| Compound of five tetrahedra | |
|---|---|
 |
|
| Type | Regular compound |
| Stellation core | icosahedron |
| Convex hull | Dodecahedron |
| Index | UC5, W24 |
| Polyhedra | 5 tetrahedra |
| Faces | 20 triangles |
| Edges | 30 |
| Vertices | 20 |
| Dual | Self-dual |
| Symmetry group | chiral icosahedral (I) |
| Subgroup restricting to one constituent | chiral tetrahedral (T) |
This compound polyhedron is also a stellation of the regular icosahedron. It was first described by Edmund Hess in 1876.
Contents |
As a compound
It can be constructed by arranging five tetrahedra in icosahedral symmetry (I), as colored in the upper right model. It is one of five regular compounds which can be constructed from identical Platonic solids. It shares the same vertex arrangement as a regular dodecahedron. There are two enantiomorphous forms (the same figure but having opposite chirality) of this compound polyhedron. Both forms together create the reflection symmetric compound of ten tetrahedra. 
- Transparent Models
As a stellation
It can also be obtained by stellating the icosahedron, and is given as Wenninger model index 24. The stellation facets for construction are:
An unusual dual property
This compound is unusual, in that the dual figure is the enantiomorph of the original. This property seems to have led to a widespread idea that the dual of any chiral figure has the opposite chirality. The idea is generally quite false: a chiral dual nearly always has the same chirality as its twin. For example if a polyhedron has a right hand twist, then its dual will also have a right hand twist. In the case of the compound of five tetrahedra, if the faces are twisted to the right then the vertices are twisted to the left. When we dualise, the faces dualise to right-twisted vertices and the verices dualise to left-twisted faces, giving the chiral twin. Figures with this property are extremely rare.
References
- Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9.
- Coxeter, HSM, Regular Polytopes, 3rd Edn., Dover 1973.
