The compound of five

tetrahedra In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...

is one of the five regular polyhedral compounds. This compound

polyhedron In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer ...

is also a

stellation In geometry, stellation is the process of extending a polygon in two dimensions, a polyhedron in three dimensions, or, in general, a polytope in ''n'' dimensions to form a new figure. Starting with an original figure, the process extends specific ...

of the regular

icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrical tha ...

. It was first described by

Edmund Hess Edmund Hess (17 February 1843 – 24 December 1903) was a German mathematician who discovered several regular polytopes. Publications *''Über die zugleich gleicheckigen und gleichflächigen Polyeder.'' In: Sitzungsberichte der Gesellscha ...

in 1876. It can be seen as a

faceting Stella octangula as a faceting of the cube In geometry, faceting (also spelled facetting) is the process of removing parts of a polygon, polyhedron or polytope, without creating any new Vertex (geometry), vertices. New edges of a faceted po ...

of a

regular dodecahedron A regular dodecahedron or pentagonal dodecahedronStrictly speaking, a pentagonal dodecahedron need not be composed of regular pentagons. The name "pentagonal dodecahedron" therefore covers a wider class of solids than just the Platonic solid, the ...

As a compound

It can be constructed by arranging five

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 solid In geometry, a Platonic solid is a Convex polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (id ...

s. It shares the same vertex arrangement as a

. 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. It has a

density Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...

of higher than 1.

As a stellation

It can also be obtained by stellating the

, and is given as Wenninger model index 24.

As a faceting

It is a

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 In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual of th ...

''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

Second compound stellation of icosahedron

This compound is unusual, in that the dual figure is the

enantiomorph In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by Rotation (mathematics), rotations and Translation (geometry), translations a ...

of the original. 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.

In 4-dimensional space

The compound of five tetrahedra is related to the regular

5-cell In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional space, four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, hypertetrahedron, pentachoron, pentatope, pe ...

, the 4-simplex

regular 4-polytope In mathematics, a regular 4-polytope or regular polychoron is a regular polytope, regular 4-polytope, four-dimensional polytope. They are the four-dimensional analogues of the Regular polyhedron, regular polyhedra in three dimensions and the regul ...

, which is also composed of 5 regular tetrahedra. In the 5-cell the tetrahedra are bonded face-to-face such that each triangular face is shared by two tetrahedral cells. The compound of five tetrahedra occurs embedded in 4-dimensional space, inscribed in the 120 dodecahedral cells of the

120-cell In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called a C120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, hec ...

. The 120-cell is the largest and most comprehensive regular 4-polytope; the regular 5-cell is the smallest and most elemental. The 120-cell contains inscribed within itself instances of every other regular 4-polytope. In each of the 120-cell's dodecahedral cells, there are two inscribed instances of the compound of 5 tetrahedra (in other words, one instance of the compound of ten tetrahedra). The 5 tetrahedra of each compound of five occur as cells of another regular 4-polytope inscribed within the 120-cell, the

600-cell In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also known as the C600, hexacosichoron and hexacosihedroid. It is also called a tetraplex (abbreviated from ...

, which has 600 regular tetrahedra as its cells. The 120-cell is a compound of 5 disjoint 600-cells, and each of its dodecahedral cells is a compound of 5 tetrahedral cells, one cell from each of the 5 disjoint 600-cells.

Citations

References

* * *

External links

*
Metal Sculpture of Five Tetrahedra Compound
* VRML model

Compounds of 5 and 10 Tetrahedra
by Sándor Kabai,

The Wolfram Demonstrations Project The Wolfram Demonstrations Project is an open-source collection of interactive programmes called Demonstrations. It is hosted by Wolfram Research. At its launch, it contained 1300 demonstrations but has grown to over 10,000. The site won a Pa ...

. * {{Icosahedron stellations Polyhedral stellation Polyhedral compounds Chiral polyhedra