| bgcolor=#e7dcc3 colspan=2 | Compound of ten tetrahedra | |
|---|---|---|
| align=center colspan=2 | ||
| bgcolor=#e7dcc3 width=50% | Type | regular compound |
| Coxeter symbol | 2[10{3,3}]2[1] | |
| Index | UC6, W25 | |
| Elements (As a compound) | 10 tetrahedra: F = 40, E = 60, V = 20 | |
| Dual compound | Self-dual | |
| Symmetry group | icosahedral (Ih) | |
| Subgroup restricting to one constituent | chiral tetrahedral (T) |
It can be seen as a faceting of a regular dodecahedron.
It can also be seen as the compound of ten tetrahedra with full icosahedral symmetry (Ih). 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 therefore be seen as a "compound of two compounds of five tetrahedra").
It can be made from the compound of five cubes by replacing each cube with a stella octangula on the cube's vertices (which results in a "compound of five compounds of two tetrahedra").
This polyhedron is a stellation of the icosahedron, and given as Wenninger model index 25.
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.
If it is treated as a simple non-convex polyhedron without self-intersecting surfaces, it has 180 faces (120 triangles and 60 concave quadrilaterals), 122 vertices (60 with degree 3, 30 with degree 4, 12 with degree 5, and 20 with degree 12), and 300 edges, giving an Euler characteristic of 122-300+180 = +2.
. Magnus Wenninger . Polyhedron Models . Cambridge University Press . 1974 . 0-521-09859-9 .