A polyhedron (plural polyhedra or polyhedrons) is often defined as a geometric solid with flat faces and straight edges (the word polyhedron comes from the Classical Greek πολύεδρον, from poly, stem of πολύς, "many," + edron, form of έδρα, "base", "seat", or "face").
This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory. Grünbaum (1994, p. 43) observed, "The Original Sin in the theory of polyhedra goes back to Euclid, and through Kepler, Poinsot, Cauchy and many others ... [in that] at each stage ... the writers failed to define what are the 'polyhedra' ...." Mathematicians still do not agree as to exactly what makes something a polyhedron.
Contents

Any polyhedron can be built up from different kinds of element or entity, each associated with a different number of dimensions:
More generally in mathematics and other disciplines, "polyhedron" is used to refer to a variety of related constructs, some geometric and others purely algebraic or abstract.
A defining characteristic of almost all kinds of polyhedra is that just two faces join along any common edge. This ensures that the polyhedral surface is continuously connected and does not end abruptly or split off in different directions.
A polyhedron is a 3dimensional example of the more general polytope in any number of dimensions.
Polyhedra are often named according to the number of faces. The naming system is again based on Classical Greek, for example tetrahedron (4), pentahedron (5), hexahedron (6), heptahedron (7), triacontahedron (30), and so on.
Often this is qualified by a description of the kinds of faces present, for example the Rhombic dodecahedron vs. the Pentagonal dodecahedron.
Other common names indicate that some operation has been performed on a simpler polyhedron, for example the truncated cube looks like a cube with its corners cut off, and has 14 faces (so it is also an example of a tetrakaidecahedron).
Some special polyhedra have grown their own names over the years, such as Miller's monster or the Szilassi polyhedron.
Edges have two important characteristics (unless the polyhedron is complex):
These two characteristics are dual to each other.
The Euler characteristic χ relates the number of vertices V, edges E, and faces F of a polyhedron:
For a simply connected polyhedron, χ = 2. For a detailed discussion, see Proofs and Refutations by Imre Lakatos.
Some polyhedra, such as all convex polyhedra, have two distinct sides to their surface, for example one side can consistently be coloured black and the other white. We say that the figure is orientable.
But for some polyhedra this is not possible, and the figure is said to be nonorientable. All polyhedra with oddnumbered Euler characteristic are nonorientable. A given figure with even χ < 2 may or may not be orientable.
For every vertex one can define a vertex figure, which describes the local structure of the figure around the vertex. If the vertex figure is a regular polygon, then the vertex itself is said to be regular.
For every polyhedron we can construct a dual polyhedron having:
For a convex polyhedron the dual can be obtained by the process of polar reciprocation.
The volume of an orientable polyhedron having an identifiable centroid can be calculated using Green's theorem:
by choosing the function
where (x,y,z) is the centroid of the surface enclosing the volume under consideration. As we have,
Hence the volume can be calculated as:
where the normal of the surface pointing outwards is given by:
The final expression can be written as
where S is the surface area of the polyhedron.
In geometry, a polyhedron is traditionally a threedimensional shape that is made up of a finite number of polygonal faces which are parts of planes; the faces meet in pairs along edges which are straightline segments, and the edges meet in points called vertices. Cubes, prisms and pyramids are examples of polyhedra. The polyhedron surrounds a bounded volume in threedimensional space; sometimes this interior volume is considered to be part of the polyhedron, sometimes only the surface is considered, and occasionally only the skeleton of edges.
A polyhedron is said to be convex if its surface (comprising its faces, edges and vertices) does not intersect itself and the line segment joining any two points of the polyhedron is contained in the interior or surface.
Many of the most studied polyhedra are highly symmetrical.
Of course it is easy to distort such polyhedra so they are no longer symmetrical. But where a polyhedral name is given, such as icosidodecahedron, the most symmetrical geometry is almost always implied, unless otherwise stated.
Some of the most common names in particular are often used with "regular" in front or implied because for each there are different types which have little in common except for having the same number of faces. These are the triangular pyramid or tetrahedron, cube or hexahedron, octahedron, dodecahedron and icosahedron:
Polyhedra of the highest symmetries have all of some kind of element  faces, edges and/or vertices, within a single symmetry orbit. There are various classes of such polyhedra:
A polyhedron can belong to the same overall symmetry group as one of higher symmetry, but will have several groups of elements (for example faces) in different symmetry orbits.
Uniform polyhedra are vertextransitive and every face is a regular polygon. They may be regular, quasiregular, or semiregular, and may be convex or starry.
The uniform duals are facetransitive and every vertex figure is a regular polygon.
Facetransitivity of a polyhedron corresponds to vertextransitivity of the dual and conversely, and edgetransitivity of a polyhedron corresponds to edgetransitivity of the dual. In most duals of uniform polyhedra, faces are irregular polygons. The regular polyhedra are an exception, because they are dual to each other.
Each uniform polyhedron shares the same symmetry as its dual, with the symmetries of faces and vertices simply swapped over. Because of this some authorities regard the duals as uniform too. But this idea is not held widely: a polyhedron and its symmetries are not the same thing.
The uniform polyhedra and their duals are traditionally classified according to their degree of symmetry, and whether they are convex or not.
Convex uniform  Convex uniform dual  Star uniform  Star uniform dual  

Regular  Platonic solids  KeplerPoinsot polyhedra  
Quasiregular  Archimedean solids  Catalan solids  (no special name)  (no special name) 
Semiregular  (no special name)  (no special name)  
Prisms  Dipyramids  Star Prisms  Star Dipyramids  
Antiprisms  Trapezohedra  Star Antiprisms  Star Trapezohedra 
A noble polyhedron is both isohedral (equalfaced) and isogonal (equalcornered). Besides the regular polyhedra, there are many other examples.
The dual of a noble polyhedron is also noble.
The polyhedral symmetry groups (using Schoenflies notation) are all point groups and include:
Those with chiral symmetry do not have reflection symmetry and hence have two enantiomorphous forms which are reflections of each other. The snub Archimedean polyhedra have this property.
A few families of polyhedra, where every face is the same kind of polygon:
There exists no polyhedron whose faces are all identical and are regular polygons with six or more sides because the vertex of three regular hexagons defines a plane. (See infinite skew polyhedron for exceptions with zigzagging vertex figures.)
A deltahedron (plural deltahedra) is a polyhedron whose faces are all equilateral triangles. There are infinitely many deltahedra, but only eight of these are convex:
Norman Johnson sought which nonuniform polyhedra had regular faces. In 1966, he published a list of 92 convex solids, now known as the Johnson solids, and gave them their names and numbers. He did not prove there were only 92, but he did conjecture that there were no others. Victor Zalgaller in 1969 proved that Johnson's list was complete.
Pyramids include some of the most timehonoured and famous of all polyhedra.
Stellation of a polyhedron is the process of extending the faces (within their planes) so that they meet to form a new polyhedron.
It is the exact reciprocal to the process of facetting which is the process of removing parts of a polyhedron without creating any new vertices.
A zonohedron is a convex polyhedron where every face is a polygon with inversion symmetry or, equivalently, symmetry under rotations through 180°.
A toroidal polyhedron is a polyhedra with an Euler characteristic of 0 or smaller, representing a torus surface.
Polyhedral compounds are formed as compounds of two or more polyhedra.
These compounds often share the same vertices as other polyhedra and are often formed by stellation. Some are listed in the list of Wenninger polyhedron models.
An orthogonal polyhedron is one all of whose faces meet at right angles, and all of whose edges are parallel to axes of a Cartesian coordinate system. Aside from a rectangular box, orthogonal polyhedra are nonconvex. They are the 3D analogs of 2D orthogonal polygons, also known as rectilinear polygons. Orthogonal polyhedra are used in computational geometry, where their constrained structure has enabled advances on problems unsolved for arbitrary polyhedra, for example, unfolding the surface of a polyhedron to a polygonal net.
The name 'polyhedron' has come to be used for a variety of objects having similar structural properties to traditional polyhedra.
A classical polyhedral surface comprises finite, bounded plane regions, joined in pairs along edges. If such a surface extends indefinitely it is called an apeirohedron. Examples include:
See also: Apeirogon  infinite regular polygon: {∞}
A complex polyhedron is one which is constructed in complex Hilbert 3space. This space has six dimensions: three real ones corresponding to ordinary space, with each accompanied by an imaginary dimension. See for example Coxeter (1974).
Some fields of study allow polyhedra to have curved faces and edges.
The surface of a sphere may be divided by line segments into bounded regions, to form a spherical polyhedron. Much of the theory of symmetrical polyhedra is most conveniently derived in this way.
Spherical polyhedra have a long and respectable history:
Some polyhedra, such as hosohedra and dihedra, exist only as spherical polyhedra and have no flatfaced analogue.
Two important types are:
More recently mathematics has defined a polyhedron as a set in real affine (or Euclidean) space of any dimensional n that has flat sides. It could be defined as the union of a finite number of convex polyhedra, where a convex polyhedron is any set that is the intersection of a finite number of halfspaces. It may be bounded or unbounded. In this meaning, a polytope is a bounded polyhedron.
All traditional polyhedra are general polyhedra, and in addition there are examples like:
It is not necessary to fill in the face of a figure before we can call it a polyhedron. For example Leonardo da Vinci devised frame models of the regular solids, which he drew for Pacioli's book Divina Proportione. In modern times, Branko Grünbaum (1994) made a special study of this class of polyhedra, in which he developed an early idea of abstract polyhedra. He defined a face as a cyclically ordered set of vertices, and allowed faces to be skew as well as planar.
Various mathematical constructs have been found to have properties also present in traditional polyhedra.
A topological polytope is a topological space given along with a specific decomposition into shapes that are topologically equivalent to convex polytopes and that are attached to each other in a regular way.
Such a figure is called simplicial if each of its regions is a simplex, i.e. in an ndimensional space each region has n+1 vertices. The dual of a simplicial polytope is called simple. Similarly, a widely studied class of polytopes (polyhedra) is that of cubical polyhedra, when the basic building block is an ndimensional cube.
An abstract polyhedron is a partially ordered set (poset) of elements whose partial ordering obeys certain rules. Theories differ in detail, but essentially the elements of the set correspond to the body, faces, edges and vertices of the polyhedron. The empty set corresponds to the null polytope, or nullitope, which has a dimensionality of −1. These posets belong to the larger family of abstract polytopes in any number of dimensions.
Any polyhedron gives rise to a graph, or skeleton, with corresponding vertices and edges. Thus graph terminology and properties can be applied to polyhedra. For example:
Stones carved in shapes showing the symmetries of various polyhedra have been found in Scotland and may be as much a 4,000 years old. These stones show not only the form of various symmetrical polyehdra, but also the relations of duality amongst some of them (that is, that the centres of the faces of the cube gives the vertices of an octahedron, and so on). Examples of these stones are on display in the John Evans room of the Ashmolean Museum at Oxford University. It is impossible to know why these objects were made, or how the sculptor gained the inspiration for them.
Other polyhedra have of course made their mark in architecture  cubes and cuboids being obvious examples, with the earliest foursided pyramids of ancient Egypt also dating from the Stone Age.
The Etruscans preceded the Greeks in their awareness of at least some of the regular polyhedra, as evidenced by the discovery near Padua (in Northern Italy) in the late 1800s of a dodecahedron made of soapstone, and dating back more than 2,500 years (Lindemann, 1987). Pyritohedric crystals are found in northern Italy^{[citation needed]}.
The earliest known written records of these shapes come from Classical Greek authors, who also gave the first known mathematical description of them. The earlier Greeks were interested primarily in the convex regular polyhedra, which came to be known as the Platonic solids. Pythagoras knew at least three of them, and Theaetetus (circa 417 B. C.) described all five. Eventually, Euclid described their construction in his Elements. Later, Archimedes expanded his study to the convex uniform polyhedra which now bear his name. His original work is lost and his solids come down to us through Pappus.
After the end of the Classical era, Islamic scholars continued to make advances, for example in the tenth century Abu'l Wafa described the convex regular and quasiregular spherical polyhedra. Meanwhile in China, dissection of the cube into its characteristic tetrahedron (orthoscheme) and related solids was used as the basis for calculating volumes of earth to be moved during engineering excavations.
As with other areas of Greek thought maintained and enhanced by Islamic scholars, Western interest in polyhedra revived during the Renaissance. Much to be said here: Piero della Francesca, Pacioli, Leonardo Da Vinci, Wenzel Jamnitzer, Durer, etc. leading up to Kepler.
For almost 2,000 years, the concept of a polyhedron had remained as developed by the ancient Greek mathematicians.
Johannes Kepler realised that star polygons could be used to build star polyhedra, which have nonconvex regular polygons, typically pentagrams as faces. Some of these star polyhedra may have been discovered before Kepler's time, but he was the first to recognise that they could be considered "regular" if one removed the restriction that regular polytopes be convex. Later, Louis Poinsot realised that star vertex figures (circuits around each corner) can also be used, and discovered the remaining two regular star polyhedra. Cauchy proved Poinsot's list complete, and Cayley gave them their accepted English names: (Kepler's) the small stellated dodecahedron and great stellated dodecahedron, and (Poinsot's) the great icosahedron and great dodecahedron. Collectively they are called the KeplerPoinsot polyhedra.
The KeplerPoinsot polyhedra may be constructed from the Platonic solids by a process called stellation. Most stellations are not regular. The study of stellations of the Platonic solids was given a big push by H. S. M. Coxeter and others in 1938, with the now famous paper The 59 icosahedra. This work has recently been republished (Coxeter, 1999).
The reciprocal process to stellation is called facetting (or faceting). Every stellation of one polytope is dual, or reciprocal, to some facetting of the dual polytope. The regular star polyhedra can also be obtained by facetting the Platonic solids. Bridge 1974 listed the simpler facettings of the dodecahedron, and reciprocated them to discover a stellation of the icosahedron that was missing from the famous "59". More have been discovered since, and the story is not yet ended.
See also:
For natural occurrences of regular polyhedra, see Regular polyhedron: Regular polyhedra in nature.
Irregular polyhedra appear in nature as crystals.


Fundamental convex regular and uniform polytopes in dimensions 210  

n  nSimplex  nHypercube  nOrthoplex  nDemicube  1_{k2}  2_{k1}  k_{21}  
Family  A_{n}  BC_{n}  D_{n}  E_{n}  F_{4}  H_{n}  
Regular 2polytope  Triangle  Square  Pentagon  
Uniform 3polytope  Tetrahedron  Cube  Octahedron  Tetrahedron  Dodecahedron • Icosahedron  
Uniform 4polytope  5cell  Tesseract  16cell (Demitesseract)  24cell  120cell • 600cell  
Uniform 5polytope  5simplex  5cube  5orthoplex  5demicube  
Uniform 6polytope  6simplex  6cube  6orthoplex  6demicube  1_{22}  2_{21}  
Uniform 7polytope  7simplex  7cube  7orthoplex  7demicube  1_{32}  2_{31}  3_{21}  
Uniform 8polytope  8simplex  8cube  8orthoplex  8demicube  1_{42}  2_{41}  4_{21}  
Uniform 9polytope  9simplex  9cube  9orthoplex  9demicube  
Uniform 10polytope  10simplex  10cube  10orthoplex  10demicube  
Topics: Polytope families • Regular polytope • List of regular polytopes 
(There is currently no text in this page)

[[File:thumbrightMost dice are polyhedra]] A Polyhedron (one polyhedron, many Polyhedra, or Polyhedrons) is a geometrical shape. It has flat faces, and straight edges. Usually it is defined by the number of faces, or edges.
Mathematicians do not agree what makes a polyhedron.
Usually, polyhedra are named by the number of faces they have. The first polyhedra are the tetrahedron, which is made of 4 triangles, pentahedron (5 faces, can look like a 4sided pyramid), hexahedron (6 faces, usually looks like a cube if it is regular), and heptahedron (7 faces, can look like a prism based on a pentagon, or a pyramid based on a hexagon amongst others). Most Polyhedrons are allowed to chose their own name, or are named by their parents, based on their number of sides.
