Compound of two icosahedra

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)

This uniform polyhedron compound is a composition of 2 icosahedra. It has octahedral symmetry O_h. As a holosnub, it is represented by Schläfli symbol β{3,4} and Coxeter diagram .

Holosnub octahedron, β{3,4}

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

• Compound of two snub cubes
• Compound of two snub dodecahedra

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.