Some Polyhedra
Dodecahedron (Regular polyhedronIn geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and... ) 
Small stellated dodecahedron (Regular star) 
IcosidodecahedronIn geometry, 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...
(Uniform/QuasiregularIn geometry an Archimedean solid is a highly symmetric, semiregular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices... ) 
Great cubicuboctahedronIn geometry, the great cubicuboctahedron is a nonconvex uniform polyhedron, indexed as U14. Related polyhedra :It shares the vertex arrangement with the convex truncated cube and two other nonconvex uniform polyhedra...
(Uniform starA uniform polyhedron is a polyhedron which has regular polygons as faces and is vertextransitive... ) 
Rhombic triacontahedronIn geometry, the rhombic triacontahedron is a convex polyhedron with 30 rhombic faces. It is an Archimedean dual solid, or a Catalan solid. It is the polyhedral dual of the icosidodecahedron, and it is a zonohedron....
(Uniform/Quasiregular dualIn mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. The Catalan solids are named for the Belgian mathematician, Eugène Catalan, who first described them in 1865.... ) 
Elongated pentagonal cupolaIn geometry, the elongated pentagonal cupola is one of the Johnson solids . As the name suggests, it can be constructed by elongating a pentagonal cupola by attaching a decagonal prism to its base...
(Convex regularfacedIn geometry, a Johnson solid is a strictly convex polyhedron, each face of which is a regular polygon, but which is not uniform, i.e., not a Platonic solid, Archimedean solid, prism or antiprism. There is no requirement that each face must be the same polygon, or that the same polygons join around... ) 
Octagonal prismIn geometry, the octagonal prism is the sixth in an infinite set of prisms, formed by square sides and two regular octagon caps.If faces are all regular, it is a semiregular polyhedron. Use :...
(Uniform prism) 
Square antiprismIn geometry, the square antiprism is the second in an infinite set of antiprisms formed by an evennumbered sequence of triangle sides closed by two polygon caps...
(Uniform antiprism) 

In elementary geometry a polyhedron (plural polyhedra or polyhedrons) is a
geometricGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of premodern mathematics, the other being the study of numbers ....
solid in
three dimensionsThreedimensional space is a geometric 3parameters model of the physical universe in which we live. These three dimensions are commonly called length, width, and depth , although any three directions can be chosen, provided that they do not lie in the same plane.In physics and mathematics, a...
with flat faces and straight edges. The word polyhedron comes from the
Classical GreekGreek is an independent branch of the IndoEuropean family of languages. Native to the southern Balkans, it has the longest documented history of any IndoEuropean language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...
πολύεδρον, as poly (stem of πολύς, "many") + hedron (form of έδρα, "base", "seat", or "face").
A polyhedron is a 3dimensional example of the more general
polytopeIn elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions...
in any number of dimensions.
Basis for definition
Defining a polyhedron as a solid bounded by flat faces and straight edges is not very precise and, to a modern mathematician, quite unsatisfactory.
GrünbaumBranko Grünbaum is a Croatianborn mathematician and a professor emeritus at the University of Washington in Seattle. He received his Ph.D. in 1957 from Hebrew University of Jerusalem in Israel....
(1994, p. 43) observed, "The
Original SinOriginal sin is, according to a Christian theological doctrine, humanity's state of sin resulting from the Fall of Man. This condition has been characterized in many ways, ranging from something as insignificant as a slight deficiency, or a tendency toward sin yet without collective guilt, referred...
in the theory of polyhedra goes back to
EuclidEuclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
, and through Kepler, Poinsot, Cauchy and many others ... [in that] at each stage ... the writers failed to define what are the 'polyhedra' ...." Since then rigorous definitions of "polyhedron" have been given within particular contexts. However such definitions are not always compatible in other contexts.
Any polyhedron can be built up from different kinds of element or entity, each associated with a different number of dimensions:
 3 dimensions: The body is bounded by the faces, and is usually the volume enclosed by them.
 2 dimensions: A face
In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube...
is a polygonIn geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...
bounded by a circuit of edges, and usually including the flat (plane) region inside the boundary. These polygonal faces together make up the polyhedral surface.
 1 dimension: An edge
In geometry, an edge is a onedimensional line segment joining two adjacent zerodimensional vertices in a polygon. Thus applied, an edge is a connector for a onedimensional line segment and two zerodimensional objects....
joins one vertex to another and one face to another, and is usually a lineThe notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...
segment. The edges together make up the polyhedral skeleton.
 0 dimensions: A vertex
In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.Of an angle:...
(plural vertices) is a corner pointIn geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zerodimensional; i.e., they do not have volume, area, length, or any other higherdimensional analogue. In branches of mathematics...
.
 1 dimension: The null polytope is a kind of nonentity required by abstract theories.
More generally in
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
and other disciplines, "polyhedron" is used to refer to a variety of related constructs, some geometric and others purely algebraic or abstract.
Polyhedral surface
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.
Edges
Edges have two important characteristics (unless the polyhedron is complex):
 An edge joins just two vertices.
 An edge joins just two faces.
These two characteristics are dual to each other.
Euler characteristic
The
Euler characteristicIn mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...
χ relates the number of vertices V, edges E, and faces F of a polyhedron:
For a convex polyhedron or more generally for any
simply connectedIn topology, a topological space is called simply connected if it is pathconnected and every path between two points can be continuously transformed, staying within the space, into any other path while preserving the two endpoints in question .If a space is not simply connected, it is convenient...
polyhedron whose faces are also simply connected and whose boundary is a
manifoldIn mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
, χ = 2. For a detailed discussion, see
Proofs and RefutationsProofs and Refutations is a book by the philosopher Imre Lakatos expounding his view ofthe progress of mathematics. The book is written as a series of Socratic dialogues involving a group of students who debate the proof of the Euler characteristic defined for the polyhedron...
by
Imre LakatosImre Lakatos was a Hungarian philosopher of mathematics and science, known for his thesis of the fallibility of mathematics and its 'methodology of proofs and refutations' in its preaxiomatic stages of development, and also for introducing the concept of the 'research programme' in his...
.
Orientability
Some polyhedra, such as all
convex polyhedraA convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the ndimensional space Rn...
, 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
orientableIn mathematics, orientability is a property of surfaces in Euclidean space measuring whether or not it is possible to make a consistent choice of surface normal vector at every point. A choice of surface normal allows one to use the righthand rule to define a "clockwise" direction of loops in the...
.
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.
Vertex figure
For every vertex one can define a
vertex figureIn geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.Definitions  theme and variations:...
, which describes the local structure of the figure around the vertex. If the vertex figure is a
regular polygonA regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.General properties:...
, then the vertex itself is said to be regular.
Duality
For every polyhedron there exists a
dual polyhedronIn geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another...
having:
 faces in place of the original's vertices and vice versa,
 the same number of edges
 the same Euler characteristic and orientability
The dual of a convex polyhedron can be obtained by the process of polar reciprocation.
Volume
The volume of an
orientableIn mathematics, orientability is a property of surfaces in Euclidean space measuring whether or not it is possible to make a consistent choice of surface normal vector at every point. A choice of surface normal allows one to use the righthand rule to define a "clockwise" direction of loops in the...
polyhedron can be calculated using the
divergence theoremIn vector calculus, the divergence theorem, also known as Gauss' theorem , Ostrogradsky's theorem , or Gauss–Ostrogradsky theorem is a result that relates the flow of a vector field through a surface to the behavior of the vector field inside the surface.More precisely, the divergence theorem...
. Consider the vector field
, whose divergence is identically 1. The divergence theorem implies that the volume of any region Ω is
When Ω is the region enclosed by a polyhedron, since the faces of a polyhedron are planar and have piecewise constant normals, this simplifies to
where for the ith face,
is any point on the face,
is the normal vector, and
is area of the face.
Names of polyhedra
Polyhedra are often named according to the number of faces. The naming system is again based on Classical Greek, for example
tetrahedronIn geometry, a tetrahedron is a polyhedron composed of four triangular 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...
(4),
pentahedronIn geometry, a pentahedron is a polyhedron with five faces. Since there are no facetransitive polyhedra with five sides and there are two distinct topological types, this term is less frequently used than tetrahedron or octahedron....
(5),
hexahedronA hexahedron is any polyhedron with six faces, although usually implies the cube as a regular hexahedron with all its faces square, and three squares around each vertex....
(6),
heptahedronA heptahedron is a polyhedron having seven sides, or faces.A heptahedron can take a surprising number of different basic forms, or topologies. Probably most familiar are the hexagonal pyramid and the pentagonal prism. Also notable is the tetrahemihexahedron, whose seven equilateral triangle faces...
(7), triacontahedron (30), and so on.
Often this is qualified by a description of the kinds of faces present, for example the
Rhombic dodecahedronIn geometry, the rhombic dodecahedron is a convex polyhedron with 12 rhombic faces. It is an Archimedean dual solid, or a Catalan solid. Its dual is the cuboctahedron.Properties:...
vs. the
Pentagonal dodecahedron.
Other common names indicate that some operation has been performed on a simpler polyhedron, for example the
truncated cubeIn geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces , 36 edges, and 24 vertices....
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 polyhedronThe Szilassi polyhedron is a nonconvex polyhedron, topologically a torus, with seven hexagonal faces.Each face of this polyhedron shares an edge with each other face. As a result, it requires seven colours to colour each adjacent face, providing the lower bound for the seven colour theorem...
.
Traditional polyhedra
In
geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of premodern mathematics, the other being the study of numbers ....
, a polyhedron is traditionally a threedimensional shape that is made up of a finite number of
polygonIn geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...
al
facesIn geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube...
which are parts of
planesIn mathematics, a plane is a flat, twodimensional surface. A plane is the two dimensional analogue of a point , a line and a space...
; the faces meet in pairs along
edgesIn geometry, an edge is a onedimensional line segment joining two adjacent zerodimensional vertices in a polygon. Thus applied, an edge is a connector for a onedimensional line segment and two zerodimensional objects....
which are straightline segments, and the edges meet in points called
verticesIn geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.Of an angle:...
. Cubes,
prismsIn geometry, a prism is a polyhedron with an nsided polygonal base, a translated copy , and n other faces joining corresponding sides of the two bases. All crosssections parallel to the base faces are the same. Prisms are named for their base, so a prism with a pentagonal base is called a...
and
pyramidsIn geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle. It is a conic solid with polygonal base....
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.
Symmetrical polyhedra
Many of the most studied polyhedra are highly
symmetricalSymmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...
.
Of course it is easy to distort such polyhedra so they are no longer symmetrical. But where a polyhedral name is given, such as
icosidodecahedronIn geometry, 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...
, 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
tetrahedronIn geometry, a tetrahedron is a polyhedron composed of four triangular 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...
,
cubeIn geometry, a cube is a threedimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
or hexahedron,
octahedronIn geometry, 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 vertex....
,
dodecahedron and
icosahedronIn geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....
:
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:
 Isogonal or Vertextransitive
In geometry, a polytope is isogonal or vertextransitive if, loosely speaking, all its vertices are the same...
if all vertices are the same, in the sense that for any two vertices there exists a symmetryThe symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...
of the polyhedron mapping the first isometricallyIn mathematics, an isometry is a distancepreserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
onto the second.
 Isotoxal or Edgetransitive if all edges are the same, in the sense that for any two edges there exists a symmetry of the polyhedron mapping the first isometrically onto the second.
 Isohedral or Facetransitive if all faces are the same, in the sense that for any two faces there exists a symmetry of the polyhedron mapping the first isometrically onto the second.
 Regular
A regular polyhedron is a polyhedron whose faces are congruent regular polygons which are assembled in the same way around each vertex. A regular polyhedron is highly symmetrical, being all of edgetransitive, vertextransitive and facetransitive  i.e. it is transitive on its flags...
if it is vertextransitive, edgetransitive and facetransitive (this implies that every face is the same regular polygonA regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.General properties:...
; it also implies that every vertex is regular).
 Quasiregular
In geometry, a quasiregular polyhedron is a semiregular polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are edgetransitive and hence step closer to regularity than the semiregular which are merely vertextransitive.There are only two convex...
if it is vertextransitive and edgetransitive (and hence has regular faces) but not facetransitive. A quasiregular dual is facetransitive and edgetransitive (and hence every vertex is regular) but not vertextransitive.
 Semiregular
The term semiregular polyhedron is used variously by different authors.In its original definition, it is a polyhedron with regular faces and a symmetry group which is transitive on its vertices, which is more commonly referred to today as a uniform polyhedron...
if it is vertextransitive but not edgetransitive, and every face is a regular polygon. (This is one of several definitions of the term, depending on author. Some definitions overlap with the quasiregular class). A semiregular dual is facetransitive but not vertextransitive, and every vertex is regular.
 Uniform
A uniform polyhedron is a polyhedron which has regular polygons as faces and is vertextransitive...
if it is vertextransitive and every face is a regular polygon, i.e. it is regular, quasiregular or semiregular. A uniform dual is facetransitive and has regular vertices, but is not necessarily vertextransitive).
 Noble
A noble polyhedron is one which is isohedral and isogonal . They were first studied in any depth by Hess and Bruckner around the turn of the century , and later by Grünbaum.Classes of noble polyhedra:...
if it is facetransitive and vertextransitive (but not necessarily edgetransitive). The regular polyhedra are also noble; they are the only noble uniform 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 and their duals
Uniform polyhedra are
vertextransitiveIn geometry, a polytope is isogonal or vertextransitive if, loosely speaking, all its vertices are the same...
and every face is a
regular polygonA regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.General properties:...
.
They may be
regularA regular polyhedron is a polyhedron whose faces are congruent regular polygons which are assembled in the same way around each vertex. A regular polyhedron is highly symmetrical, being all of edgetransitive, vertextransitive and facetransitive  i.e. it is transitive on its flags...
,
quasiregularIn geometry, a quasiregular polyhedron is a semiregular polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are edgetransitive and hence step closer to regularity than the semiregular which are merely vertextransitive.There are only two convex...
, or
semiregularThe term semiregular polyhedron is used variously by different authors.In its original definition, it is a polyhedron with regular faces and a symmetry group which is transitive on its vertices, which is more commonly referred to today as a uniform polyhedron...
, and may be convex or starry.
The uniform
dualsIn geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another...
are facetransitive and every
vertex figureIn geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.Definitions  theme and variations:...
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. The dual of a regular polyhedron is also regular. The dual of a nonregular uniform polyhedron (called a
Catalan solidIn mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. The Catalan solids are named for the Belgian mathematician, Eugène Catalan, who first described them in 1865....
if convex) has irregular faces.
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 

RegularA regular polyhedron is a polyhedron whose faces are congruent regular polygons which are assembled in the same way around each vertex. A regular polyhedron is highly symmetrical, being all of edgetransitive, vertextransitive and facetransitive  i.e. it is transitive on its flags...

Platonic solidIn geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and... s 
KeplerPoinsot polyhedra 
QuasiregularIn geometry, a quasiregular polyhedron is a semiregular polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are edgetransitive and hence step closer to regularity than the semiregular which are merely vertextransitive.There are only two convex...

Archimedean solidIn geometry an Archimedean solid is a highly symmetric, semiregular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices... s 
Catalan solids 
(no special name) 
(no special name) 
Semiregular The term semiregular polyhedron is used variously by different authors.In its original definition, it is a polyhedron with regular faces and a symmetry group which is transitive on its vertices, which is more commonly referred to today as a uniform polyhedron...

(no special name) 
(no special name) 
PrismsIn geometry, a prism is a polyhedron with an nsided polygonal base, a translated copy , and n other faces joining corresponding sides of the two bases. All crosssections parallel to the base faces are the same. Prisms are named for their base, so a prism with a pentagonal base is called a...

Dipyramids 
Star PrismsIn geometry, a prism is a polyhedron with an nsided polygonal base, a translated copy , and n other faces joining corresponding sides of the two bases. All crosssections parallel to the base faces are the same. Prisms are named for their base, so a prism with a pentagonal base is called a...

Star Dipyramids 
AntiprismIn geometry, an nsided antiprism is a polyhedron composed of two parallel copies of some particular nsided polygon, connected by an alternating band of triangles... s 
Trapezohedra 
Star AntiprismIn geometry, an nsided antiprism is a polyhedron composed of two parallel copies of some particular nsided polygon, connected by an alternating band of triangles... s 
Star Trapezohedra 
Noble polyhedra
A
nobleA noble polyhedron is one which is isohedral and isogonal . They were first studied in any depth by Hess and Bruckner around the turn of the century , and later by Grünbaum.Classes of noble polyhedra:...
polyhedron is both isohedral (equalfaced) and isogonal (equalcornered). Besides the regular polyhedra, there are many other examples.
The
dualIn geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another...
of a noble polyhedron is also noble.
Symmetry groups
The polyhedral
symmetry groupThe symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...
s (using
Schoenflies notationThe Schoenflies notation or Schönflies notation, named after the German mathematician Arthur Moritz Schoenflies, is one of two conventions commonly used to describe Point groups. This notation is used in spectroscopy. The other convention is the Hermann–Mauguin notation, also known as the...
) are all point groups and include:
 T  chiral tetrahedral symmetry
150pxrightthumbA regular [[tetrahedron]], an example of a solid with full tetrahedral symmetryA regular tetrahedron has 12 rotational symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation.The group of all symmetries is isomorphic to the group...
; the rotation group for a regular tetrahedronIn geometry, a tetrahedron is a polyhedron composed of four triangular 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...
; order 12.
 T_{d}  full tetrahedral symmetry
150pxrightthumbA regular [[tetrahedron]], an example of a solid with full tetrahedral symmetryA regular tetrahedron has 12 rotational symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation.The group of all symmetries is isomorphic to the group...
; the symmetry group for a regular tetrahedronIn geometry, a tetrahedron is a polyhedron composed of four triangular 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...
; order 24.
 T_{h}  pyritohedral symmetry
150pxrightthumbA regular [[tetrahedron]], an example of a solid with full tetrahedral symmetryA regular tetrahedron has 12 rotational symmetries, and a symmetry order of 24 including transformations that combine a reflection and a rotation.The group of all symmetries is isomorphic to the group...
; order 24. The symmetry of a pyritohedronIn geometry, a pyritohedron is an irregular dodecahedron with pyritohedral symmetry. Like the regular dodecahedron, it has twelve identical pentagonal faces, with three meeting in each of the 20 vertices. However, the pentagons are not regular, and the structure has no fivefold symmetry axes...
.
 O  chiral octahedral symmetry
150pxthumbrightThe [[cube]] is the most common shape with octahedral symmetryA regular octahedron has 24 rotational symmetries, and a symmetry order of 48 including transformations that combine a reflection and a rotation...
;the rotation group of the cube and octahedronIn geometry, 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 vertex....
; order 24.
 O_{h}  full octahedral symmetry
150pxthumbrightThe [[cube]] is the most common shape with octahedral symmetryA regular octahedron has 24 rotational symmetries, and a symmetry order of 48 including transformations that combine a reflection and a rotation...
; the symmetry group of the cube and octahedronIn geometry, 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 vertex....
; order 48.
 I  chiral icosahedral symmetry
A regular icosahedron has 60 rotational symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation...
; the rotation group of the icosahedronIn geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....
and the dodecahedron; order 60.
 I_{h}  full icosahedral symmetry
A regular icosahedron has 60 rotational symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation...
; the symmetry group of the icosahedronIn geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....
and the dodecahedron; order 120.
 C_{nv}  nfold pyramidal symmetry
This article deals with the four infinite series of point groups in three dimensions with nfold rotational symmetry about one axis , and no other rotational symmetry :Chiral:*Cn of order n  nfold rotational symmetry...
 D_{nh}  nfold prismatic symmetry
 D_{nv}  nfold antiprismatic symmetry.
Those with
chiralIn geometry, a figure is chiral 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. For example, a right shoe is different from a left shoe, and clockwise is different from counterclockwise.A chiral object...
symmetry do not have
reflection symmetryReflection symmetry, reflectional symmetry, line symmetry, mirror symmetry, mirrorimage symmetry, or bilateral symmetry is symmetry with respect to reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry.In 2D there is a line of symmetry, in 3D a...
and hence have two
enantiomorphousIn geometry, a figure is chiral 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. For example, a right shoe is different from a left shoe, and clockwise is different from counterclockwise.A chiral object...
forms which are reflections of each other. The snub Archimedean polyhedra have this property.
Equal regular faces
A few families of polyhedra, where every face is the same kind of polygon:
 Deltahedra have equilateral triangles for faces.
 With regard to polyhedra whose faces are all squares: if coplanar faces are not allowed, even if they are disconnected, there is only the cube. Otherwise there is also the result of pasting six cubes to the sides of one, all seven of the same size; it has 30 square faces (counting disconnected faces in the same plane as separate). This can be extended in one, two, or three directions: we can consider the union of arbitrarily many copies of these structures, obtained by translations of (expressed in cube sizes) (2,0,0), (0,2,0), and/or (0,0,2), hence with each adjacent pair having one common cube. The result can be any connected set of cubes with positions (a,b,c), with integers a,b,c of which at most one is even.
 There is no special name for polyhedra whose faces are all equilateral pentagons or pentagrams. There are infinitely many of these, but only one is convex: the dodecahedron. The rest are assembled by (pasting) combinations of the regular polyhedra described earlier: the dodecahedron, the small stellated dodecahedron, the great stellated dodecahedron and the great icosahedron.
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 polyhedronIn geometry, an infinite skew polyhedron is an extension of the idea of a polyhedron, consisting of regular polygon faces with nonplanar vertex figures, allowing the figure to extend indefinitely without folding round to form a closed surface....
for exceptions with zigzagging
vertex figureIn geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.Definitions  theme and variations:...
s.)
Deltahedra
A
deltahedronA deltahedron is a polyhedron whose faces are all equilateral triangles. The name is taken from the Greek majuscule delta , which has the shape of an equilateral triangle. There are infinitely many deltahedra, but of these only eight are convex, having 4, 6, 8, 10, 12, 14, 16 and 20 faces...
(plural deltahedra) is a polyhedron whose faces are all equilateral triangles. There are infinitely many deltahedra, but only eight of these are convex:
 3 regular convex polyhedra (3 of the Platonic solids)
 Tetrahedron
In geometry, a tetrahedron is a polyhedron composed of four triangular 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...
 Octahedron
In geometry, 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 vertex....
 Icosahedron
In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....
 5 nonuniform convex polyhedra (5 of the Johnson solids)
 Triangular dipyramid
In geometry, the triangular bipyramid is the first in the infinite set of facetransitive bipyramids. It is the dual of the triangular prism with 6 isosceles triangle faces....
 Pentagonal dipyramid
In geometry, the pentagonal bipyramid is third of the infinite set of facetransitive bipyramids.Each bipyramid is the dual of a uniform prism.If the faces are equilateral triangles, it is a deltahedron and a Johnson solid...
 Snub disphenoid
In geometry, the snub disphenoid is one of the Johnson solids . It is a threedimensional solid that has only equilateral triangles as faces, and is therefore a deltahedron. It is not a regular polyhedron because some vertices have four faces and others have five...
 Triaugmented triangular prism
In geometry, the triaugmented triangular prism is one of the Johnson solids . As the name suggests, it can be constructed by attaching square pyramids to each of the three equatorial faces of the triangular prism...
 Gyroelongated square dipyramid
In geometry, the gyroelongated square bipyramid is one of the Johnson solids . As the name suggests, it can be constructed by gyroelongating an octahedron by inserting a square antiprism between its congruent halves...
.
Johnson solids
Norman Johnson sought which convex nonuniform polyhedra had regular faces. In 1966, he published a list of 92 such solids, gave them names and numbers, and conjectured that there were no others.
Victor ZalgallerVictor Abramovich Zalgaller is a mathematician in the fields of geometry and optimization. He is best known for his results on convex polyhedra, linear and dynamic programming, isoperimetry, and differential geometry....
proved in 1969 that the list of these
Johnson solidIn geometry, a Johnson solid is a strictly convex polyhedron, each face of which is a regular polygon, but which is not uniform, i.e., not a Platonic solid, Archimedean solid, prism or antiprism. There is no requirement that each face must be the same polygon, or that the same polygons join around...
s was complete.
Pyramids
Pyramids include some of the most timehonoured and famous of all polyhedra.
Stellations and facettings
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
facettingIn 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...
which is the process of removing parts of a polyhedron without creating any new vertices.
Zonohedra
A zonohedron is a convex polyhedron where every face is a
polygonIn geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...
with inversion
symmetrySymmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...
or, equivalently, symmetry under
rotationA rotation is a circular movement of an object around a center of rotation. A threedimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
s through 180°.
Toroidal polyhedra
A toroidal polyhedron is a polyhedron with an
Euler characteristicIn mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...
of 0 or smaller, representing a
torusIn geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
surface.
Compounds
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.
Orthogonal polyhedra
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 polygonA rectilinear polygon is a polygon all of whose edges meet at right angles. Thus the interior angle at each vertex is either 90° or 270°. Rectilinear polygons are a special case of isothetic polygons....
s. Orthogonal polyhedra are used in
computational geometryComputational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems are also considered to be part of computational...
, 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.
Generalisations of polyhedra
The name 'polyhedron' has come to be used for a variety of objects having similar structural properties to traditional polyhedra.
Apeirohedra
A classical polyhedral surface comprises finite, bounded plane regions, joined in pairs along edges. If such a surface extends indefinitely it is called an
apeirohedronAn apeirohedron is a polyhedron having infinitely many faces. Like an ordinary polyhedron it forms a surface with no border. But where an ordinary polyhedral surface has no border because it folds round to close back on itself, an apeirohedron has no border because its surface is unbounded.Two main...
. Examples include:
 Tilings or tessellation
A tessellation or tiling of the plane is a pattern of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible. Tessellations frequently appeared in the art...
s of the plane.
 Spongelike structures called infinite skew polyhedra
In geometry, an infinite skew polyhedron is an extension of the idea of a polyhedron, consisting of regular polygon faces with nonplanar vertex figures, allowing the figure to extend indefinitely without folding round to form a closed surface....
.
See also:
ApeirogonAn apeirogon is a degenerate polygon with a countably infinite number of sides. It is the limit of a sequence of polygons with more and more sides.Like any polygon, it is a sequence of line segments and angles...
 infinite regular polygon: {∞}
Complex polyhedra
A
complex polyhedronA complex polytope is a generalization of a polytope in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one....
is one which is constructed in complex
HilbertThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the twodimensional Euclidean plane and threedimensional space to spaces with any finite or infinite number of dimensions...
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).
Spherical polyhedra
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:
 The first known manmade polyhedra are spherical polyhedra carved in stone.
 Poinsot used spherical polyhedra to discover the four regular star polyhedra.
 Coxeter used them to enumerate all but one of the uniform polyhedra.
Some polyhedra, such as
hosohedra and
dihedraA dihedron is a type of polyhedron, made of two polygon faces which share the same set of edges. In threedimensional Euclidean space, it is degenerate if its faces are flat, while in threedimensional spherical space, a dihedron with flat faces can be thought of as a lens, an example of which is...
, exist only as spherical polyhedra and have no flatfaced analogue.
Curved spacefilling polyhedra
Two important types are:
 Bubbles in froths and foams, such as WeairePhelan bubbles.
 Spacefilling forms used in architecture. See for example Pearce (1978).
General polyhedra
More recently
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
has defined a polyhedron as a set in
realIn mathematics, a real number is a value that represents a quantity along a continuum, such as 5 , 4/3 , 8.6 , √2 and π...
affineIn mathematics affine geometry is the study of geometric properties which remain unchanged by affine transformations, i.e. nonsingular linear transformations and translations...
(or
EuclideanEuclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
) space of any dimensional n that has flat sides. It may alternatively 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
halfspaceIn geometry, a halfspace is either of the two parts into which a plane divides the threedimensional euclidean space. More generally, a halfspace is either of the two parts into which a hyperplane divides an affine space...
s. It may be bounded or unbounded. In this meaning, a
polytopeIn elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions...
is a bounded polyhedron.
Analytically, such a convex polyhedron is expressed as the solution set for a system of linear inequalities. Defining polyhedra (and more generally
polytopeIn elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions...
s) in this way provides a geometric perspective for problems in
Linear programmingLinear programming is a mathematical method for determining a way to achieve the best outcome in a given mathematical model for some list of requirements represented as linear relationships...
.
Many traditional polyhedral forms are general polyhedra. Other examples include:
 A quadrant in the plane. For instance, the region of the cartesian plane consisting of all points above the horizontal axis and to the right of the vertical axis: { ( x, y ) : x ≥ 0, y ≥ 0 }. Its sides are the two positive axes.
 An octant in Euclidean 3space, { ( x, y, z ) : x ≥ 0, y ≥ 0, z ≥ 0 }.
 A prism of infinite extent. For instance a doubly infinite square prism in 3space, consisting of a square in the xyplane swept along the zaxis: { ( x, y, z ) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 }.
 Each cell
In geometry, a cell is a threedimensional element that is part of a higherdimensional object. In polytopes :A cell is a threedimensional polyhedron element that is part of the boundary of a higherdimensional polytope, such as a polychoron or honeycomb For example, a cubic honeycomb is made...
in a Voronoi tessellationIn mathematics, a Voronoi diagram is a special kind of decomposition of a given space, e.g., a metric space, determined by distances to a specified family of objects in the space...
is a convex polyhedron. In the Voronoi tessellation of a set S, the cell A corresponding to a point c∈S is bounded (hence a traditional polyhedron) when c lies in the interiorIn mathematics, specifically in topology, the interior of a set S of points of a topological space consists of all points of S that do not belong to the boundary of S. A point that is in the interior of S is an interior point of S....
of the convex hullIn 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....
of S, and otherwise (when c lies on the boundaryIn topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary...
of the convex hull of S) A is unbounded.
Hollow faced or skeletal polyhedra
It is not necessary to fill in the face of a figure before we can call it a polyhedron. For example
Leonardo da VinciLeonardo di ser Piero da Vinci was an Italian Renaissance polymath: painter, sculptor, architect, musician, scientist, mathematician, engineer, inventor, anatomist, geologist, cartographer, botanist and writer whose genius, perhaps more than that of any other figure, epitomized the Renaissance...
devised frame models of the regular solids, which he drew for Pacioli's book Divina Proportione. In modern times,
Branko GrünbaumBranko Grünbaum is a Croatianborn mathematician and a professor emeritus at the University of Washington in Seattle. He received his Ph.D. in 1957 from Hebrew University of Jerusalem in Israel....
(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
skewIn geometry, a skew polygon is a polygon whose vertices do not lie in a plane. Skew polygons must have at least 4 vertices.A regular skew polygon is a skew polygon with equal edge lengths and which is vertextransitive....
as well as planar.
Nongeometric polyhedra
Various mathematical constructs have been found to have properties also present in traditional polyhedra.
Topological polyhedra
A topological polytope is a topological space given along with a specific decomposition into shapes that are topologically equivalent to
convex polytopeA convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the ndimensional space Rn...
s and that are attached to each other in a regular way.
Such a figure is called simplicial if each of its regions is a
simplexIn geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an nsimplex is an ndimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2simplex is a triangle, a 3simplex is a tetrahedron,...
, 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.
Abstract polyhedra
An abstract polyhedron is a
partially ordered setIn mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the...
(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 polytopeIn mathematics, an abstract polytope, informally speaking, is a structure which considers only the combinatorial properties of a traditional polytope, ignoring many of its other properties, such as angles, edge lengths, etc...
s in any number of dimensions.
Polyhedra as graphs
Any polyhedron gives rise to a
graphIn mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...
, or skeleton, with corresponding vertices and edges. Thus
graph terminologyGraph theory is a growing area in mathematical research, and has a large specialized vocabulary. Some authors use the same word with different meanings. Some authors use different words to mean the same thing. This page attempts to keep up with current usage....
and properties can be applied to polyhedra. For example:
 Due to Steinitz theorem convex polyhedra are in onetoone correspondence with 3connected planar graphs.
 The tetrahedron
In geometry, a tetrahedron is a polyhedron composed of four triangular 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...
gives rise to a complete graphIn the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge.Properties:...
(K_{4}). It is the only polyhedron to do so.
 The octahedron
In geometry, 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 vertex....
gives rise to a strongly regular graphIn graph theory, a discipline within mathematics, a strongly regular graph is defined as follows. Let G = be a regular graph with v vertices and degree k...
, because adjacent vertices always have two common neighbors, and nonadjacent vertices have four.
 The Archimedean solid
In geometry an Archimedean solid is a highly symmetric, semiregular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices...
s give rise to regular graphIn graph theory, a regular graph is a graph where each vertex has the same number of neighbors; i.e. every vertex has the same degree or valency. A regular directed graph must also satisfy the stronger condition that the indegree and outdegree of each vertex are equal to each other...
s: 7 of the Archimedean solids are of degreeIn graph theory, the degree of a vertex of a graph is the number of edges incident to the vertex, with loops counted twice. The degree of a vertex v is denoted \deg. The maximum degree of a graph G, denoted by Δ, and the minimum degree of a graph, denoted by δ, are the maximum and minimum degree...
3, 4 of degree 4, and the remaining 2 are chiral pairs of degree 5.
Prehistory
Stones carved in shapes showing the symmetries of various polyhedra have been found in
ScotlandScotland is a country that is part of the United Kingdom. Occupying the northern third of the island of Great Britain, it shares a border with England to the south and is bounded by the North Sea to the east, the Atlantic Ocean to the north and west, and the North Channel and Irish Sea to the...
and may be as much as 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 MuseumThe Ashmolean Museum on Beaumont Street, Oxford, England, is the world's first university 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
architectureArchitecture is both the process and product of planning, designing and construction. Architectural works, in the material form of buildings, are often perceived as cultural and political symbols and as works of art...
—cubes and cuboids being obvious examples, with the earliest foursided pyramids of ancient
EgyptEgypt , officially the Arab Republic of Egypt, Arabic: , is a country mainly in North Africa, with the Sinai Peninsula forming a land bridge in Southwest Asia. Egypt is thus a transcontinental country, and a major power in Africa, the Mediterranean Basin, the Middle East and the Muslim world...
also dating from the Stone Age.
The
EtruscansEtruscan civilization is the modern English name given to a civilization of ancient Italy in the area corresponding roughly to Tuscany. The ancient Romans called its creators the Tusci or Etrusci...
preceded the Greeks in their awareness of at least some of the regular polyhedra, as evidenced by the discovery near
PaduaPadua is a city and comune in the Veneto, northern Italy. It is the capital of the province of Padua and the economic and communications hub of the area. Padua's population is 212,500 . The city is sometimes included, with Venice and Treviso, in the PaduaTrevisoVenice Metropolitan Area, having...
(in Northern
ItalyItaly , officially the Italian Republic languages]] under the European Charter for Regional or Minority Languages. In each of these, Italy's official name is as follows:;;;;;;;;), is a unitary parliamentary republic in SouthCentral Europe. To the north it borders France, Switzerland, Austria and...
) in the late 19th century of a
dodecahedron made of
soapstoneSoapstone is a metamorphic rock, a talcschist. It is largely composed of the mineral talc and is thus rich in magnesium. It is produced by dynamothermal metamorphism and metasomatism, which occurs in the areas where tectonic plates are subducted, changing rocks by heat and pressure, with influx...
, and dating back more than 2,500 years (Lindemann, 1987). Pyritohedric crystals are found in northern Italy.
Greeks
The earliest known written records of these shapes come from Classical
GreekAncient Greece is a civilization belonging to a period of Greek history that lasted from the Archaic period of the 8th to 6th centuries BC to the end of antiquity. Immediately following this period was the beginning of the Early Middle Ages and the Byzantine era. Included in Ancient Greece is the...
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 solidIn geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...
s.
PythagorasPythagoras of Samos was an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so very little reliable information is known about him...
knew at least three of them, and
TheaetetusTheaetetus, Theaitētos, of Athens, possibly son of Euphronius, of the Athenian deme Sunium, was a classical Greek mathematician...
(circa 417 B. C.) described all five. Eventually,
EuclidEuclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
described their construction in his
ElementsEuclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...
. Later,
ArchimedesArchimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...
expanded his study to the
convex uniform polyhedraA uniform polyhedron is a polyhedron which has regular polygons as faces and is vertextransitive...
which now bear his name. His original work is lost and his solids come down to us through
PappusPappus of Alexandria was one of the last great Greek mathematicians of Antiquity, known for his Synagoge or Collection , and for Pappus's Theorem in projective geometry...
.
Chinese
By 236 AD, in China Liu Hui was describing the dissection of the cube into its characteristic tetrahedron (orthoscheme) and related solids, using assemblages of these solids as the basis for calculating volumes of earth to be moved during engineering excavations.
Islamic
After the end of the Classical era, scholars in the Islamic civilisation continued to take the Greek knowledge forward (see Mathematics in medieval Islam).
The 9th century scholar
Thabit ibn Qurra' was a mathematician, physician, astronomer and translator of the Islamic Golden Age.Ibn Qurra made important discoveries in algebra, geometry and astronomy...
gave formulae for calculating the volumes of polyhedra such as truncated pyramids.
Then in the 10th century Abu'l Wafa described the convex regular and quasiregular spherical polyhedra.
Renaissance
As with other areas of Greek thought maintained and enhanced by Islamic scholars, Western interest in polyhedra revived during the Italian
RenaissanceThe Renaissance was a cultural movement that spanned roughly the 14th to the 17th century, beginning in Italy in the Late Middle Ages and later spreading to the rest of Europe. The term is also used more loosely to refer to the historical era, but since the changes of the Renaissance were not...
. Artists constructed skeletal polyhedra, depicting them from life as a part of their investigations into
perspective Literally, in visual topics :* Perspective , the way in which objects appear to the eye.* Perspective , representing the effects of visual perspective in graphic arts Metaphorically, in relation to cognitive topics :...
. Several appear in marquetry panels of the period. Piero della Francesca gave the first written description of direct geometrical construction of such perspective views of polyhedra. Leonardo da Vinci made skeletal models of several polyhedra and drew illustrations of them for a book by Pacioli. A painting by an anonymous artist of Pacioli and a pupli depicts a glass
rhombicuboctahedronIn geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces. There are 24 identical vertices, with one triangle and three squares meeting at each. Note that six of the squares only share vertices with the triangles...
halffilled with water.
As the Renaissance spread beyond Italy, later artists such as Wenzel Jamnitzer, Dürer and others also depicted polyhedra of various kinds, many of them novel, in imaginative etchings.
Star polyhedra
For almost 2,000 years, the concept of a polyhedron as a convex solid had remained as developed by the ancient Greek mathematicians.
During the
RenaissanceThe Renaissance was a cultural movement that spanned roughly the 14th to the 17th century, beginning in Italy in the Late Middle Ages and later spreading to the rest of Europe. The term is also used more loosely to refer to the historical era, but since the changes of the Renaissance were not...
star forms were discovered. A marble tarsia in the floor of St. Mark's Basilica, Venice, depicts a stellated dodecahedron. Artists such as Wenzel Jamnitzer delighted in depicting novel starlike forms of increasing complexity.
Johannes KeplerJohannes Kepler was a German mathematician, astronomer and astrologer. A key figure in the 17th century scientific revolution, he is best known for his eponymous laws of planetary motion, codified by later astronomers, based on his works Astronomia nova, Harmonices Mundi, and Epitome of Copernican...
realised that
star polygons, typically
pentagramA pentagram is the shape of a fivepointed star drawn with five straight strokes...
s, could be used to build star polyhedra. 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 PoinsotLouis Poinsot was a French mathematician and physicist. Poinsot was the inventor of geometrical mechanics, showing how a system of forces acting on a rigid body could be resolved into a single force and a couple.Life:...
realised that star
vertex figureIn geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.Definitions  theme and variations:...
s (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
stellationStellation is a process of constructing new polygons , new polyhedra 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...
. 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
facettingIn 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...
(or faceting). Every stellation of one polytope is
dualIn geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another...
, or reciprocal, to some facetting of the dual polytope. The regular star polyhedra can also be obtained by facetting the Platonic solids. 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:
 Regular polyhedron: History
 Regular polytope: History of discovery.
Polyhedra in nature
For natural occurrences of regular polyhedra, see Regular polyhedron: Regular polyhedra in nature.
Irregular polyhedra appear in nature as
crystalA crystal or crystalline solid is a solid material whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions. The scientific study of crystals and crystal formation is known as crystallography...
s.
General theory
Lists and databases of polyhedra
Software
Resources for making models, and models for sale
Miscellaneous