Compound of ten tetrahedra
| Compound of ten tetrahedra | |
|---|---|
 | |
| Type | regular compound |
| Coxeter symbol | 2{5,3}[10{3,3}]2{3,5}[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) |
The compound of ten tetrahedra 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
As a spherical tiling
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.
As a stellation
This polyhedron is a stellation of the icosahedron, and given as Wenninger model index 25.
| Stellation diagram | Stellation core | Convex hull |
|---|---|---|
 |
 Icosahedron |
 Dodecahedron |
As a facetting
Ten tetrahedra in a dodecahedron.
It is also a facetting of the dodecahedron, as shown at left.
See also
References
- ↑ Regular polytopes, p.98
- Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9.
- Coxeter, Harold Scott MacDonald; Du Val, P.; Flather, H. T.; Petrie, J. F. (1999). The fifty-nine icosahedra (3rd ed.). Tarquin. ISBN 978-1-899618-32-3. MR 676126. (1st Edn University of Toronto (1938))
- H.S.M. Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, 3.6 The five regular compounds, pp.47-50, 6.2 Stellating the Platonic solids, pp.96-104
External links
- Weisstein, Eric W., "Tetrahedron 10-Compound", MathWorld.
- VRML model:
- Compounds of 5 and 10 Tetrahedra by Sándor Kabai, The Wolfram Demonstrations Project.
- Richard Klitzing, 3D compound,
| Notable stellations of the icosahedron | |||||||||
| Regular | Uniform duals | Regular compounds | Regular star | Others | |||||
| (Convex) icosahedron | Small triambic icosahedron | Medial triambic icosahedron | Great triambic icosahedron | Compound of five octahedra | Compound of five tetrahedra | Compound of ten tetrahedra | Great icosahedron | Excavated dodecahedron | Final stellation |
|---|---|---|---|---|---|---|---|---|---|
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| The stellation process on the icosahedron creates a number of related polyhedra and compounds with icosahedral symmetry. | |||||||||