compound of two icosahedra
{{Short description|Polyhedral compound}}
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!bgcolor=#e7dcc3 colspan=2|Compound of two icosahedra | |
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| bgcolor=#e7dcc3|Type | Uniform compound |
| bgcolor=#e7dcc3|Index | UC46 |
| bgcolor=#e7dcc3 width=150|Schläfli symbols | β{3,4} βr{3,3} |
| bgcolor=#e7dcc3|Coxeter diagrams | {{CDD|node_h3|3|node_h3|4|node}} {{CDD|node_h3|3|node_h3|3|node_h3}} |
| bgcolor=#e7dcc3|Polyhedra | 2 icosahedra |
| bgcolor=#e7dcc3|Faces | 16+24 triangles |
| bgcolor=#e7dcc3|Edges | 60 |
| bgcolor=#e7dcc3|Vertices | 24 |
| bgcolor=#e7dcc3|Symmetry group | octahedral (Oh) |
| bgcolor=#e7dcc3|Subgroup restricting to one constituent | pyritohedral (Th) |
This uniform polyhedron compound is a composition of 2 icosahedra. It has octahedral symmetry Oh. As a holosnub, it is represented by Schläfli symbol β{3,4} and Coxeter diagram {{CDD|node_h3|3|node_h3|4|node}}.
The triangles in this compound decompose into two orbits under action of the symmetry group: 16 of the triangles lie in coplanar pairs in octahedral planes, while the other 24 lie in unique planes.
It shares the same vertex arrangement as a nonuniform truncated octahedron, having irregular hexagons alternating with long and short edges.
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Nonuniform and uniform truncated octahedra. The first shares its vertex arrangement with this compound. |
The icosahedron, as a uniform snub tetrahedron25px, is similar to these snub-pair compounds: compound of two snub cubes and compound of two snub dodecahedra.
Together with its convex hull, it represents the icosahedron-first projection of the nonuniform snub tetrahedral antiprism.
Cartesian coordinates
Cartesian coordinates for the vertices of this compound are all the permutations of
: (±1, 0, ±τ)
where τ = (1+{{radic|5}})/2 is the golden ratio (sometimes written φ).
Compound of two dodecahedra
The dual compound has two dodecahedra as pyritohedra in dual positions:
See also
References
- {{citation|first=John|last=Skilling|title=Uniform Compounds of Uniform Polyhedra|journal=Mathematical Proceedings of the Cambridge Philosophical Society|volume=79|pages=447–457|year=1976|doi=10.1017/S0305004100052440|mr=0397554|issue=3}}.
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