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Small stellated dodecahedron
In elementary geometry, a polyhedron (plural polyhedra or polyhedrons) is a solid in three dimensions with flat faces, straight edges and sharp corners or vertices. The word polyhedron comes from the Classical Greek πολύεδρον, as poly- (stem of πολύς, "many") + -hedron (form of ἕδρα, "base" or "seat").
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.
A polyhedron is a 3-dimensional example of the more general polytope in any number of dimensions.
In elementary geometry, the polygonal faces are regions of planes, meeting in pairs along the edges which are straight-line segments, and with the edges meeting in vertex points. Treating a polyhedron as a solid bounded by flat faces and straight edges is not very precise, for example it is difficult to reconcile with star polyhedra. 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' ...." Many definitions of "polyhedron" have been given within particular contexts, some more rigorous than others. For example definitions based on the idea of a bounding surface rather than a solid are common. However such definitions are not always compatible in other mathematical contexts.
One modern approach treats a geometric polyhedron as an injection into real space, a realisation, of some abstract polyhedron. Any such polyhedron can be built up from different kinds of element or entity, each associated with a different number of dimensions:
Different approaches - and definitions - may require different realisations. Sometimes the interior volume is considered to be part of the polyhedron, sometimes only the surface is considered, and occasionally only the skeleton of edges or even just the set of vertices.
In such elementary geometric and set-based definitions, a polyhedron is typically understood as a three-dimensional example of the more general polytope in any number of dimensions. For example a polygon has a two-dimensional body and no faces, while a 4-polytope has a four-dimensional body and an additional set of three-dimensional "cells".
In other mathematical disciplines, the term "polyhedron" may be used to refer to a variety of specialised constructs, some geometric and others purely algebraic or abstract. The term is sometimes used in such contexts not for a kind of polytope but for something different.
A defining characteristic of almost all kinds of polyhedra is that just two faces join along any common edge. Likewise any edge meets just two vertices, one at each end. These two characteristics are dual to each other and they ensure that the polyhedral surface is continuously connected and does not end abruptly or split off in different directions.
Every simple (non-self-intersecting) polyhedron has at least two faces with the same number of edges.:p.224,#105
Polyhedra may be classified and are often named according to the number of faces. The naming system is based on Classical Greek, for example tetrahedron (4), pentahedron (5), hexahedron (6), triacontahedron (30), and so on.
The topological class of a polyhedron is defined by its Euler characteristic and orientability.
From this perspective, any polyhedron may be classed as certain kind of topological manifold. For example a convex or simply-connected polyhedron is a topological sphere or ball (depending on whether its body is taken into account).
The Euler characteristic χ relates the number of vertices V, edges E, and faces F of a polyhedron:
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, such as the tetrahemihexahedron, this is not possible and the surface is effectively one-sided. The polyhedron is said to be non-orientable.
All polyhedra with odd-numbered Euler characteristic are non-orientable. A given figure with even χ < 2 may or may not be orientable. For example the simple toroid and the Klein bottle both have χ = 0, with the first being orientable and the other not.
For every polyhedron there exists a dual polyhedron having:
The dual of a convex polyhedron and of many other polyhedra can be obtained by the process of polar reciprocation.
Dual polyhedra exist in pairs. The dual of a dual is just the original polyhedron again. Some polyhedra are self-dual, meaning that the dual of the polyhedron is congruent to the original polyhedron.
For every vertex one can define a vertex figure, which describes the local structure of the figure around the vertex. Precise definitions vary, but a vertex figure can be thought of as the polygon exposed where a slice through the polyhedron cuts off a corner. If the vertex figure is a regular polygon, then the vertex itself is said to be regular.
Any regular polyhedron can be divided up into congruent pyramids, with each pyramid having a face of the polyhedron as its base and the centre of the polyhedron as its apex. The height of a pyramid is equal to the inradius of the polyhedron. If the area of a face is and the in-radius is then the volume of the pyramid is one-third of the base times the height, or . For a regular polyhedron with faces, its volume is then simply
For instance, a cube with edges of length has six faces, each face being a square with area . The inradius from the center of the face to the center of the cube is . Then the volume is given by
the usual formula for the volume of a cube.
The volume of any 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 is equal to a surface integral of :
When Ω is the region enclosed by a polyhedron, since the faces of a polyhedron are planar and have piecewise constant normal vectors, this simplifies to
where is the ith face's barycenter, is its normal vector, and is its area. Once the faces are decomposed in a set of non-overlapping triangles with surface normals pointing away from the volume, the volume is one sixth of the sum over the triple products of the nine Cartesian vertex coordinates of the triangles.
Since it may be difficult to enumerate the faces, volume computation may be challenging, and hence there exist specialized algorithms to determine the volume (many of these generalize to convex polytopes in higher dimensions).
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. A convex polyhedron is sometimes defined as a convex set of points in space, the intersection of a set of half-spaces, or the convex hull of a set of points. However many such definitions cannot easily be extended to include self-intersecting figures such as star polyhedra.
Many of the most studied polyhedra are highly symmetrical.
A symmetrical polyhedron can be rotated and superimposed on its original position such that its faces and so on have changed position. All the elements which can be superimposed on each other in this way are said to lie in a given "symmetry orbit". For example all the faces of a cube lie in one orbit, while all the edges lie in another. If all the elements of a given dimension, say all the faces, lie in the same orbit, the figure is said to be "transitive" on that orbit. For example a cube has one kind of face so it face-transitive, while a truncated cube has two kinds of face and is not.
Such polyhedra can be distorted so that 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.
There are several types of highly symmetric polyhedron, classified by which kind of element - faces, edges and/or vertices - belong to a single symmetry orbit:
A polyhedron can belong to the same overall symmetry group as one of higher symmetry, but will be of lower symmetry if it has several groups of elements in different symmetry orbits. For example the truncated cube has its triangles and octagons in different orbits.
Regular polyhedra are the most highly symmetrical. Altogether there are nine regular polyhedra.
The five convex examples have been known since antiquity and are called the Platonic solids. Plato did not discover them, but he was the first to give instructions on how to construct them all. These are the triangular pyramid or tetrahedron, cube (regular hexahedron), octahedron, dodecahedron and icosahedron:
There are also four regular star polyhedra, known as the Kepler-Poinsot polyhedra after their discoverers.
The dual of a regular polyhedron is also regular.
The uniform duals have irregular faces but are face-transitive and every vertex figure is a regular polygon. A uniform polyhedron has the same symmetry orbits as its dual, with the faces and vertices simply swapped over. The duals of the convex Archimedean polyhedra are sometimes called the Catalan solids.
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||Kepler-Poinsot 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|
Symmetrical pyramids include some of the most time-honoured and famous of all polyhedra, such as the four-sided Egyptian pyramids.
The dual of a noble polyhedron is also noble.
Besides the regular and uniform polyhedra, there are some other classes which have regular faces but lower overall symmetry.
Convex polyhedra where every face is the same kind of regular polygon may be found among three families:
Polyhedra with congruent regular faces of six or more sides are all non-convex, because the vertex of three regular hexagons defines a plane.
The total number of convex polyhedra with equal regular faces is thus ten, comprising the five Platonic solids and the five non-uniform deltahedra.
There are infinitely many non-convex examples. Infinite sponge-like examples called infinite skew polyhedra exist in some of these families.
Norman Johnson sought which convex non-uniform polyhedra had regular faces, although not necessarily all alike. 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.
Stellation of a polyhedron is the process of extending the faces (within their planes) so that they meet to form a new polyhedron.
It is the exact reciprocal to the process of facetting which is the process of removing parts of a polyhedron without creating any new vertices.
A spacefilling polyhedron packs with copies of itself to fill space. Such a close-packing or spacefilling is often called a tessellation of space or a honeycomb. Some honeycombs involve more than one kind of polyhedron.
A polyhedral compound is made of two or more polyhedra sharing a common centre.
Symmetrical compounds often share the same vertices as other well-known polyhedra and may often also be formed by stellation. Some are listed in the list of Wenninger polyhedron models.
An orthogonal polyhedron is one all of whose faces meet at right angles, and all of whose edges are parallel to axes of a Cartesian coordinate system. Aside from a rectangular box, orthogonal polyhedra are nonconvex. They are the 3D analogs of 2D orthogonal polygons, also known as rectilinear polygons. Orthogonal polyhedra are used in computational geometry, where their constrained structure has enabled advances on problems unsolved for arbitrary polyhedra, for example, unfolding the surface of a polyhedron to a polygonal net.
The name 'polyhedron' has come to be used for a variety of objects having similar structural properties to traditional polyhedra.
A classical polyhedral surface comprises finite, bounded plane regions, joined in pairs along edges. If such a surface extends indefinitely it is called an apeirohedron. Examples include:
A complex polyhedron is one which is constructed in complex Hilbert 3-space. This space has six dimensions: three real ones corresponding to ordinary space, with each accompanied by an imaginary dimension. A complex polyhedron is mathematically more closely related to configurations than to real polyhedra.
Some fields of study allow polyhedra to have curved faces and edges.
The surface of a sphere may be divided by line segments into bounded regions, to form a spherical polyhedron. Much of the theory of symmetrical polyhedra is most conveniently derived in this way.
Spherical polyhedra have a long and respectable history:
Two important types are:
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.
From the latter half of the twentieth century, various mathematical constructs have been found to have properties also present in traditional polyhedra. Rather than confining the term "polyhedron" to describe a three-dimensional polytope, it has been adopted to describe various related but distinct kinds of structure.
A polyhedron has been defined as a set of points in real affine (or Euclidean) space of any dimension 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 point set that is the intersection of a finite number of half-spaces. Unlike an elementary polyhedron, 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 in this way provides a geometric perspective for problems in linear programming.
Many traditional polyhedral forms are general polyhedra. Other examples include:
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 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.
An abstract polytope is a partially ordered set (poset) of elements whose partial ordering obeys certain rules of incidence (connectivity) and ranking. The elements of the set correspond to the vertices, edges, faces and so on of the polytope: vertices have rank 0, edges rank 1, etc. with the partially ordered ranking corresponding to the dimensionality of the geometric elements. The empty set, required by set theory, has a rank of −1 and is sometimes said to correspond to the null polytope. An abstract polyhedron is an abstract polytope having the following ranking:
Any geometric polyhedron is then said to be a "realization" in real space of the abstract poset.
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).
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.
Cubical gaming dice in China have been dated back as early as 600 B.C.
By 236 AD, 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.
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.
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 pupil depicts a glass rhombicuboctahedron half-filled 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.
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 star-like forms of increasing complexity.
Johannes Kepler (1571 - 1630) used star polygons, typically pentagrams, to build star polyhedra. Some of these figures may have been discovered before Kepler's time, but he was the first to recognize that they could be considered "regular" if one removed the restriction that regular polytopes must 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 Kepler-Poinsot polyhedra.
The Kepler-Poinsot 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.
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 set of "59". More have been discovered since, and the story is not yet ended.
For natural occurrences of regular polyhedra, see Regular polyhedron: Regular polyhedra in nature.
Irregular polyhedra appear in nature as crystals.
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|Fundamental convex regular and uniform polytopes in dimensions 2–10|
|Family||An||Bn||I2(p) / Dn||E6 / E7 / E8 / F4 / G2||Hn|
|Uniform polyhedron||Tetrahedron||Octahedron • Cube||Demicube||Dodecahedron • Icosahedron|
|Uniform 4-polytope||5-cell||16-cell • Tesseract||Demitesseract||24-cell||120-cell • 600-cell|
|Uniform 5-polytope||5-simplex||5-orthoplex • 5-cube||5-demicube|
|Uniform 6-polytope||6-simplex||6-orthoplex • 6-cube||6-demicube||122 • 221|
|Uniform 7-polytope||7-simplex||7-orthoplex • 7-cube||7-demicube||132 • 231 • 321|
|Uniform 8-polytope||8-simplex||8-orthoplex • 8-cube||8-demicube||142 • 241 • 421|
|Uniform 9-polytope||9-simplex||9-orthoplex • 9-cube||9-demicube|
|Uniform 10-polytope||10-simplex||10-orthoplex • 10-cube||10-demicube|
|Uniform n-polytope||n-simplex||n-orthoplex • n-cube||n-demicube||1k2 • 2k1 • k21||n-pentagonal polytope|
|Topics: Polytope families • Regular polytope • List of regular polytopes|