Compound of two icosahedra | |
---|---|
![]() | |
Type | Uniform compound |
Index | UC46 |
Schläfli symbols | β{3,4} βr{3,3} |
Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Polyhedra | 2 icosahedra |
Faces | 16+24 triangles |
Edges | 60 |
Vertices | 24 |
Symmetry group | octahedral (Oh) |
Subgroup restricting to one constituent | pyritohedral (Th) |
![](http://upload.wikimedia.org/wikipedia/commons/thumb/1/14/Holosnub_octahedron.png/220px-Holosnub_octahedron.png)
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 .
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.
![]() ![]() Nonuniform and uniform truncated octahedra. The first shares its vertex arrangement with this compound. |
The icosahedron, as a uniform snub tetrahedron, 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+√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
- Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society, 79 (3): 447–457, doi:10.1017/S0305004100052440, MR 0397554.