|2}} + sφ−1), ±3s)
where φ = (1 + {{radic|5}})/2 is the golden ratio and s is either +1 or −1. Setting s = −1 gives UC15, while s = +1 gives UC16.
See also
References
- {{citation|first=John|last=Skilling|title=Uniform Compounds of Uniform Polyhedra|journal=Mathematical Proceedings of the Cambridge Philosophical Society|volume=79|issue=3|pages=447–457|year=1976|doi=10.1017/S0305004100052440|mr=0397554}}.
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