 Polyhedron

"Polyhedra" redirects here. For the relational database system, see Polyhedra DBMS.For the game magazine, see Polyhedron (magazine). For the scientific journal, see Polyhedron (journal).
Some Polyhedra
Dodecahedron
(Regular polyhedron)
Small stellated dodecahedron
(Regular star)
Icosidodecahedron
(Uniform/Quasiregular)
Great cubicuboctahedron
(Uniform star)
Rhombic triacontahedron
(Uniform/Quasiregular dual)
Elongated pentagonal cupola
(Convex regularfaced)
Octagonal prism
(Uniform prism)
Square antiprism
(Uniform antiprism)In elementary geometry a polyhedron (plural polyhedra or polyhedrons) is a geometric solid in three dimensions with flat faces and straight edges. The word polyhedron comes from the Classical Greek 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 polytope 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. Grünbaum (1994, p. 43) observed, "The Original Sin in the theory of polyhedra goes back to Euclid, and through Kepler, Poinsot, Cauchy and many others ... [in that] at each stage ... the writers failed to define what are the 'polyhedra' ...." 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 is a polygon 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 joins one vertex to another and one face to another, and is usually a line segment. The edges together make up the polyhedral skeleton.
 0 dimensions: A vertex (plural vertices) is a corner point.
 1 dimension: The null polytope is a kind of nonentity 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.
Characteristics
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 characteristic 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 connected polyhedron whose faces are also simply connected and whose boundary is a manifold, χ = 2. For a detailed discussion, see Proofs and Refutations by Imre Lakatos.
Orientability
Some polyhedra, such as all convex polyhedra, have two distinct sides to their surface, for example one side can consistently be coloured black and the other white. We say that the figure is orientable.
But for some polyhedra this is not possible, and the figure is said to be nonorientable. All polyhedra with oddnumbered Euler characteristic are nonorientable. A given figure with even χ < 2 may or may not be orientable.
Vertex figure
For every vertex one can define a vertex figure, which describes the local structure of the figure around the vertex. If the vertex figure is a regular polygon, then the vertex itself is said to be regular.
Duality
For every polyhedron there exists a dual polyhedron 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 orientable polyhedron can be calculated using 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 i'th face, is any point on the face, is the normal vector, and A_{i} 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 tetrahedron (4), pentahedron (5), hexahedron (6), heptahedron (7), triacontahedron (30), and so on.
Often this is qualified by a description of the kinds of faces present, for example the Rhombic dodecahedron vs. the Pentagonal dodecahedron.
Other common names indicate that some operation has been performed on a simpler polyhedron, for example the truncated cube looks like a cube with its corners cut off, and has 14 faces (so it is also an example of a tetrakaidecahedron).
Some special polyhedra have grown their own names over the years, such as Miller's monster or the Szilassi polyhedron.
Traditional polyhedra
In geometry, a polyhedron is traditionally a threedimensional shape that is made up of a finite number of polygonal faces which are parts of planes; the faces meet in pairs along edges which are straightline segments, and the edges meet in points called vertices. Cubes, prisms and pyramids are examples of polyhedra. The polyhedron surrounds a bounded volume in threedimensional space; sometimes this interior volume is considered to be part of the polyhedron, sometimes only the surface is considered, and occasionally only the skeleton of edges.
A polyhedron is said to be convex if its surface (comprising its faces, edges and vertices) does not intersect itself and the line segment joining any two points of the polyhedron is contained in the interior or surface.
Symmetrical polyhedra
Many of the most studied polyhedra are highly symmetrical.
Of course it is easy to distort such polyhedra so they are no longer symmetrical. But where a polyhedral name is given, such as icosidodecahedron, the most symmetrical geometry is almost always implied, unless otherwise stated.
Some of the most common names in particular are often used with "regular" in front or implied because for each there are different types which have little in common except for having the same number of faces. These are the triangular pyramid or tetrahedron, cube or hexahedron, octahedron, dodecahedron and icosahedron:
Polyhedra of the highest symmetries have all of some kind of element  faces, edges and/or vertices, within a single symmetry orbit. There are various classes of such polyhedra:
 Isogonal or Vertextransitive 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.
 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 if it is vertextransitive, edgetransitive and facetransitive (this implies that every face is the same regular polygon; it also implies that every vertex is regular).
 Quasiregular 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 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 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 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
Main article: Uniform polyhedronUniform polyhedra are vertextransitive and every face is a regular polygon. They may be regular, quasiregular, or semiregular, and may be convex or starry.
The uniform duals are facetransitive and every vertex figure is a regular polygon.
Facetransitivity of a polyhedron corresponds to vertextransitivity of the dual and conversely, and edgetransitivity of a polyhedron corresponds to edgetransitivity of the dual. The dual of a regular polyhedron is also regular. The dual of a nonregular uniform polyhedron (called a Catalan solid 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 Regular Platonic solids KeplerPoinsot polyhedra Quasiregular Archimedean solids Catalan solids (no special name) (no special name) Semiregular (no special name) (no special name) Prisms Dipyramids Star Prisms Star Dipyramids Antiprisms Trapezohedra Star Antiprisms Star Trapezohedra Noble polyhedra
Main article: Noble polyhedronA noble polyhedron is both isohedral (equalfaced) and isogonal (equalcornered). Besides the regular polyhedra, there are many other examples.
The dual of a noble polyhedron is also noble.
Symmetry groups
The polyhedral symmetry groups (using Schoenflies notation) are all point groups and include:
 T  chiral tetrahedral symmetry; the rotation group for a regular tetrahedron; order 12.
 T_{d}  full tetrahedral symmetry; the symmetry group for a regular tetrahedron; order 24.
 T_{h}  pyritohedral symmetry; order 24. The symmetry of a pyritohedron.
 O  chiral octahedral symmetry;the rotation group of the cube and octahedron; order 24.
 O_{h}  full octahedral symmetry; the symmetry group of the cube and octahedron; order 48.
 I  chiral icosahedral symmetry; the rotation group of the icosahedron and the dodecahedron; order 60.
 I_{h}  full icosahedral symmetry; the symmetry group of the icosahedron and the dodecahedron; order 120.
 C_{nv}  nfold pyramidal symmetry
 D_{nh}  nfold prismatic symmetry
 D_{nv}  nfold antiprismatic symmetry.
Those with chiral symmetry do not have reflection symmetry and hence have two enantiomorphous forms which are reflections of each other. The snub Archimedean polyhedra have this property.
Other polyhedra with regular faces
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 polyhedron for exceptions with zigzagging vertex figures.)
Deltahedra
A deltahedron (plural deltahedra) is a polyhedron whose faces are all equilateral triangles. There are infinitely many deltahedra, but only eight of these are convex:
 3 regular convex polyhedra (3 of the Platonic solids)
 5 nonuniform convex polyhedra (5 of the Johnson solids)
Johnson solids
Main article: Johnson solidNorman 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 Zalgaller proved in 1969 that the list of these Johnson solids was complete.
Other important families of polyhedra
Pyramids
Main article: Pyramid (geometry)Pyramids include some of the most timehonoured and famous of all polyhedra.
Stellations and facettings
Main article: StellationStellation of a polyhedron is the process of extending the faces (within their planes) so that they meet to form a new polyhedron.
It is the exact reciprocal to the process of facetting which is the process of removing parts of a polyhedron without creating any new vertices.
Zonohedra
Main article: ZonohedronA zonohedron is a convex polyhedron where every face is a polygon with inversion symmetry or, equivalently, symmetry under rotations through 180°.
Toroidal polyhedra
Main article: Toroidal polyhedronA toroidal polyhedron is a polyhedron with an Euler characteristic of 0 or smaller, representing a torus surface.
Compounds
Main article: Polyhedral compoundPolyhedral 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 polygons. Orthogonal polyhedra are used in computational geometry, where their constrained structure has enabled advances on problems unsolved for arbitrary polyhedra, for example, unfolding the surface of a polyhedron to a polygonal net.
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.
 Spongelike structures called infinite skew polyhedra.
See also: Apeirogon  infinite regular polygon: {∞}
Complex polyhedra
A complex polyhedron is one which is constructed in complex Hilbert 3space. This space has six dimensions: three real ones corresponding to ordinary space, with each accompanied by an imaginary dimension. See for example Coxeter (1974).
Curved polyhedra
Some fields of study allow polyhedra to have curved faces and edges.
Spherical polyhedra
Main article: Spherical polyhedronThe 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 dihedra, 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 mathematics has defined a polyhedron as a set in real affine (or Euclidean) 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 halfspaces. It may be bounded or unbounded. In this meaning, a polytope 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 polytopes) in this way provides a geometric perspective for problems in Linear programming.
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 a Voronoi tessellation 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 interior of the convex hull of S, and otherwise (when c lies on 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 Vinci devised frame models of the regular solids, which he drew for Pacioli's book Divina Proportione. In modern times, Branko Grünbaum (1994) made a special study of this class of polyhedra, in which he developed an early idea of abstract polyhedra. He defined a face as a cyclically ordered set of vertices, and allowed faces to be skew as well as planar.
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 polytopes and that are attached to each other in a regular way.
Such a figure is called simplicial if each of its regions is a simplex, i.e. in an ndimensional space each region has n+1 vertices. The dual of a simplicial polytope is called simple. Similarly, a widely studied class of polytopes (polyhedra) is that of cubical polyhedra, when the basic building block is an ndimensional cube.
Abstract polyhedra
An abstract polyhedron is a partially ordered set (poset) of elements whose partial ordering obeys certain rules. Theories differ in detail, but essentially the elements of the set correspond to the body, faces, edges and vertices of the polyhedron. The empty set corresponds to the null polytope, or nullitope, which has a dimensionality of −1. These posets belong to the larger family of abstract polytopes in any number of dimensions.
Polyhedra as graphs
Any polyhedron gives rise to a graph, or skeleton, with corresponding vertices and edges. Thus graph terminology and properties can be applied to polyhedra. For example:
 Due to Steinitz theorem convex polyhedra are in onetoone correspondence with 3connected planar graphs.
 The tetrahedron gives rise to a complete graph (K_{4}). 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 nonadjacent vertices have four.
 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.
History
Prehistory
Stones carved in shapes showing the symmetries of various polyhedra have been found in Scotland 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 Museum at Oxford University. It is impossible to know why these objects were made, or how the sculptor gained the inspiration for them.
Other polyhedra have of course made their mark in architecture—cubes and cuboids being obvious examples, with the earliest foursided pyramids of ancient Egypt also dating from the Stone Age.
The Etruscans preceded the Greeks in their awareness of at least some of the regular polyhedra, as evidenced by the discovery near Padua (in Northern Italy) in the late 19th century of a dodecahedron made of soapstone, and dating back more than 2,500 years (Lindemann, 1987). Pyritohedric crystals are found in northern Italy^{[citation needed]}.
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 earlier Greeks were interested primarily in the convex regular polyhedra, which came to be known as the Platonic solids. Pythagoras knew at least three of them, and Theaetetus (circa 417 B. C.) described all five. Eventually, Euclid described their construction in his Elements. Later, Archimedes expanded his study to the convex uniform polyhedra which now bear his name. His original work is lost and his solids come down to us through Pappus.
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 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 Renaissance. Artists constructed skeletal polyhedra, depicting them from life as a part of their investigations into perspective. 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 rhombicuboctahedron 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 Renaissance 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 Kepler realised that star polygons, typically pentagrams, 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 Poinsot realised that star vertex figures (circuits around each corner) can also be used, and discovered the remaining two regular star polyhedra. Cauchy proved Poinsot's list complete, and Cayley gave them their accepted English names: (Kepler's) the small stellated dodecahedron and great stellated dodecahedron, and (Poinsot's) the great icosahedron and great dodecahedron. Collectively they are called the KeplerPoinsot polyhedra.
The KeplerPoinsot polyhedra may be constructed from the Platonic solids by a process called stellation. Most stellations are not regular. The study of stellations of the Platonic solids was given a big push by H. S. M. Coxeter and others in 1938, with the now famous paper The 59 icosahedra. This work has recently been republished (Coxeter, 1999).
The reciprocal process to stellation is called facetting (or faceting). Every stellation of one polytope is dual, or reciprocal, to some facetting of the dual polytope. The regular star polyhedra can also be obtained by facetting the Platonic solids. Bridge 1974 listed the simpler facettings of the dodecahedron, and reciprocated them to discover a stellation of the icosahedron that was missing from the famous "59". More have been discovered since, and the story is not yet ended.
See also:
Polyhedra in nature
For natural occurrences of regular polyhedra, see Regular polyhedron: Regular polyhedra in nature.
Irregular polyhedra appear in nature as crystals.
See also
 Antiprism
 Archimedean solid
 Bipyramid
 Conway polyhedron notation (a notation for describing construction of polyhedra)
 Defect
 Deltahedron
 Deltohedron
 Escher
 Flexible polyhedra
 Johnson solid
 KeplerPoinsot polyhedra
 List of polyhedral images
 Nearmiss Johnson solid
 Net (polyhedron)
 Platonic solid
 Polychoron (4 dimensional analogues to polyhedra)
 Polyhedral compound
 Polyhedron models
 Prism
 Semiregular polyhedron
 Schlegel diagram
 Spidron
 Tessellation
 Trapezohedron
 Uniform polyhedron
 Waterman polyhedron
 Zonohedron
 Extension of a polyhedron
References
 Coxeter, H.S.M.; Regular complex Polytopes, CUP (1974).
 Cromwell, P.;Polyhedra, CUP hbk (1997), pbk. (1999).
 Grünbaum, B.; Polyhedra with Hollow Faces, Proc of NATOASI Conference on Polytopes ... 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 GoodmanPollack 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
Main article: List of books about polyhedraExternal links
General theory
 Weisstein, Eric W., "Polyhedron" from MathWorld.
 Polyhedra Pages
 uniform solution for uniform polyhedra by Dr. Zvi Har'El
 Symmetry, Crystals and Polyhedra
 Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons
Lists and databases of polyhedra
 The Uniform Polyhedra
 Virtual Reality Polyhedra  The Encyclopedia of Polyhedra
 Interactive 3D polyhedra in Java
 Electronic Geometry Models  Contains a peer reviewed selection of polyhedra with unusual properties.
 Origami Polyhedra  Models made with Modular Origami
 Polyhedra Collection  Various virtual and physical polyhedra models.
 Polyhedra plaited with paper strips  Polyhedra models constructed without use of glue.
 Rotatable polyhedron models (uses Javascript)
 Rotatable selfintersecting polyhedra (uses Java)
 Polyhedron Models  Virtual polyhedra
 Java Applets for Visualizing Polyhedra  Systematically explicit formulae given!
Software
 A Plethora of Polyhedra – An interactive and free collection of polyhedra in Java. Features includes nets, planar sections, duals, truncations and stellations of more than 300 polyhedra.
 Stella: Polyhedron Navigator  Software for exploring polyhedra and printing nets for their physical construction. Includes uniform polyhedra, stellations, compounds, Johnson solids, etc.
 World of Polyhedra  Comprehensive polyhedra in Flash applet, showing vertices and edges (but not shaded faces)
 Hyperspace Star Polytope Slicer  Explorer java applet, includes a variety of 3d viewer options.
 HEDRON Polyhedron modelling software
 Uniform Polyhedra Java Applets with sources
Resources for making models, and models for sale
 Making Polyhedra
 Paper Models of Polyhedra Free nets of polyhedra
 Paper Models of Uniform (and other) Polyhedra
 Kits and instructions for making models of polyhedra out of bamboo
 Polyhedra software, diecast models, & posters
 Simple instructions for building over 30 paper polyhedra
 Kits to print out, cut, and fold into various polyhedra
 openSCAD. Free crossplatform software for programmers. Polyhedra are just one of the things you can model. The openSCAD User Manual is also available.
Miscellaneous
 PictureSpice  A site that lets you make polyhedra with your own uploaded pictures.
 Lattice Polyhedra
Polyhedron navigator Platonic solids (regular) Archimedean solids
(Semiregular/Uniform)Catalan solids
(Dual semiregular)triakis tetrahedron · rhombic dodecahedron · triakis octahedron · tetrakis cube · deltoidal icositetrahedron · disdyakis dodecahedron · pentagonal icositetrahedron · rhombic triacontahedron · triakis icosahedron · pentakis dodecahedron · deltoidal hexecontahedron · disdyakis triacontahedron · pentagonal hexecontahedronDihedral regular Dihedral uniform Duals of dihedral uniform Dihedral others Degenerate polyhedra are in italics.Fundamental convex regular and uniform polytopes in dimensions 2–10 Family A_{n} BC_{n} D_{n} E_{6} / E_{7} / E_{8} / F_{4} / G_{2} H_{n} Regular polygon Triangle Square Hexagon Pentagon Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron Uniform polychoron 5cell 16cell • Tesseract Demitesseract 24cell 120cell • 600cell Uniform 5polytope 5simplex 5orthoplex • 5cube 5demicube Uniform 6polytope 6simplex 6orthoplex • 6cube 6demicube 1_{22} • 2_{21} Uniform 7polytope 7simplex 7orthoplex • 7cube 7demicube 1_{32} • 2_{31} • 3_{21} Uniform 8polytope 8simplex 8orthoplex • 8cube 8demicube 1_{42} • 2_{41} • 4_{21} Uniform 9polytope 9simplex 9orthoplex • 9cube 9demicube Uniform 10polytope 10simplex 10orthoplex • 10cube 10demicube npolytopes nsimplex northoplex • ncube ndemicube 1_{k2} • 2_{k1} • k_{21} pentagonal polytope Topics: Polytope families • Regular polytope • List of regular polytopes
Categories:
Wikimedia Foundation. 2010.