Compound of five tetrahedra

Compound of five tetrahedra
Type	Regular compound
Index	UC5, W24
Elements (As a compound)	5 tetrahedra: F = 20, E = 30, V = 20
Dual compound	Self-dual
Symmetry group	chiral icosahedral (I)
Subgroup restricting to one constituent	chiral tetrahedral (T)
Stellation diagram Stellation core Convex hull Icosahedron Dodecahedron

This compound polyhedron is also a stellation of the regular icosahedron. It was first described by Edmund Hess in 1876.

As a compound

Physical model of compound of 5 tetrahedra (Animation).

It can be constructed by arranging five tetrahedra in rotational 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 (Animation)

As a stellation

It can also be obtained by stellating the icosahedron, and is given as Wenninger model index 24.

As a facetting

Five tetrahedra in a dodecahedron.

It is a facetting 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 (rotational) icosahedral group I of order 60, while the stabilizer of a single chosen tetrahedron is the (rotational) tetrahedral group T of order 12, and the orbit space I/T (of order 60/12 = 5) is naturally identified with the 5 tetrahedra – the coset gT corresponds to which tetrahedron g sends the chosen tetrahedron to.

An unusual dual property

Compound of five tetrahedra

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 vertices dualise to left-twisted faces, giving the chiral twin. Figures with this property are extremely rare.

See also

Compound of ten tetrahedra

References

• Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9. 
• Coxeter, HSM, Regular Polytopes, 3rd Edn., Dover 1973.
• 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. MR676126  (1st Edn University of Toronto (1938))

External links

• Weisstein, Eric W., "Tetrahedron 5-Compound" from MathWorld.
• Metal Sculpture of Five Tetrahedra Compound
• VRML model: [1]
• Compounds of 5 and 10 Tetrahedra by Sándor Kabai, The Wolfram Demonstrations Project.

Notable stellations of the icosahedron
Regular	Uniform duals		Regular compounds			Regular star	Others
Icosahedron	Small triambic icosahedron	Great triambic icosahedron	Compound of five octahedra	Compound of five tetrahedra	Compound of ten tetrahedra	Great icosahedron	Excavated dodecahedron	Final stellation
The stellation process on the icosahedron creates a number of related polyhedra and compounds with icosahedral symmetry.