Polyhedral Compound
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geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a polyhedral compound is a figure that is composed of several
polyhedra In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on t ...
sharing a common
centre Center or centre may refer to: Mathematics * Center (geometry), the middle of an object * Center (algebra), used in various contexts ** Center (group theory) ** Center (ring theory) * Graph center, the set of all vertices of minimum eccentri ...
. They are the three-dimensional analogs of polygonal compounds such as the
hexagram , can be seen as a compound composed of an upwards (blue here) and downwards (pink) facing equilateral triangle, with their intersection as a regular hexagon (in green). A hexagram ( Greek language, Greek) or sexagram (Latin) is a six-pointed ...
. The outer vertices of a compound can be connected to form a
convex polyhedron A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...
called its convex hull. A compound is a
facetting 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 vertices. New edges of a faceted polyhedron may be cre ...
of its convex hull. Another convex polyhedron is formed by the small central space common to all members of the compound. This polyhedron can be used as the core for a set of stellations.


Regular compounds

A regular polyhedral compound can be defined as a compound which, like a
regular polyhedron A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equival ...
, is
vertex-transitive In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in ...
,
edge-transitive In geometry, a polytope (for example, a polygon or a polyhedron) or a tiling is isotoxal () or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given t ...
, and
face-transitive In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congrue ...
. Unlike the case of polyhedra, this is not equivalent to the symmetry group acting transitively on its
flags A flag is a piece of textile, fabric (most often rectangular or quadrilateral) with a distinctive design and colours. It is used as a symbol, a signalling device, or for decoration. The term ''flag'' is also used to refer to the graphic desi ...
; the compound of two tetrahedra is the only regular compound with that property. There are five regular compounds of polyhedra: Best known is the regular compound of two
tetrahedra In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
, often called the
stella octangula The stellated octahedron is the only stellation of the octahedron. It is also called the stella octangula (Latin for "eight-pointed star"), a name given to it by Johannes Kepler in 1609, though it was known to earlier geometers. It was depict ...
, a name given to it by
Kepler Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws o ...
. The vertices of the two tetrahedra define a
cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...
, and the intersection of the two define a regular
octahedron In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...
, which shares the same face-planes as the compound. Thus the compound of two tetrahedra is a stellation of the octahedron, and in fact, the only finite stellation thereof. The regular
compound of five tetrahedra The compound of five tetrahedra is one of the five regular polyhedral compounds. This compound polyhedron is also a stellation of the regular icosahedron. It was first described by Edmund Hess in 1876. It can be seen as a faceting of a regular d ...
comes in two
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 rotations and translations alone. An object that is not chiral is said to ...
ic versions, which together make up the regular compound of ten tetrahedra. The regular compound of ten tetrahedra can also be constructed with five Stellae octangulae. Each of the regular tetrahedral compounds is self-dual or dual to its chiral twin; the regular compound of five cubes and the regular compound of five octahedra are dual to each other. Hence, regular polyhedral compounds can also be regarded as dual-regular compounds. Coxeter's notation for regular compounds is given in the table above, incorporating Schläfli symbols. The material inside the square brackets, 'd'' denotes the components of the compound: ''d'' separate 's. The material ''before'' the square brackets denotes the vertex arrangement of the compound: ''c'' 'd''is a compound of ''d'' 's sharing the vertices of counted ''c'' times. The material ''after'' the square brackets denotes the facet arrangement of the compound: 'd'''e'' is a compound of ''d'' 's sharing the faces of counted ''e'' times. These may be combined: thus ''c'' 'd'''e'' is a compound of ''d'' 's sharing the vertices of counted ''c'' times ''and'' the faces of counted ''e'' times. This notation can be generalised to compounds in any number of dimensions.


Dual compounds

A dual compound is composed of a polyhedron and its dual, arranged reciprocally about a common
midsphere In geometry, the midsphere or intersphere of a polyhedron is a sphere which is tangent to every edge of the polyhedron. That is to say, it touches any given edge at exactly one point. Not every polyhedron has a midsphere, but for every convex po ...
, such that the edge of one polyhedron intersects the dual edge of the dual polyhedron. There are five dual compounds of the regular polyhedra. The core is the rectification of both solids. The hull is the dual of this rectification, and its rhombic faces have the intersecting edges of the two solids as diagonals (and have their four alternate vertices). For the convex solids, this is the convex hull. The tetrahedron is self-dual, so the dual compound of a tetrahedron with its dual is the regular stellated octahedron. The octahedral and icosahedral dual compounds are the first stellations of the
cuboctahedron A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it ...
and
icosidodecahedron In geometry, an icosidodecahedron is a polyhedron with twenty (''icosi'') triangular faces and twelve (''dodeca'') pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 i ...
, respectively. The small stellated dodecahedral (or great dodecahedral) dual compound has the great dodecahedron completely interior to the small stellated dodecahedron.


Uniform compounds

In 1976 John Skilling published ''Uniform Compounds of Uniform Polyhedra'' which enumerated 75 compounds (including 6 as infinite
prismatic An optical prism is a transparent optical element with flat, polished surfaces that are designed to refract light. At least one surface must be angled — elements with two parallel surfaces are ''not'' prisms. The most familiar type of optical ...
sets of compounds, #20-#25) made from uniform polyhedra with rotational symmetry. (Every vertex is
vertex-transitive In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in ...
and every vertex is transitive with every other vertex.) This list includes the five regular compounds above

The 75 uniform compounds are listed in the Table below. Most are shown singularly colored by each polyhedron element. Some chiral pairs of face groups 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 Dihedral symmetry in three dimensions, prism symmetry, * 26-45: Prism symmetry embedded in
octahedral In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet a ...
or
icosahedral symmetry 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 polyhedr ...
, * 46-67: Tetrahedral symmetry embedded in octahedral or icosahedral symmetry, * 68-75:
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 rotations and translations alone. An object that is not chiral is said to ...
pairs


Other compounds

*
Compound of three octahedra In mathematics, the compound of three octahedra or octahedron 3-compound is a polyhedral compound formed from three regular octahedra, all sharing a common center but rotated with respect to each other. Although appearing earlier in the mathemati ...
*
Compound of four cubes The compound of four cubes or Bakos compound is a face-transitive polyhedron compound of four cubes with octahedral symmetry. It is the dual of the compound of four octahedra. Its surface area is 687/77 square lengths of the edge. Its Cartesian c ...
Two polyhedra that are compounds but have their elements rigidly locked into place are the
small complex icosidodecahedron In geometry, the small complex icosidodecahedron is a degenerate uniform star polyhedron. Its edges are doubled, making it degenerate. The star has 32 faces (20 triangles and 12 pentagons), 60 (doubled) edges and 12 vertices and 4 sharing faces. ...
(compound of icosahedron and
great dodecahedron In geometry, the great dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol and Coxeter–Dynkin diagram of . It is one of four nonconvex regular polyhedra. It is composed of 12 pentagonal faces (six pairs of parallel pentagon ...
) and the
great complex icosidodecahedron In geometry, the great complex icosidodecahedron is a degenerate uniform star polyhedron. It has 12 vertices, and 60 (doubled) edges, and 32 faces, 12 pentagrams and 20 triangles. All edges are doubled (making it degenerate), sharing 4 faces, but ...
(compound of
small stellated dodecahedron In geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol . It is one of four nonconvex regular polyhedra. It is composed of 12 pentagrammic faces, with five pentagrams meeti ...
and
great icosahedron In geometry, the great icosahedron is one of four Kepler–Poinsot polyhedra (nonconvex regular polyhedra), with Schläfli symbol and Coxeter-Dynkin diagram of . It is composed of 20 intersecting triangular faces, having five triangles meeti ...
). If the definition of a
uniform polyhedron In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent. Uniform polyhedra may be regular (if also fa ...
is generalised, they are uniform. The section for enantiomorph pairs in Skilling's list does not contain the compound of two great snub dodecicosidodecahedra, as the
pentagram A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle arou ...
faces would coincide. Removing the coincident faces results in the
compound of twenty octahedra The compound of twenty octahedra is a uniform polyhedron compound. It's composed of a symmetric arrangement of 20 octahedra (considered as triangular antiprisms). It is a special case of the compound of 20 octahedra with rotational freedom, in ...
.


4-polytope compounds

In 4-dimensions, there are a large number of regular compounds of regular polytopes.
Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington to ...
lists a few of these in his book
Regular Polytopes In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...
.Regular polytopes, Table VII, p. 305 McMullen added six in his paper ''New Regular Compounds of 4-Polytopes''.McMullen, Peter (2018), ''New Regular Compounds of 4-Polytopes'', New Trends in Intuitive Geometry, 27: 307–320 Self-duals: Dual pairs: Uniform compounds and duals with convex 4-polytopes: The superscript (var) in the tables above indicates that the labeled compounds are distinct from the other compounds with the same number of constituents.


Compounds with regular star 4-polytopes

Self-dual star compounds: Dual pairs of compound stars: Uniform compound stars and duals:


Compounds with duals

Dual positions:


Group theory

In terms of
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, if ''G'' is the symmetry group of a polyhedral compound, and the group acts transitively on the polyhedra (so that each polyhedron can be sent to any of the others, as in uniform compounds), then if ''H'' is the stabilizer of a single chosen polyhedron, the polyhedra can be identified with the
orbit space In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a ...
''G''/''H'' – the coset ''gH'' corresponds to which polyhedron ''g'' sends the chosen polyhedron to.


Compounds of tilings

There are eighteen two-parameter families of regular compound tessellations of the Euclidean plane. In the hyperbolic plane, five one-parameter families and seventeen isolated cases are known, but the completeness of this listing has not been enumerated. The Euclidean and hyperbolic compound families 2 (4 ≤ ''p'' ≤ ∞, ''p'' an integer) are analogous to the spherical
stella octangula The stellated octahedron is the only stellation of the octahedron. It is also called the stella octangula (Latin for "eight-pointed star"), a name given to it by Johannes Kepler in 1609, though it was known to earlier geometers. It was depict ...
, 2 . A known family of regular Euclidean compound honeycombs in any number of dimensions is an infinite family of compounds of hypercubic honeycombs, all sharing vertices and faces with another hypercubic honeycomb. This compound can have any number of hypercubic honeycombs. There are also ''dual-regular'' tiling compounds. A simple example is the E2 compound of a hexagonal tiling and its dual
triangular tiling In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilate ...
, which shares its edges with the
deltoidal trihexagonal tiling In geometry, the rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. There are one triangle, two squares, and one hexagon on each vertex. It has Schläfli symbol of rr. John Conway calls it a rhombihexadeltille.Conway, 200 ...
. The Euclidean compounds of two hypercubic honeycombs are both regular and dual-regular.


Footnotes


External links


MathWorld: Polyhedron Compound
– from Virtual Reality Polyhedra *

*http://users.skynet.be/polyhedra.fleurent/Compounds_2/Compounds_2.htm

*


References

*. *. *. *. *. *. * ''
Regular Polytopes In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...
'', (3rd edition, 1973), Dover edition, * p. 87 Five regular compounds *{{citation, first=Peter, last=McMullen, title=New Regular Compounds of 4-Polytopes, journal=New Trends in Intuitive Geometry, series=Bolyai Society Mathematical Studies , volume=27, pages=307–320, year=2018, doi=10.1007/978-3-662-57413-3_12, isbn=978-3-662-57412-6 .