Compound of two icosahedra

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Compound of two icosahedra
UC46-2 icosahedra.png
Type Uniform compound
Index UC46
Schläfli symbols β{3,4}
Coxeter diagrams CDel node h3.pngCDel 3.pngCDel node h3.pngCDel 4.pngCDel node.png
CDel node h3.pngCDel 3.pngCDel node h3.pngCDel 3.pngCDel node h3.png
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 Oh. As a holosnub, it is represented by Schläfli symbol β{3,4} and Coxeter diagram CDel node h3.pngCDel 3.pngCDel node h3.pngCDel 4.pngCDel node.png.

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 polyhedron-33-t012.pngUniform polyhedron-33-t012.png

Nonuniform and uniform truncated octahedra. The first shares its vertex arrangement with this compound.

The icosahedron, as a uniform snub tetrahedronSnub tetrahedron.png, is similar to these snub-pair compounds: compound of two snub cubes and compound of two snub dodecahedra.

Cartesian coordinates[edit]

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[edit]

The dual compound has two dodecahedra as pyritohedrons in dual positions:

Compound pyritohedron and dual.png


  • 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.

External links[edit]