Compound of five tetrahedra
The compound of five tetrahedra is one of the five regular polyhedral compounds. This compound polyhedron is also a stellation of the regular icosahedron. It was first described by Edmund Hess in 1876.
It can be seen as a [|faceting] of a regular dodecahedron.
As a compound
It can be constructed by arranging five tetrahedra in rotational icosahedral symmetry, 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 of this compound polyhedron. Both forms together create the reflection symmetric compound of ten tetrahedra.
It has a density of higher than 1.
As a spherical tiling | Transparent Models :image:CompoundOfFiveTetrahedra.gif| | Five interlocked tetrahedra |
As a stellation
It can also be obtained by stellating the icosahedron, and is given as Wenninger model index 24.| Stellation diagram | Stellation core | Convex hull |
Icosahedron | Dodecahedron |
As a facetting
It is a faceting of a dodecahedron, as shown at left.Group theory
The compound of five tetrahedra is a geometric illustration of the notion of orbits and stabilizers, as follows.The symmetry group of the compound is the icosahedral group I of order 60, while the stabilizer of a single chosen tetrahedron is the tetrahedral group T of order 12, and the orbit space I/T is naturally identified with the 5 tetrahedra – the coset gT corresponds to which tetrahedron g sends the chosen tetrahedron to.