Compound of ten tetrahedra


The compound of ten tetrahedra, also known as the icosicosahedron, is one of the five regular polyhedral compounds. This polyhedron can be seen as either a stellation of the icosahedron or a compound. This compound was first described by Edmund Hess in 1876.
It can be seen as a [|faceting] of a regular dodecahedron.

As a compound

It can also be seen as the compound of ten tetrahedra with full icosahedral symmetry. It is one of five regular compounds constructed from identical Platonic solids.
It shares the same vertex arrangement as a dodecahedron.
The compound of five tetrahedra represents two chiral halves of this compound.
It can be made from the compound of five cubes by replacing each cube with a stella octangula on the cube's vertices.

As a stellation

This polyhedron is a stellation of the icosahedron, and given as Wenninger model index 25.
Stellation diagramStellation coreConvex hull

Icosahedron
Dodecahedron

As a facetting

It is also a facetting of the dodecahedron, as shown at left. Concave pentagrams can be seen on the compound where the pentagonal faces of the dodecahedron are positioned.

As a simple polyhedron

If it is treated as a simple non-convex polyhedron without self-intersecting surfaces, it has 180 faces, 122 vertices, and 300 edges, giving an Euler characteristic of 122-300+180 = +2.