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In elementary
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -dimensional polytope or -polytope. For example, a two-dimensional
polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two ...
is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In this context, "flat sides" means that the sides of a -polytope consist of -polytopes that may have -polytopes in common. Some theories further generalize the idea to include such objects as unbounded apeirotopes and
tessellation A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of ...
s, decompositions or tilings of curved
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s including
spherical polyhedra In geometry, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. Much of the theory of symmetrical polyhedra is most co ...
, and set-theoretic abstract polytopes. Polytopes of more than three dimensions were first discovered by Ludwig Schläfli before 1853, who called such a figure a polyschem. The
German German(s) may refer to: * Germany (of or related to) **Germania (historical use) * Germans, citizens of Germany, people of German ancestry, or native speakers of the German language ** For citizens of Germany, see also German nationality law **Ge ...
term ''polytop'' was coined by the mathematician
Reinhold Hoppe Ernst Reinhold Eduard Hoppe (November 18, 1816 – May 7, 1900) was a German mathematician who worked as a professor at the University of Berlin. Education and career Hoppe was a student of Johann August Grunert at the University of Greifswald, gr ...
, and was introduced to English mathematicians as ''polytope'' by
Alicia Boole Stott Alicia Boole Stott (8 June 1860 – 17 December 1940) was an Irish mathematician. Despite never holding an academic position, she made a number of valuable contributions to the field, receiving an honorary doctorate from the University of Gr ...
.


Approaches to definition

Nowadays, the term ''polytope'' is a broad term that covers a wide class of objects, and various definitions appear in the mathematical literature. Many of these definitions are not equivalent to each other, resulting in different overlapping sets of objects being called ''polytopes''. They represent different approaches to generalizing the
convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the w ...
s to include other objects with similar properties. The original approach broadly followed by Ludwig Schläfli,
Thorold Gosset John Herbert de Paz Thorold Gosset (16 October 1869 – December 1962) was an English lawyer and an amateur mathematician. In mathematics, he is noted for discovering and classifying the semiregular polytopes in dimensions four and higher, ...
and others begins with the extension by analogy into four or more dimensions, of the idea of a polygon and polyhedron respectively in two and three dimensions.Coxeter (1973) Attempts to generalise the
Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological spac ...
of polyhedra to higher-dimensional polytopes led to the development of
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
and the treatment of a decomposition or CW-complex as analogous to a polytope. In this approach, a polytope may be regarded as a
tessellation A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of ...
or decomposition of some given
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
. An example of this approach defines a polytope as a set of points that admits a simplicial decomposition. In this definition, a polytope is the union of finitely many simplices, with the additional property that, for any two simplices that have a nonempty intersection, their intersection is a vertex, edge, or higher dimensional face of the two.Grünbaum (2003) However this definition does not allow star polytopes with interior structures, and so is restricted to certain areas of mathematics. The discovery of
star polyhedra In geometry, a star polyhedron is a polyhedron which has some repetitive quality of nonconvexity giving it a star-like visual quality. There are two general kinds of star polyhedron: *Polyhedra which self-intersect in a repetitive way. *Concave ...
and other unusual constructions led to the idea of a polyhedron as a bounding surface, ignoring its interior. In this light convex polytopes in ''p''-space are equivalent to tilings of the (''p''−1)-sphere, while others may be tilings of other elliptic, flat or toroidal (''p''−1)-surfaces – see
elliptic tiling In geometry, a (globally) projective polyhedron is a tessellation of the real projective plane. These are projective analogs of spherical polyhedra – tessellations of the sphere – and toroidal polyhedra – tessellations of the toroids. Proje ...
and toroidal polyhedron. A
polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all o ...
is understood as a surface whose faces are
polygons In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two ...
, a 4-polytope as a hypersurface whose facets (
cells Cell most often refers to: * Cell (biology), the functional basic unit of life Cell may also refer to: Locations * Monastic cell, a small room, hut, or cave in which a religious recluse lives, alternatively the small precursor of a monastery w ...
) are polyhedra, and so forth. The idea of constructing a higher polytope from those of lower dimension is also sometimes extended downwards in dimension, with an (
edge Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed ...
) seen as a 1-polytope bounded by a point pair, and a point or
vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet *Vertex (computer graphics), a data structure that describes the position ...
as a 0-polytope. This approach is used for example in the theory of abstract polytopes. In certain fields of mathematics, the terms "polytope" and "polyhedron" are used in a different sense: a ''polyhedron'' is the generic object in any dimension (referred to as ''polytope'' in this article) and ''polytope'' means a bounded polyhedron. This terminology is typically confined to polytopes and polyhedra that are
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...
. With this terminology, a convex polyhedron is the intersection of a finite number of halfspaces and is defined by its sides while a convex polytope is the
convex hull In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
of a finite number of points and is defined by its vertices. Polytopes in lower numbers of dimensions have standard names:


Elements

A polytope comprises elements of different dimensionality such as vertices, edges, faces, cells and so on. Terminology for these is not fully consistent across different authors. For example, some authors use ''face'' to refer to an (''n'' − 1)-dimensional element while others use ''face'' to denote a 2-face specifically. Authors may use ''j''-face or ''j''-facet to indicate an element of ''j'' dimensions. Some use ''edge'' to refer to a ridge, while
H. S. M. Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...
uses ''cell'' to denote an (''n'' − 1)-dimensional element. The terms adopted in this article are given in the table below: An ''n''-dimensional polytope is bounded by a number of (''n'' − 1)-dimensional '' facets''. These facets are themselves polytopes, whose facets are (''n'' − 2)-dimensional '' ridges'' of the original polytope. Every ridge arises as the intersection of two facets (but the intersection of two facets need not be a ridge). Ridges are once again polytopes whose facets give rise to (''n'' − 3)-dimensional boundaries of the original polytope, and so on. These bounding sub-polytopes may be referred to as faces, or specifically ''j''-dimensional faces or ''j''-faces. A 0-dimensional face is called a ''vertex'', and consists of a single point. A 1-dimensional face is called an ''edge'', and consists of a line segment. A 2-dimensional face consists of a
polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two ...
, and a 3-dimensional face, sometimes called a '' cell'', consists of a
polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all o ...
.


Important classes of polytopes


Convex polytopes

A polytope may be ''convex''. The convex polytopes are the simplest kind of polytopes, and form the basis for several different generalizations of the concept of polytopes. A convex polytope is sometimes defined as the intersection of a set of half-spaces. This definition allows a polytope to be neither bounded nor finite. Polytopes are defined in this way, e.g., in
linear programming Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linear programming is ...
. A polytope is ''bounded'' if there is a ball of finite radius that contains it. A polytope is said to be ''pointed'' if it contains at least one vertex. Every bounded nonempty polytope is pointed. An example of a non-pointed polytope is the set \. A polytope is ''finite'' if it is defined in terms of a finite number of objects, e.g., as an intersection of a finite number of half-planes. It is an
integral polytope In geometry and polyhedral combinatorics, an integral polytope is a convex polytope whose vertices all have integer Cartesian coordinates. That is, it is a polytope that equals the convex hull of its integer points. Integral polytopes are also c ...
if all of its vertices have integer coordinates. A certain class of convex polytopes are ''reflexive'' polytopes. An integral \mathcal is reflexive if for some integral matrix \mathbf, \mathcal = \, where \mathbf denotes a vector of all ones, and the inequality is component-wise. It follows from this definition that \mathcal is reflexive if and only if (t+1)\mathcal^\circ \cap \mathbb^d = t\mathcal \cap \mathbb^d for all t \in \mathbb_. In other words, a of \mathcal differs, in terms of integer lattice points, from a of \mathcal only by lattice points gained on the boundary. Equivalently, \mathcal is reflexive if and only if its dual polytope \mathcal^* is an integral polytope.


Regular polytopes

Regular polytopes have the highest degree of symmetry of all polytopes. The symmetry group of a regular polytope acts transitively on its flags; hence, the dual polytope of a regular polytope is also regular. There are three main classes of regular polytope which occur in any number of dimensions: * Simplices, including the
equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...
and the
regular tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all th ...
. *
Hypercube In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, p ...
s or measure polytopes, including the square and the
cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only ...
. * Orthoplexes or cross polytopes, including the square and
regular octahedron In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet a ...
. Dimensions two, three and four include regular figures which have fivefold symmetries and some of which are non-convex stars, and in two dimensions there are infinitely many regular polygons of ''n''-fold symmetry, both convex and (for ''n'' ≥ 5) star. But in higher dimensions there are no other regular polytopes. In three dimensions the convex
Platonic solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all e ...
s include the fivefold-symmetric dodecahedron and
icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...
, and there are also four star Kepler-Poinsot polyhedra with fivefold symmetry, bringing the total to nine regular polyhedra. In four dimensions the regular 4-polytopes include one additional convex solid with fourfold symmetry and two with fivefold symmetry. There are ten star Schläfli-Hess 4-polytopes, all with fivefold symmetry, giving in all sixteen regular 4-polytopes.


Star polytopes

A non-convex polytope may be self-intersecting; this class of polytopes include the star polytopes. Some regular polytopes are stars.


Properties


Euler characteristic

Since a (filled) convex polytope ''P'' in d dimensions is
contractible In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within th ...
to a point, the
Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological spac ...
\chi of its boundary ∂P is given by the alternating sum: :\chi = n_0 - n_1 + n_2 - \cdots \plusmn n_ = 1 + (-1)^, where n_j is the number of j-dimensional faces. This generalizes
Euler's formula for polyhedra In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space' ...
.


Internal angles

The Gram–Euler theorem similarly generalizes the alternating sum of internal angles \sum \varphi for convex polyhedra to higher-dimensional polytopes:M. A. Perles and G. C. Shephard. 1967. "Angle sums of convex polytopes". ''Math. Scandinavica'', Vol 21, No 2. March 1967. pp. 199–218. : \sum \varphi = (-1)^


Generalisations of a polytope


Infinite polytopes

Not all manifolds are finite. Where a polytope is understood as a tiling or decomposition of a manifold, this idea may be extended to infinite manifolds. plane tilings, space-filling ( honeycombs) and hyperbolic tilings are in this sense polytopes, and are sometimes called apeirotopes because they have infinitely many cells. Among these, there are regular forms including the regular skew polyhedra and the infinite series of tilings represented by the regular
apeirogon In geometry, an apeirogon () or infinite polygon is a generalized polygon with a countably infinite number of sides. Apeirogons are the two-dimensional case of infinite polytopes. In some literature, the term "apeirogon" may refer only to t ...
, square tiling, cubic honeycomb, and so on.


Abstract polytopes

The theory of abstract polytopes attempts to detach polytopes from the space containing them, considering their purely combinatorial properties. This allows the definition of the term to be extended to include objects for which it is difficult to define an intuitive underlying space, such as the 11-cell. An abstract polytope is a
partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
of elements or members, which obeys certain rules. It is a purely algebraic structure, and the theory was developed in order to avoid some of the issues which make it difficult to reconcile the various geometric classes within a consistent mathematical framework. A geometric polytope is said to be a realization in some real space of the associated abstract polytope.


Complex polytopes

Structures analogous to polytopes exist in complex
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s \Complex^n where ''n'' real dimensions are accompanied by ''n'' imaginary ones.
Regular complex polytope In geometry, a complex polytope is a generalization of a polytope in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one. A complex polytope may be understood as a collect ...
s are more appropriately treated as configurations.


Duality

Every ''n''-polytope has a dual structure, obtained by interchanging its vertices for facets, edges for ridges, and so on generally interchanging its (''j'' − 1)-dimensional elements for (''n'' − ''j'')-dimensional elements (for ''j'' = 1 to ''n'' − 1), while retaining the connectivity or incidence between elements. For an abstract polytope, this simply reverses the ordering of the set. This reversal is seen in the
Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to mo ...
s for regular polytopes, where the symbol for the dual polytope is simply the reverse of the original. For example, is dual to . In the case of a geometric polytope, some geometric rule for dualising is necessary, see for example the rules described for dual polyhedra. Depending on circumstance, the dual figure may or may not be another geometric polytope. If the dual is reversed, then the original polytope is recovered. Thus, polytopes exist in dual pairs.


Self-dual polytopes

If a polytope has the same number of vertices as facets, of edges as ridges, and so forth, and the same connectivities, then the dual figure will be similar to the original and the polytope is self-dual. Some common self-dual polytopes include: *Every regular ''n''-
simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
, in any number of dimensions, with
Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to mo ...
. These include the
equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...
,
regular tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all th ...
, and
5-cell In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, pentachoron, pentatope, pentahedroid, or tetrahedral pyramid. It ...
. *Every
hypercubic honeycomb In geometry, a hypercubic honeycomb is a family of regular honeycombs (tessellations) in -dimensional spaces with the Schläfli symbols and containing the symmetry of Coxeter group (or ) for . The tessellation is constructed from 4 - hypercu ...
, in any number of dimensions. These include the
apeirogon In geometry, an apeirogon () or infinite polygon is a generalized polygon with a countably infinite number of sides. Apeirogons are the two-dimensional case of infinite polytopes. In some literature, the term "apeirogon" may refer only to t ...
, square tiling and cubic honeycomb . *Numerous compact, paracompact and noncompact hyperbolic tilings, such as the icosahedral honeycomb , and order-5 pentagonal tiling . *In 2 dimensions, all regular polygons (regular 2-polytopes) *In 3 dimensions, the canonical polygonal pyramids and
elongated pyramid In geometry, the elongated pyramids are an infinite set of polyhedra, constructed by adjoining an pyramid to an prism. Along with the set of pyramids, these figures are topologically self-dual. There are three ''elongated pyramids'' that are ...
s, and tetrahedrally diminished dodecahedron. *In 4 dimensions, the 24-cell, with
Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to mo ...
. Also the great 120-cell and
grand stellated 120-cell In geometry, the grand stellated 120-cell or grand stellated polydodecahedron is a regular star 4-polytope with Schläfli symbol . It is one of 10 regular Schläfli-Hess polytopes. It is also one of two such polytopes that is self-dual. Relat ...
.


History

Polygons and polyhedra have been known since ancient times. An early hint of higher dimensions came in 1827 when August Ferdinand Möbius discovered that two mirror-image solids can be superimposed by rotating one of them through a fourth mathematical dimension. By the 1850s, a handful of other mathematicians such as
Arthur Cayley Arthur Cayley (; 16 August 1821 – 26 January 1895) was a prolific British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics. As a child, Cayley enjoyed solving complex maths problem ...
and
Hermann Grassmann Hermann Günther Grassmann (german: link=no, Graßmann, ; 15 April 1809 – 26 September 1877) was a German polymath known in his day as a linguist and now also as a mathematician. He was also a physicist, general scholar, and publisher. His mat ...
had also considered higher dimensions. Ludwig Schläfli was the first to consider analogues of polygons and polyhedra in these higher spaces. He described the six convex regular 4-polytopes in 1852 but his work was not published until 1901, six years after his death. By 1854,
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first ...
's '' Habilitationsschrift'' had firmly established the geometry of higher dimensions, and thus the concept of ''n''-dimensional polytopes was made acceptable. Schläfli's polytopes were rediscovered many times in the following decades, even during his lifetime. In 1882
Reinhold Hoppe Ernst Reinhold Eduard Hoppe (November 18, 1816 – May 7, 1900) was a German mathematician who worked as a professor at the University of Berlin. Education and career Hoppe was a student of Johann August Grunert at the University of Greifswald, gr ...
, writing in German, coined the word '' polytop'' to refer to this more general concept of polygons and polyhedra. In due course
Alicia Boole Stott Alicia Boole Stott (8 June 1860 – 17 December 1940) was an Irish mathematician. Despite never holding an academic position, she made a number of valuable contributions to the field, receiving an honorary doctorate from the University of Gr ...
, daughter of logician
George Boole George Boole (; 2 November 1815 – 8 December 1864) was a largely self-taught English mathematician, philosopher, and logician, most of whose short career was spent as the first professor of mathematics at Queen's College, Cork in ...
, introduced the anglicised ''polytope'' into the English language. In 1895,
Thorold Gosset John Herbert de Paz Thorold Gosset (16 October 1869 – December 1962) was an English lawyer and an amateur mathematician. In mathematics, he is noted for discovering and classifying the semiregular polytopes in dimensions four and higher, ...
not only rediscovered Schläfli's regular polytopes but also investigated the ideas of semiregular polytopes and space-filling
tessellation A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of ...
s in higher dimensions. Polytopes also began to be studied in non-Euclidean spaces such as hyperbolic space. An important milestone was reached in 1948 with
H. S. M. Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...
's book '' Regular Polytopes'', summarizing work to date and adding new findings of his own. Meanwhile, the French mathematician
Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "Th ...
had developed the topological idea of a polytope as the piecewise decomposition (e.g. CW-complex) of a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
.
Branko Grünbaum Branko Grünbaum ( he, ברנקו גרונבאום; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descentConvex Polytopes ''Convex Polytopes'' is a graduate-level mathematics textbook about convex polytopes, higher-dimensional generalizations of three-dimensional convex polyhedra. It was written by Branko Grünbaum, with contributions from Victor Klee, Micha Perl ...
'' in 1967. In 1952 Geoffrey Colin Shephard generalised the idea as
complex polytope In geometry, a complex polytope is a generalization of a polytope in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one. A complex polytope may be understood as a colle ...
s in complex space, where each real dimension has an imaginary one associated with it. Coxeter developed the theory further. The conceptual issues raised by complex polytopes, non-convexity, duality and other phenomena led Grünbaum and others to the more general study of abstract combinatorial properties relating vertices, edges, faces and so on. A related idea was that of incidence complexes, which studied the incidence or connection of the various elements with one another. These developments led eventually to the theory of abstract polytopes as partially ordered sets, or posets, of such elements.
Peter McMullen Peter McMullen (born 11 May 1942) is a British mathematician, a professor emeritus of mathematics at University College London. Education and career McMullen earned bachelor's and master's degrees from Trinity College, Cambridge, and studied at ...
and Egon Schulte published their book ''Abstract Regular Polytopes'' in 2002. Enumerating the
uniform polytope In geometry, a uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform Facet (mathematics), facets. The uniform polytopes in two dimensions are the regular polygons (the definition is different in 2 dimen ...
s, convex and nonconvex, in four or more dimensions remains an outstanding problem. The convex uniform 4-polytopes were fully enumerated by John Conway and Michael Guy using a computer in 1965; in higher dimensions this problem was still open as of 1997. The full enumeration for nonconvex uniform polytopes is not known in dimensions four and higher as of 2008. In modern times, polytopes and related concepts have found many important applications in fields as diverse as
computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...
, optimization,
search engine A search engine is a software system designed to carry out web searches. They search the World Wide Web in a systematic way for particular information specified in a textual web search query. The search results are generally presented in a ...
s,
cosmology Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe. The term ''cosmology'' was first used in English in 1656 in Thomas Blount's ''Glossographia'', and in 1731 taken up in Latin by German philosopher ...
,
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
and numerous other fields. In 2013 the
amplituhedron In mathematics and theoretical physics (especially twistor string theory), an amplituhedron is a geometric structure introduced in 2013 by Nima Arkani-Hamed and Jaroslav Trnka. It enables simplified calculation of particle interactions in some ...
was discovered as a simplifying construct in certain calculations of theoretical physics.


Applications

In the field of optimization,
linear programming Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linear programming is ...
studies the
maxima and minima In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given r ...
of
linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
functions; these maxima and minima occur on the boundary of an ''n''-dimensional polytope. In linear programming, polytopes occur in the use of
generalized barycentric coordinates In geometry, a barycentric coordinate system is a coordinate system in which the location of a point is specified by reference to a simplex (a triangle for points in a plane, a tetrahedron for points in three-dimensional space, etc.). The baryc ...
and slack variables. In
twistor theory In theoretical physics, twistor theory was proposed by Roger Penrose in 1967 as a possible path to quantum gravity and has evolved into a branch of theoretical and mathematical physics. Penrose proposed that twistor space should be the basic ar ...
, a branch of
theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
, a polytope called the
amplituhedron In mathematics and theoretical physics (especially twistor string theory), an amplituhedron is a geometric structure introduced in 2013 by Nima Arkani-Hamed and Jaroslav Trnka. It enables simplified calculation of particle interactions in some ...
is used in to calculate the scattering amplitudes of subatomic particles when they collide. The construct is purely theoretical with no known physical manifestation, but is said to greatly simplify certain calculations.


See also

* List of regular polytopes *
Bounding volume In computer graphics and computational geometry, a bounding volume for a set of objects is a closed volume that completely contains the union of the objects in the set. Bounding volumes are used to improve the efficiency of geometrical operati ...
-discrete oriented polytope * Intersection of a polyhedron with a line *
Extension of a polyhedron In convex geometry and polyhedral combinatorics, the extension complexity is a convex polytope P is the smallest number of facets among convex polytopes Q that have P as a projection. In this context, Q is called an extended formulation of P; it ma ...
* Polytope de Montréal *
Honeycomb (geometry) In geometry, a honeycomb is a ''space filling'' or '' close packing'' of polyhedral or higher-dimensional ''cells'', so that there are no gaps. It is an example of the more general mathematical ''tiling'' or ''tessellation'' in any number of dim ...
* Opetope


References


Citations


Bibliography

*. *. *.


External links

*
"Math will rock your world"
– application of polytopes to a database of articles used to support custom news feeds via the
Internet The Internet (or internet) is the global system of interconnected computer networks that uses the Internet protocol suite (TCP/IP) to communicate between networks and devices. It is a '' network of networks'' that consists of private, p ...
– (''Business Week Online'')
Regular and semi-regular convex polytopes a short historical overview:
{{Authority control Real algebraic geometry