A polyhedron (plural polyhedra or polyhedrons) is often defined as a geometric object with flat faces and straight edges (the word polyhedron comes from the Classical Greek πολυεδρον, from poly-, stem of πολυς, "many," + -edron, form of εδρον, "base", "seat", or "face"). A dodecahedron is any Polyhedron with twelve faces but usually a regular dodecahedron is meant a Platonic solid composed of twelve regular Pentagonal In Geometry, a Platonic solid is a convex Regular polyhedron. In Geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedron. The Kepler-Poinsot polyhedra is a popular name for the regular star polyhedra. An icosidodecahedron is a Polyhedron with twenty triangular faces and twelve pentagonal faces In Geometry an Archimedean solid is a highly symmetric semi-regular convex Polyhedron composed of two or more types of Regular polygons meeting In Geometry, the great cubicuboctahedron is a nonconvex Uniform polyhedron, indexed as U14 A uniform polyhedron is a Polyhedron which has Regular polygons as faces and is Transitive on its vertices (i In Geometry, the rhombic triacontahedron is a convex Polyhedron with 30 rhombic faces In Mathematics, a Catalan solid, or Archimedean dual, is a Dual polyhedron to an Archimedean solid. In Geometry, the elongated pentagonal cupola is one of the Johnson solids ( J 20 In Geometry, a Johnson solid is a strictly convex Polyhedron, each face of which is a Regular polygon, which is not a Platonic solid In Geometry, the octagonal prism is the sixth in an infinite set of prisms formed by square sides and two regular polygon caps General right and uniform prisms A right prism is a prism in which the joining edges and faces are perpendicular to the base faces In Geometry, the square antiprism is the second in an infinite set of Antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps Cartesian coordinates Cartesian coordinates for the vertices of a right antiprism with n -gonal bases and isosceles triangles are ( \cos(k\pi/n Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Greek (el ελληνική γλώσσα or simply el ελληνικά — "Hellenic" is an Indo-European language, spoken today by 15-22 million people mainly

This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory. Grünbaum (1994, p. Branko Grünbaum (born 1929 is a Croatian born Mathematician and a professor Emeritus at the University of Washington in Seattle. 43) observed that:

The Original Sin in the theory of polyhedra goes back to Euclid, and through Kepler, Poinsot, Cauchy and many others . Original sin is according to a doctrine in Catholic theology, humanity's state of Sin resulting from the Fall of Man. Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry Johannes Kepler (ˈkɛplɚ ( December 27 1571 &ndash November 15 1630) was a German Mathematician, Astronomer Louis Poinsot (1777 - 1859 was a French Mathematician and Physicist. . . [in that] at each stage . . . the writers failed to define what are the 'polyhedra' . . .

Modern mathematicians do not even agree as to exactly what makes something a polyhedron.

## What is a polyhedron?

We can at least say that a polyhedron is 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 inside them.
• 2 dimensions: A face is bounded by a circuit of edges, and is usually a flat (plane) region called a polygon. In Geometry, a face of a Polyhedron is any of the Polygons that make up its boundaries In Geometry a polygon (ˈpɒlɨɡɒn ˈpɒliɡɒn is traditionally a plane figure that is bounded by a closed path or circuit The faces together make up the polyhedral surface.
• 1 dimension: An edge joins one vertex to another and one face to another, and is usually a line of some kind. For edge in Graph theory, see Edge (graph theory In Geometry, an edge is a one-dimensional Line segment joining The edges together make up the polyhedral skeleton.
• 0 dimensions: A vertex (plural vertices) is a corner point. In Geometry, a vertex (plural "vertices" is a special kind of point. In Geometry, Topology and related branches of mathematics a spatial point describes a specific point within a given space that consists of neither Volume
• -1 dimension: The nullity is a kind of non-entity required by abstract theories.

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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and

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 3-dimensional example of the more general polytope in any number of dimensions. In Geometry, polytope is a generic term that can refer to a two-dimensional Polygon, a three-dimensional Polyhedron, or any of the various generalizations

## Characteristics

Naming polyhedra

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. A tetrahedron (plural tetrahedra) is a Polyhedron composed of four triangular faces three of which meet at each vertex. In Geometry, a pentahedron (plural pentahedra is a Polyhedron with five faces A hexahedron (plural hexahedra is a Polyhedron with six faces A heptahedron (plural heptahedra is a Polyhedron having seven sides or faces. In Geometry, the rhombic triacontahedron is a convex Polyhedron with 30 rhombic faces

Often this is qualified by a description of the kinds of faces present, for example the Rhombic dodecahedron vs. The rhombic dodecahedron is a convex Polyhedron with 12 rhombic faces the Pentagonal dodecahedron. A dodecahedron is any Polyhedron with twelve faces but usually a regular dodecahedron is meant a Platonic solid composed of twelve regular Pentagonal

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). The truncated cube, or truncated hexahedron, is an Archimedean solid.

Some special polyhedra have grown their own names over the years, such as Miller's monster or the Szilassi polyhedron. In Geometry, the great dirhombicosidodecahedron is a nonconvex Uniform polyhedron, indexed last as U75 The Szilassi polyhedron is a nonconvex Polyhedron, topologically a Torus, with seven hexagonal faces

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 characteristic χ relates the number of vertices V, edges E, and faces F of a polyhedron:

χ = V - E + F. In Mathematics, and more specifically in Algebraic topology and Polyhedral combinatorics, the Euler characteristic is a Topological invariant

For a simply connected polyhedron, χ = 2. In Topology, a geometrical object or space is called simply connected (or 1-connected) if it is Path-connected and every path between two points can be For a detailed discussion, see Proofs and Refutations by Imre Lakatos. Proofs and Refutations is a book by the Philosopher Imre Lakatos expounding his view ofthe progress of Mathematics. Imre Lakatos ( November 9, 1922 – February 2, 1974) was a Philosopher of mathematics and science,

Duality

For every polyhedron there is a dual polyhedron having faces in place of the original's vertices and vice versa. In Geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the In most cases the dual can be obtained by the process of spherical reciprocation.

Vertex figure

For every vertex one can define a vertex figure consisting of the vertices joined to it. In Geometry a vertex figure is broadly speaking the figure exposed when a corner of a Polyhedron or Polytope is sliced off The vertex is said to be regular if this is a regular polygon and symmetrical with respect to the whole polyhedron.

A dodecahedron

In geometry, a polyhedron is traditionally a three-dimensional 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 straight-line segments, and the edges meet in points called vertices. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In Geometry a polygon (ˈpɒlɨɡɒn ˈpɒliɡɒn is traditionally a plane figure that is bounded by a closed path or circuit In Geometry, a face of a Polyhedron is any of the Polygons that make up its boundaries For edge in Graph theory, see Edge (graph theory In Geometry, an edge is a one-dimensional Line segment joining In Geometry, a vertex (plural "vertices" is a special kind of point. Cubes, prisms and pyramids are examples of polyhedra. A cube is a three-dimensional solid object bounded by six square faces facets or sides with three meeting at each vertex. General right and uniform prisms A right prism is a prism in which the joining edges and faces are perpendicular to the base faces Volume The Volume of a pyramid is V = \frac{1}{3} Bh where B is the area of the base and h the height from the base to the apex The polyhedron surrounds a bounded volume in three-dimensional 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 and surface. In Euclidean space, an object is convex if for every pair of points within the object every point on the Straight line segment that joins them is also within the

### Symmetrical polyhedra

Many of the most studied polyhedra are highly symmetrical. Symmetry 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

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. An icosidodecahedron is a Polyhedron with twenty triangular faces and twelve pentagonal faces

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 tetrahedron, cube, 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. A tetrahedron (plural tetrahedra) is a Polyhedron composed of four triangular faces three of which meet at each vertex. A cube is a three-dimensional solid object bounded by six square faces facets or sides with three meeting at each vertex. An octahedron (plural octahedra is a Polyhedron with eight faces A dodecahedron is any Polyhedron with twelve faces but usually a regular dodecahedron is meant a Platonic solid composed of twelve regular Pentagonal In Geometry, an icosahedron ( Greek: eikosaedron, from eikosi twenty + hedron seat /ˌaɪ There are various classes of such polyhedra:

• Isogonal or Vertex-transitive if all vertices are the same, in the sense that for any two vertices there exists a symmetry of the polyhedron mapping the first isometrically onto the second. The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is For the Mechanical engineering and Architecture usage see Isometric projection.
• Isotoxal or Edge-transitive 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 Face-transitive 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 if it is vertex-transitive, edge-transitive and face-transitive (this implies that every face is the same regular polygon; it also implies that every vertex is regular). A regular polyhedron is a Polyhedron whose faces are congruent (all alike Regular polygons which are assembled in the same way around each Vertex General properties These properties apply to both convex and star regular polygons
• Quasi-regular if it is vertex-transitive and edge-transitive (and hence has regular faces) but not face-transitive. A Polyhedron which has regular faces and is transitive on its edges but not transitive on its faces is said to be quasiregular. A quasi-regular dual is face-transitive and edge-transitive (and hence every vertex is regular) but not vertex-transitive.
• Semi-regular if it is vertex-transitive but not edge-transitive, and every face is a regular polygon. A semiregular polyhedron is a Polyhedron with regular faces and a symmetry group which is transitive on its vertices (This is one of several definitions of the term, depending on author. Some definitions overlap with the quasi-regular class). A semi-regular dual is face-transitive but not vertex-transitive, and every vertex is regular.
• Uniform if it is vertex-transitive and every face is a regular polygon, i. A uniform polyhedron is a Polyhedron which has Regular polygons as faces and is Transitive on its vertices (i e. it is regular, quasi-regular or semi-regular. A uniform dual is face-transitive and has regular vertices, but is not necessarily vertex-transitive).
• Noble if it is face-transitive and vertex-transitive (but not necessarily edge-transitive). A noble polyhedron is one which is Isohedral (all faces the same and Isogonal (all vertices the same 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

Main article: Uniform polyhedron

Uniform polyhedra are vertex-transitive and every face is a regular polygon. A uniform polyhedron is a Polyhedron which has Regular polygons as faces and is Transitive on its vertices (i General properties These properties apply to both convex and star regular polygons They may be regular, quasi-regular, or semi-regular, and may be convex or starry. A regular polyhedron is a Polyhedron whose faces are congruent (all alike Regular polygons which are assembled in the same way around each Vertex A Polyhedron which has regular faces and is transitive on its edges but not transitive on its faces is said to be quasiregular. A semiregular polyhedron is a Polyhedron with regular faces and a symmetry group which is transitive on its vertices

The uniform duals are face-transitive and every vertex figure is a regular polygon. In Geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the In Geometry a vertex figure is broadly speaking the figure exposed when a corner of a Polyhedron or Polytope is sliced off

Face-transitivity of a polyhedron corresponds to vertex-transitivity of the dual and conversely, and edge-transitivity of a polyhedron corresponds to edge-transitivity 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 uniformConvex uniform dualStar uniformStar uniform dual
RegularPlatonic solidsKepler-Poinsot polyhedra
QuasiregularArchimedean solidsCatalan solids(no special name)(no special name)
Semiregular(no special name)(no special name)
PrismsDipyramidsStar PrismsStar Dipyramids
AntiprismsTrapezohedraStar AntiprismsStar Trapezohedra

#### Noble polyhedra

Main article: Noble polyhedron

A noble polyhedron is both isohedral (equal-faced) and isogonal (equal-cornered). A regular polyhedron is a Polyhedron whose faces are congruent (all alike Regular polygons which are assembled in the same way around each Vertex In Geometry, a Platonic solid is a convex Regular polyhedron. The Kepler-Poinsot polyhedra is a popular name for the regular star polyhedra. A Polyhedron which has regular faces and is transitive on its edges but not transitive on its faces is said to be quasiregular. In Geometry an Archimedean solid is a highly symmetric semi-regular convex Polyhedron composed of two or more types of Regular polygons meeting In Mathematics, a Catalan solid, or Archimedean dual, is a Dual polyhedron to an Archimedean solid. A semiregular polyhedron is a Polyhedron with regular faces and a symmetry group which is transitive on its vertices General right and uniform prisms A right prism is a prism in which the joining edges and faces are perpendicular to the base faces Equilateral triangle bipyramids Only three kinds of bipyramids can have all edges of the same length (which implies that all faces are Equilateral triangles: the General right and uniform prisms A right prism is a prism in which the joining edges and faces are perpendicular to the base faces Equilateral triangle bipyramids Only three kinds of bipyramids can have all edges of the same length (which implies that all faces are Equilateral triangles: the Cartesian coordinates Cartesian coordinates for the vertices of a right antiprism with n -gonal bases and isosceles triangles are ( \cos(k\pi/n Forms Trigonal trapezohedron - 6 (rhombic faces - dual Octahedron * A Cube is a special case trigonal trapezohedron Cartesian coordinates Cartesian coordinates for the vertices of a right antiprism with n -gonal bases and isosceles triangles are ( \cos(k\pi/n Forms Trigonal trapezohedron - 6 (rhombic faces - dual Octahedron * A Cube is a special case trigonal trapezohedron A noble polyhedron is one which is Isohedral (all faces the same and Isogonal (all vertices the same A noble polyhedron is one which is Isohedral (all faces the same and Isogonal (all vertices the same Besides the regular polyhedra, there are many other examples.

The dual of a noble polyhedron is also noble. In Geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the

#### Symmetry groups

The polyhedral symmetry groups are all point groups and include:

• T - chiral tetrahedral symmetry; the rotation group for a regular tetrahedron; order 12. The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is In Geometry, a Point group in three dimensions is an Isometry group in three dimensions that leaves the origin fixed or correspondingly an isometry group A regular Tetrahedron has 12 rotational (or orientation-preserving symmetries and a total of 24 symmetries including transformations that combine a reflection and a rotation A tetrahedron (plural tetrahedra) is a Polyhedron composed of four triangular faces three of which meet at each vertex.
• Td - full tetrahedral symmetry; the symmetry group for a regular tetrahedron; order 24. A regular Tetrahedron has 12 rotational (or orientation-preserving symmetries and a total of 24 symmetries including transformations that combine a reflection and a rotation A tetrahedron (plural tetrahedra) is a Polyhedron composed of four triangular faces three of which meet at each vertex.
• Th - pyritohedral symmetry; order 24. A regular Tetrahedron has 12 rotational (or orientation-preserving symmetries and a total of 24 symmetries including transformations that combine a reflection and a rotation The symmetry of a pyritohedron. In Geometry, a pyritohedron is an irregular Dodecahedron. Like the regular dodecahedron it has twelve identical Pentagonal faces with three meeting in
• O - chiral octahedral symmetry;the rotation group of the cube and octahedron; order 24. A regular Octahedron has 24 rotational (or orientation-preserving symmetries and a total of 48 symmetries including transformations that combine a reflection and a rotation A cube is a three-dimensional solid object bounded by six square faces facets or sides with three meeting at each vertex. An octahedron (plural octahedra is a Polyhedron with eight faces
• Oh - full octahedral symmetry; the symmetry group of the cube and octahedron; order 48. A regular Octahedron has 24 rotational (or orientation-preserving symmetries and a total of 48 symmetries including transformations that combine a reflection and a rotation A cube is a three-dimensional solid object bounded by six square faces facets or sides with three meeting at each vertex. An octahedron (plural octahedra is a Polyhedron with eight faces
• I - chiral icosahedral symmetry; the rotation group of the icosahedron and the dodecahedron; order 60. A regular Icosahedron has 60 rotational (or orientation-preserving symmetries and a total of 120 symmetries including transformations that combine a reflection and a rotation In Geometry, an icosahedron ( Greek: eikosaedron, from eikosi twenty + hedron seat /ˌaɪ A dodecahedron is any Polyhedron with twelve faces but usually a regular dodecahedron is meant a Platonic solid composed of twelve regular Pentagonal
• Ih - full icosahedral symmetry; the symmetry group of the icosahedron and the dodecahedron; order 120. A regular Icosahedron has 60 rotational (or orientation-preserving symmetries and a total of 120 symmetries including transformations that combine a reflection and a rotation In Geometry, an icosahedron ( Greek: eikosaedron, from eikosi twenty + hedron seat /ˌaɪ A dodecahedron is any Polyhedron with twelve faces but usually a regular dodecahedron is meant a Platonic solid composed of twelve regular Pentagonal
• Cnv - n-fold pyramidal symmetry
• Dnh - n-fold prismatic symmetry
• Dnv - n-fold antiprismatic symmetry

Those with chiral symmetry do not have reflection symmetry and hence have two enantiomorphous forms which are reflections of each other. This article deals with the four infinite series of Point groups in three dimensions ( n &ge1 with n -fold Rotational symmetry about one axis In Mathematics, a dihedral group is the group of symmetries of a Regular polygon, including both rotations and reflections In Mathematics, a dihedral group is the group of symmetries of a Regular polygon, including both rotations and reflections In Geometry, a figure is chiral (and said to have chirality) if it is not identical to its Mirror image, or more particularly if it cannot be mapped to Reflection symmetry, line symmetry, mirror symmetry, mirror-image symmetry, or bilateral symmetry is Symmetry with respect In Geometry, a figure is chiral (and said to have chirality) if it is not identical to its Mirror image, or more particularly if it cannot be mapped to The snub Archimedean polyhedra have this property.

### Other polyhedra with regular faces

#### Equal regular faces

A few families of polyhedra, where every face is the same kind of polygon:

• 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. In Geometry, a set of points in space is coplanar if the points all lie in the same geometric plane. 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 polyhedron for exceptions with zig-zagging vertex figures. In Geometry, infinite skew polyhedra are an extended definition of polyhedra, created by Regular polygon faces and nonplanar Vertex figures In Geometry a vertex figure is broadly speaking the figure exposed when a corner of a Polyhedron or Polytope is sliced off )

##### Deltahedra

A deltahedron (plural deltahedra) is a polyhedron whose faces are all equilateral triangles. A deltahedron ( Plural deltahedra) is a Polyhedron whose faces are all Equilateral triangles The name is taken from the Greek There are infinitely many deltahedra, but only eight of these are convex:

• 3 regular convex polyhedra (3 of the Platonic solids)
• 5 non-uniform convex polyhedra (5 of the Johnson solids)

#### Johnson solids

Main article: Johnson solid

Norman Johnson sought which non-uniform polyhedra had regular faces. A tetrahedron (plural tetrahedra) is a Polyhedron composed of four triangular faces three of which meet at each vertex. An octahedron (plural octahedra is a Polyhedron with eight faces In Geometry, an icosahedron ( Greek: eikosaedron, from eikosi twenty + hedron seat /ˌaɪ For the related molecular geometry see Trigonal bipyramid molecular geometry In Geometry, the triangular dipyramid (or Bipyramid) is the first In Geometry, the pentagonal dipyramid (or Bipyramid) is third of the infinite set of Face-transitive Dipyramids The set In Geometry, the snub disphenoid is one of the Johnson solids ( J 84 In Geometry, the triaugmented triangular prism is one of the Johnson solids ( J 51 In Geometry, the gyroelongated square dipyramid is one of the Johnson solids ( J 17 In Geometry, a Johnson solid is a strictly convex Polyhedron, each face of which is a Regular polygon, which is not a Platonic solid Norman W Johnson is a Mathematician, previously at Wheaton College, Norton Massachusetts. In 1966, he published a list of 92 convex solids, now known as the Johnson solids, and gave them their names and numbers. Year 1966 ( MCMLXVI) was a Common year starting on Saturday (link will display full calendar of the 1966 Gregorian calendar. In Geometry, a Johnson solid is a strictly convex Polyhedron, each face of which is a Regular polygon, which is not a Platonic solid 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. Victor (Viktor Abramovich Zalgaller (ויקטור אבּרמוביץ' זלגלר Виктор Абрамович Залгаллер born on December 25, 1920 in

### Other important families of polyhedra

#### Pyramids

Main article: Pyramid (geometry)

Pyramids include some of the most time-honoured and famous of all polyhedra. Volume The Volume of a pyramid is V = \frac{1}{3} Bh where B is the area of the base and h the height from the base to the apex

#### Stellations and facettings

Main article: Stellation

Stellation of a polyhedron is the process of extending the faces (within their planes) so that they meet to form a new polyhedron. Stellation is a process of constructing new Polygons (in two Dimensions, new Polyhedra in three dimensions or in general new Polytopes in

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. History Facetting has not been studied as extensively as Stellation.

#### Zonohedra

Main article: Zonohedron

A zonohedron is a convex polyhedron where every face is a polygon with inversion symmetry or, equivalently, symmetry under rotations through 180°. A zonohedron is a convex Polyhedron where every face is a Polygon with point Symmetry or equivalently symmetry under Rotations through A zonohedron is a convex Polyhedron where every face is a Polygon with point Symmetry or equivalently symmetry under Rotations through In Geometry a polygon (ˈpɒlɨɡɒn ˈpɒliɡɒn is traditionally a plane figure that is bounded by a closed path or circuit Symmetry 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 A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation

#### Compounds

Main article: Polyhedral compound

Polyhedral compounds are formed as compounds of two or more polyhedra. A polyhedral compound is a Polyhedron that is itself composed of several other polyhedra sharing a common centre

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. This table contains an indexed list of the Uniform and stellated polyhedra from the book Polyhedron Models, by Magnus Wenninger

#### 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 polygons). A 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° A 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° 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 net (polyhedron). Computational geometry is a branch of Computer science devoted to the study of algorithms which can be stated in terms of Geometry. In Geometry the net of a Polyhedron is an arrangement of edge-joined Polygons in the plane which can be folded (along edges to become the faces of the polyhedron

## 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 apeirohedron. Examples include:

• Tilings or tessellations of the plane. A tessellation or tiling of the plane is a collection of Plane figures that fills the plane with no overlaps and no gaps
• Sponge-like structures called infinite skew polyhedra. In Geometry, infinite skew polyhedra are an extended definition of polyhedra, created by Regular polygon faces and nonplanar Vertex figures

### Complex polyhedra

A complex polyhedron is one which is constructed in complex Hilbert 3-space. A complex polytope is one which exists in a complex Hilbert space, where each real dimension is accompanied by an imaginary one This article assumes some familiarity with Analytic geometry and the concept of a limit. This space has six dimensions: three real ones corresponding to ordinary space, with each accompanied by an imaginary dimension. See for example Coxeter (1974).

### Curved polyhedra

Some fields of study allow polyhedra to have curved faces and edges.

#### Spherical polyhedra

Main article: Spherical polyhedron

The surface of a sphere may be divided by line segments into bounded regions, to form a spherical polyhedron. In Mathematics, the surface of a sphere may be divided by line segments into bounded regions to form a spherical tiling or 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 man-made 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, exist only as spherical polyhedra and have no flat-faced analogue.

#### Curved spacefilling polyhedra

Two important types are:

• Bubbles in froths and foams.
• Spacefilling forms used in architecture. See for example Pearce (1978).

More needs to be said about these, too.

### General polyhedra

More recently mathematics has defined a polyhedron as a set in real affine (or Euclidean) space of any dimensional n that has flat sides. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, the real numbers may be described informally in several different ways Affine geometry is a form of Geometry featuring the unique parallel line property (see the parallel postulate) but where the notion of angle is undefined and lengths Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. 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 half-spaces. In Geometry, a half-space is either of the two parts into which a plane divides the three-dimensional space It may be bounded or unbounded. In this meaning, a polytope is a bounded polyhedron. In Geometry, polytope is a generic term that can refer to a two-dimensional Polygon, a three-dimensional Polyhedron, or any of the various generalizations

All traditional polyhedra are general polyhedra, and in addition there are examples like:

• 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 3-space, { ( x, y, z ) : x ≥ 0, y ≥ 0, z ≥ 0 }.
• A prism of infinite extent. For instance a doubly-infinite square prism in 3-space, consisting of a square in the xy-plane swept along the z-axis: { ( x, y, z ) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 }.
• Each cell in a Voronoi tessellation is a convex polyhedron. In Geometry, a cell is a three- Dimensional element that is part of a higher-dimensional object In Mathematics, a Voronoi diagram, named after Georgy Voronoi, also called a Voronoi Tessellation, a Voronoi decomposition, or In the Voronoi tessellation of a set S, the cell A corresponding to a point cS is bounded (hence a traditional polyhedron) when c lies in the interior of the convex hull of S, and otherwise (when c lies on the boundary of the convex hull of S) A is unbounded. In Mathematics, the interior of a set S consists of all points of S that are intuitively "not on the edge of S " In Mathematics, the convex hull or convex envelope for a set of points X in a Real Vector space V is the minimal Convex For a different notion of boundary related to Manifolds see that article

### 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 Vinci devised frame models of the regular solids, which he drew for Pacioli's book Divina Proportione. Leonardo di ser Piero da Vinci ( April 15 1452 – May 2 1519 was an Italian Polymath, having been a scientist Mathematician, Engineer Fra Luca Bartolomeo de Pacioli (sometimes Paciolo) (1446/7&ndash1517 was an Italian Mathematician and Franciscan friar collaborator with 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. Branko Grünbaum (born 1929 is a Croatian born Mathematician and a professor Emeritus at the University of Washington in Seattle. He defined a face as a cyclically ordered set of vertices, and allowed faces to be skew as well as planar. In Geometry, a skew polygon is a Polygon whose vertices do not lie in a Plane.

### Tessellations or tilings

Tessellations or tilings of the plane are sometimes treated as polyhedra, because they have quite a lot in common. A tessellation or tiling of the plane is a collection of Plane figures that fills the plane with no overlaps and no gaps For example the regular ones can be given Schläfli symbols. In Mathematics, the Schläfli symbol is a notation of the form {pqr

## Non-geometric 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 polytopes and that are attached to each other in a regular way. In Geometry, polytope is a generic term that can refer to a two-dimensional Polygon, a three-dimensional Polyhedron, or any of the various generalizations

Such a figure is called simplicial if each of its regions is a simplex, i. In Geometry, a simplex (plural simplexes or simplices) or n -simplex is an n -dimensional analogue of a triangle e. in an n-dimensional 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 n-dimensional cube.

### Abstract polyhedra

An abstract polyhedron is a partially ordered set (poset) of elements. In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement 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. In Mathematics, an abstract polytope is a combinatorial structure with properties similar to those shared by a more classical Polytope.

### Polyhedra as graphs

Any polyhedron gives rise to a graph, or skeleton, with corresponding vertices and edges. In Mathematics and Computer science, a graph is the basic object of study in Graph theory. Thus graph terminology and properties can be applied to polyhedra. Graph theory is a growing area in mathematical research and has a large specialized vocabulary For example:

• Due to Steinitz theorem convex polyhedra are in one-to-one correspondence with 3-connected planar graphs.
• The tetrahedron gives rise to a complete graph (K4). A tetrahedron (plural tetrahedra) is a Polyhedron composed of four triangular faces three of which meet at each vertex. In the mathematical field of Graph theory, a complete graph is a Simple graph in which every pair of distinct vertices is connected by an It is the only polyhedron to do so.
• The octahedron gives rise to a strongly regular graph, because adjacent vertices always have two common neighbors, and non-adjacent vertices have four. An octahedron (plural octahedra is a Polyhedron with eight faces Let G = (VE be a Regular graph with v vertices and degree k. G is said to be strongly regular if there are also Integers
• The Archimedean solids give rise to regular graphs: 7 of the Archimedean solids are of degree 3, 4 of degree 4, and the remaining 2 are chiral pairs of degree 5. In Geometry an Archimedean solid is a highly symmetric semi-regular convex Polyhedron composed of two or more types of Regular polygons meeting In Graph theory, a regular graph is a graph where each vertex has the same number of neighbors i In Graph theory, the degree (or valency) of a vertex of a graph is the number of edges incident to the vertex

## History

### Prehistory

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. Scotland ( Gaelic: Alba) is a Country in northwest Europethat occupies the northern third of the island of Great Britain. 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. The Ashmolean Museum (in full the Ashmolean Museum of Art and Archaeology) on Beaumont Street, Oxford, England, is the world's first The University of Oxford (informally "Oxford University" or simply "Oxford" located in the city of Oxford, Oxfordshire, England is the 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 four-sided pyramids of ancient Egypt also dating from the Stone Age. The term architecture (from Greek αρχιτεκτονικήarchitektoniki) can be used to mean a process a profession or documentation This article is about the country of Egypt For a topic outline on this subject see List of basic Egypt topics.

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). Etruscan civilization is the modern English name given to the culture and way of life of a people of ancient Italy Padua ( Padova 'padova Latin: Patavium, Padoa) is a city in the Veneto, northern Italy. Italy (Italia officially the Italian Republic, (Repubblica Italiana is located on the Italian Peninsula in Southern Europe, and on the two largest A dodecahedron is any Polyhedron with twelve faces but usually a regular dodecahedron is meant a Platonic solid composed of twelve regular Pentagonal Soapstone (also known as steatite or soaprock) is a Metamorphic rock, a talc- Schist. Pyritohedric crystals are found in northern Italy.

### Greeks

The earliest known written records of these shapes come from Classical Greek authors, who also gave the first known mathematical description of them. The term ancient Greece refers to the period of Greek history lasting from the Greek Dark Ages ca The earlier Greeks were interested primarily in the convex regular polyhedra, while Archimedes later expanded his study to the convex uniform polyhedra. A regular polyhedron is a Polyhedron whose faces are congruent (all alike Regular polygons which are assembled in the same way around each Vertex Archimedes of Syracuse ( Greek:) ( c. 287 BC – c 212 BC was a Greek mathematician, Physicist, Engineer A uniform polyhedron is a Polyhedron which has Regular polygons as faces and is Transitive on its vertices (i

### Muslims and Chinese

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.

### Renaissance

Much to be said here: Piero della Francesca, Pacioli, Leonardo Da Vinci, Wenzel Jamnitzer, Durer, etc. leading up to Kepler.

### Star polyhedra

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 non-convex regular polygons, typically pentagrams as faces. Johannes Kepler (ˈkɛplɚ ( December 27 1571 &ndash November 15 1630) was a German Mathematician, Astronomer Regular star polygons In Geometry, a regular star polygon is a self-intersecting equilateral equiangular Polygon, created by connecting one Early history Sumer The first known uses of the pentagram are found in Mesopotamian writings dating to about 3000 BC 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. Louis Poinsot (1777 - 1859 was a French Mathematician and Physicist. In Geometry a vertex figure is broadly speaking the figure exposed when a corner of a Polyhedron or Polytope is sliced off 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. In Geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedron. In Geometry, the great stellated dodecahedron is a Kepler-Poinsot polyhedron. In Geometry, the great icosahedron is a Kepler-Poinsot polyhedron. In Geometry, the great dodecahedron is a Kepler-Poinsot polyhedron. Collectively they are called the Kepler-Poinsot polyhedra. The Kepler-Poinsot polyhedra is a popular name for the regular star polyhedra.

The Kepler-Poinsot polyhedra may be constructed from the Platonic solids by a process called stellation. Stellation is a process of constructing new Polygons (in two Dimensions, new Polyhedra in three dimensions or in general new Polytopes in 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. Harold Scott MacDonald "Donald" Coxeter CC ( February 9, 1907 – March 31, 2003) is regarded as one of the great This work has recently been re-published (Coxeter, 1999).

The reciprocal process to stellation is called facetting (or faceting). History Facetting has not been studied as extensively as Stellation. Every stellation of one polytope is dual, or reciprocal, to some facetting of the dual polytope. In Geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the 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.

• Regular polyhedron: History
• Regular polytope: History of discovery. A regular polyhedron is a Polyhedron whose faces are congruent (all alike Regular polygons which are assembled in the same way around each Vertex In Mathematics, a regular polytope is a Polytope whose Symmetry is transitive on its flags, thus giving it the highest degree of symmetry

## Polyhedra in nature

For natural occurrences of regular polyhedra, see Regular polyhedron: History. A regular polyhedron is a Polyhedron whose faces are congruent (all alike Regular polygons which are assembled in the same way around each Vertex

Irregular polyhedra appear in nature as crystals. In Materials science, a crystal is a Solid in which the constituent Atoms Molecules or Ions are packed in a regularly ordered repeating

## References

• Coxeter, H.S.M.; Regular complex Polytopes, CUP (1974). Harold Scott MacDonald "Donald" Coxeter CC ( February 9, 1907 – March 31, 2003) is regarded as one of the great
• Cromwell, P. ;Polyhedra, CUP hbk (1997), pbk. (1999).
• Grünbaum, B.; Polyhedra with Hollow Faces, Proc of NATO-ASI Conference on Polytopes . Branko Grünbaum (born 1929 is a Croatian born Mathematician and a professor Emeritus at the University of Washington in Seattle. . . etc. (Toronto 1993), ed T. Bisztriczky et al, Kluwer Academic (1994) pp. 43-70.
• Grünbaum, B. ; Are your polyhedra the same as my polyhedra? Discrete and comput. geom: the Goodman-Pollack festschrift, ed. Aronov et al. Springer (2003) pp. 461-488. (pdf)
• Pearce, P. ; Structure in nature is a strategy for design, MIT (1978)

## Books on polyhedra

### Introductory books, also suitable for school use

• Cromwell, P. ; Polyhedra, CUP hbk (1997), pbk. (1999).
• Cundy, H. M. & Rollett, A. P. ; Mathematical models, 1st Edn. hbk OUP (1951), 2nd Edn. hbk OUP (1961), 3rd Edn. pbk Tarquin (1981).
• Holden; Shapes, space and symmetry, (1971), Dover pbk (1991).
• Pearce, P and Pearce, S: Polyhedra primer, Van Nost. Reinhold (May 1979), ISBN-10: 0442264968, ISBN-13: 978-0442264963.
• Tarquin publications: books of cut-out and make card models.
• Senechal, M. & Fleck, G. ; Shaping Space a Polyhedral Approch, Birhauser (1988), ISBN-10: 0817633510
• Wenninger, M. ; Polyhedron models for the classroom, pbk (1974)
• Wenninger, M. ; Polyhedron models, CUP hbk (1971), pbk (1974).
• Wenninger, M. ; Spherical models, CUP.
• Wenninger, M. ; Dual models, CUP.

• Coxeter, H.S.M. DuVal, Flather & Petrie; The fifty-nine icosahedra, 3rd Edn. Harold Scott MacDonald "Donald" Coxeter CC ( February 9, 1907 – March 31, 2003) is regarded as one of the great Tarquin.
• Coxeter, H.S.M. Twelve geometric essays. Harold Scott MacDonald "Donald" Coxeter CC ( February 9, 1907 – March 31, 2003) is regarded as one of the great Republished as The beauty of geometry, Dover.
• Thompson, Sir D'A. W. On growth and form, (1943). (not sure if this is the right category for this one, I haven't read it).

### Design and architecture bias

• Critchlow, K. ; Order in space.
• Pearce, P. ; Structure in nature is a strategy for design, MIT (1978)
• Williams, R. ; The geometrical foundation of natural structure, Dover (1979).

• Coxeter, H.S.M.; Regular Polytopes 3rd Edn. Harold Scott MacDonald "Donald" Coxeter CC ( February 9, 1907 – March 31, 2003) is regarded as one of the great Dover (1973).
• Coxeter, H.S.M.; Regular complex polytopes, CUP (1974). Harold Scott MacDonald "Donald" Coxeter CC ( February 9, 1907 – March 31, 2003) is regarded as one of the great
• Lakatos, Imre; Proofs and Refutations, Cambridge University Press (1976) - discussion of proof of Euler characteristic
• Several more to add here. Imre Lakatos ( November 9, 1922 – February 2, 1974) was a Philosopher of mathematics and science, Proofs and Refutations is a book by the Philosopher Imre Lakatos expounding his view ofthe progress of Mathematics.

### Historical books

• Brückner, Vielecke und Vielflache (Polygons and polyhedra), (1900).
• Fejes Toth, L. ;