Polyhedral compound

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Polyhedral compound

A polyhedral compound is a polyhedron

Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
that is itself composed of several other polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compound

Star polygon

A star polygon is a non-convex polygon which looks in some way like a star. Only the regular ones have been studied in any depth; star polygons in general have never been formally defined....
s such as the hexagram

Hexagram

A hexagram is a six-pointed geometric star figure, or 2, the compound of two equilateral triangle s. The intersection is a regular hexagon.While generally recognized as a symbol of Jewish identity it is used also in other historical, religious and cultural contexts, for example in #Use of the Star by Arabs and Muslims, and #Occurrence in...
.

Neighbouring vertices of a compound can be connected to form a convex polyhedron called the convex hull

Convex hull

In mathematics, the convex hull or convex envelope for a Set of points X in a real vector space V is the minimal convex set containing X....
. The compound is a facetting

Facetting

|}In geometry, facetting is the process of removing parts of a polygon, polyhedron or polytope, without creating any new vertices.Facetting is the reciprocal or dual process to stellation....
of the convex hull.

Another convex polyhedron is formed by the small central space common to all members of the compound.

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Encyclopedia

A polyhedral compound is a polyhedron

Polyhedron

Star polygon

Hexagram

Convex hull

In mathematics, the convex hull or convex envelope for a Set of points X in a real vector space V is the minimal convex set containing X....
. The compound is a facetting

Facetting

Stellation

Stellation is a process of constructing new polygons , new polyhedron in three dimensions, or, in general, new polytopes in n dimensions. The process consists of extending elements such as edges or face planes, usually in a symmetrical way, until they meet each other again....
s including this compound. (See List of Wenninger polyhedron models

List of Wenninger polyhedron models

This table contains an indexed list of the Uniform and stellated polyhedra from the book Polyhedron Models, by Magnus Wenninger.The book was written as a guide book to building polyhedra as physical models....
for these compounds and more stellations.)

Regular compounds

A regular polyhedron compound can be defined as a compound which, like a regular polyhedron, is vertex-transitive

Vertex-transitive

Components	Convex hull Convex hull In mathematics, the convex hull or convex envelope for a Set of points X in a real vector space V is the minimal convex set containing X....	Core	Symmetry Symmetry group The symmetry group of an object is the group of all isometries under which it is invariant with Function composition as the operation. It is a subgroup of the isometry group of the space concerned....	Dual
Compound of two tetrahedra, or Stella octangula Stella octangula The stella octangula, also known as the stellated octahedron, Star Tetrahedron, eight-pointed star, or 2D geometric model as the Star of David....	Cube	Octahedron Octahedron An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....	O_h	Self-dual
Compound of five tetrahedra Compound of five tetrahedra This Polyhedron compound polyhedron is also a stellation of the regular icosahedron. It was first described by Edmund Hess in 1876....	Dodecahedron Dodecahedron A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....	Icosahedron Icosahedron In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....	I	enantiomorph Chirality (mathematics) In geometry, a figure is chiral if it is not identical to its mirror image, or more particularly if it cannot be mapped to its mirror image by rotations and translations alone.... , or chiral Chirality (mathematics) In geometry, a figure is chiral if it is not identical to its mirror image, or more particularly if it cannot be mapped to its mirror image by rotations and translations alone.... twin
Compound of ten tetrahedra Compound of ten tetrahedra This polyhedron can be seen as either a polyhedral stellation or a Polyhedron compound. This compound was first described by Edmund Hess in 1876....	Dodecahedron Dodecahedron A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....	Icosahedron Icosahedron In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....	I_h	Self-dual
Compound of five cubes Compound of five cubes This polyhedral compound is a symmetric arrangement of five cubes. This compound was first described by Edmund Hess in 1876.It is one of five Polyhedral_compound#Regular_compounds, and dual to the compound of five octahedra....	Dodecahedron Dodecahedron A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....	Rhombic triacontahedron Rhombic triacontahedron In geometry, the rhombic triacontahedron is a convex set polyhedron with 30 rhombus faces. It is an Archimedean solid solid, or a Catalan solid....	I_h	Compound of five octahedra Compound of five octahedra This polyhedron can be seen as either a polyhedral stellation or a Polyhedron compound. This compound was first described by Edmund Hess in 1876....
Compound of five octahedra Compound of five octahedra This polyhedron can be seen as either a polyhedral stellation or a Polyhedron compound. This compound was first described by Edmund Hess in 1876....	Icosidodecahedron Icosidodecahedron An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon....	Icosahedron Icosahedron In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....	I_h	Compound of five cubes Compound of five cubes This polyhedral compound is a symmetric arrangement of five cubes. This compound was first described by Edmund Hess in 1876.It is one of five Polyhedral_compound#Regular_compounds, and dual to the compound of five octahedra....

Best known is the compound of two tetrahedra

Tetrahedron

A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....
, often called the stella octangula

Stella octangula

The stella octangula, also known as the stellated octahedron, Star Tetrahedron, eight-pointed star, or 2D geometric model as the Star of David....
, a name given to it by Kepler

Johannes Kepler

Johannes Kepler was a Germans mathematician, astronomer and astrologer, and key figure in the 17th century Scientific revolution. He is best known for his eponymous Kepler's laws of planetary motion, codified by later astronomers based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican Astrononomy....
. The vertices of the two tetrahedra define a cube and the intersection of the two an octahedron

Octahedron

An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....
, which shares the same face-planes as the compound. Thus it is a stellation

Stellation

Stellation is a process of constructing new polygons , new polyhedron in three dimensions, or, in general, new polytopes in n dimensions. The process consists of extending elements such as edges or face planes, usually in a symmetrical way, until they meet each other again....
of the octahedron, and in fact, the only finite stellation thereof.

The stella octangula can also be regarded as a dual-regular compound.

The compound of five tetrahedra

Compound of five tetrahedra

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

Chirality (mathematics)

In geometry, a figure is chiral if it is not identical to its mirror image, or more particularly if it cannot be mapped to its mirror image by rotations and translations alone....
ic versions, which together make up the compound of 10 tetrahedra. Each of the tetrahedral compounds is self-dual, and the compound of 5 cubes is dual to the compound of 5 octahedra.

Dual-regular compounds

A dual-regular compound is composed of a regular polyhedron (one of the Platonic solid

Platonic solid

In geometry, a Platonic solid is a convex set polyhedron that is regular polyhedron, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruence regular polygons, with the same number of faces meeting at each vertex....
s or Kepler-Poinsot polyhedra) and its regular dual, arranged reciprocally about a common intersphere or midsphere, such that the edge of one polyhedron intersects the dual edge of the dual polyhedron. There are five such compounds.

Components	Convex hull Convex hull In mathematics, the convex hull or convex envelope for a Set of points X in a real vector space V is the minimal convex set containing X....	Core	Symmetry Symmetry group The symmetry group of an object is the group of all isometries under which it is invariant with Function composition as the operation. It is a subgroup of the isometry group of the space concerned....
Compound of two tetrahedra	Cube Cube A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....	Octahedron Octahedron An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....	O_h
Compound of cube and octahedron Compound of cube and octahedron This polyhedron can be seen as either a polyhedral stellation or a Polyhedron compound....	Rhombic dodecahedron Rhombic dodecahedron The rhombic dodecahedron is a convex set polyhedron with 12 rhombus faces. It is an Archimedean solid solid, or a Catalan solid. Its dual is the cuboctahedron....	Cuboctahedron Cuboctahedron In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....	O_h
Compound of dodecahedron and icosahedron Compound of dodecahedron and icosahedron This polyhedron can be seen as either a polyhedral stellation or a Polyhedron compound....	Rhombic triacontahedron Rhombic triacontahedron In geometry, the rhombic triacontahedron is a convex set polyhedron with 30 rhombus faces. It is an Archimedean solid solid, or a Catalan solid....	Icosidodecahedron Icosidodecahedron An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon....	I_h
Compound of great icosahedron and great stellated dodecahedron Compound of great icosahedron and great stellated dodecahedron This polyhedron can be seen as either a polyhedral stellation or a Polyhedron compound....	Dodecahedron Dodecahedron A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....	Icosahedron Icosahedron In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....	I_h
Compound of small stellated dodecahedron and great dodecahedron	Icosahedron Icosahedron In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....	Dodecahedron Dodecahedron A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....	I_h

The dual-regular compound of a tetrahedron with its dual polyhedron

Dual polyhedron

In geometry, polyhedron are associated into pairs called duals, where the wikt:vertex of one correspond to the face s of the other. The dual of the dual is the original polyhedron....
is also the regular Stella octangula

Stella octangula

The stella octangula, also known as the stellated octahedron, Star Tetrahedron, eight-pointed star, or 2D geometric model as the Star of David....
, since the tetrahedron is self-dual.

The cube-octahedron and dodecahedron-icosahedron dual-regular compounds are the first stellations of the cuboctahedron

Cuboctahedron

In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square....
and icosidodecahedron

Icosidodecahedron

An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon....
, respectively.

The compound of the small stellated dodecahedron

Small stellated dodecahedron

In geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedra. It is one of four nonconvex regular polyhedra. It is composed of 12 pentagrammic faces, with five pentagrams meeting at each vertex....
and great dodecahedron

Great dodecahedron

In geometry, the great dodecahedron is a Kepler-Poinsot polyhedra. It is one of four nonconvex regular polyhedra. It is composed of 12 pentagonal faces , with five pentagons meeting at each vertex, intersecting each other making a pentagrammic path....
looks outwardly the same as the small stellated dodecahedron, because the great dodecahedron is completely contained inside.

Uniform compounds

In 1976 John Skilling

John Skilling

John Skilling was a civil engineer and architect, best known for being the chief structural engineer of the World Trade Center....
published Uniform Compounds of Uniform Polyhedra which enumerated 75 compounds (including 6 as infinite prismatic

Prism (geometry)

In geometry, an n-sided prism is a polyhedron made of an n-sided polygon base, a Translation copy, and n faces joining corresponding sides....
sets of compounds, #20-#25) made from uniform polyhedra with rotational symmetry. (Every vertex is vertex-transitive

Vertex-transitive

In geometry, a polytope is isogonal or vertex-transitive if all its vertex are the same. That is, each vertex is surrounded by the same kinds of face in the same order, and with the same angles between corresponding faces....
and every vertex is transitive with every other vertex.) This list includes the five regular compounds above.

Here is a pictorial table of the 75 uniform compounds as listed by Skilling. Most are singularly colored by each polyhedron element. Some as chiral pairs are colored by symmetry of the faces within each polyhedron.

1-19: Miscellaneous (4,5,6,9,17 are the 5 regular compounds)

20-25: Prism symmetry embedded in prism symmetry
Dihedral symmetry in three dimensions

This article deals with three infinite series of point groups in three dimensions which have a symmetry group which as abstract group is a dihedral group Dihn ....
,

26-45: Prism symmetry embedded in octahedral
Octahedral symmetry

A regular octahedron has 24 rotational symmetries, and a total of 48 symmetries including transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the dual polyhedron of an octahedron....
or icosahedral symmetry
Icosahedral symmetry

File:Soccer ball.svgA regular icosahedron has 60 rotational symmetries, and a total of 120 symmetries including transformations that combine a reflection and a rotation....
,

46-67: Tetrahedral symmetry embedded in octahedral or icosahedral symmetry,

68-75: enantiomorph pairs

External links

- Software used to create the images on this page. Can print nets for many compounds.
– from Virtual Reality Polyhedra