Compound of five tetrahedra

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Compound of five tetrahedra
Compound of five tetrahedra.png
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
Second compound stellation of icosahedron facets.png Icosahedron.png
Icosahedron
Dodecahedron.png
Dodecahedron

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

As a compound[edit]

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.

CompoundOfFiveTetrahedra.jpg

Transparent Models (Animation)

As a stellation[edit]

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

As a facetting[edit]

Five tetrahedra in a dodecahedron.

It is a facetting of a dodecahedron, as shown at left.

Group theory[edit]

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[edit]

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[edit]

Compound of ten tetrahedra

References[edit]

External links[edit]

Notable stellations of the icosahedron
Regular Uniform duals Regular compounds Regular star Others
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
Zeroth stellation of icosahedron.png First stellation of icosahedron.png Ninth stellation of icosahedron.png First compound stellation of icosahedron.png Second compound stellation of icosahedron.png Third compound stellation of icosahedron.png Sixteenth stellation of icosahedron.png Third stellation of icosahedron.png Seventeenth stellation of icosahedron.png
Zeroth stellation of icosahedron facets.png First stellation of icosahedron facets.png Ninth stellation of icosahedron facets.png First compound stellation of icosahedron facets.png Second compound stellation of icosahedron facets.png Third compound stellation of icosahedron facets.png Sixteenth stellation of icosahedron facets.png Third stellation of icosahedron facets.png Seventeenth stellation of icosahedron facets.png
The stellation process on the icosahedron creates a number of related polyhedra and compounds with icosahedral symmetry.