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| {{Distinguish|polytrope}}
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| [[File:A 2-dimensional polytope.svg|thumb|A 2-dimensional polytope.]]
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| In elementary [[geometry]], a '''polytope''' is a geometric object with flat sides, which exists in any general number of dimensions. A [[polygon]] is a polytope in two dimensions, a [[polyhedron]] in three dimensions,<ref>Note that some authors use ''polytope'' and ''polyhedron'' in a different sense, as follows: a ''polyhedron'' is the generic object in any dimension (which is referred to as ''polytope'' on this wikipedia article) and ''polytope'' means a [[bounded]] polyhedron; c.f. Definition 2.2 in Nemhauser and Wolsey in "Integer and Combinatorial Optimization" ISBN 978-0471359432 1999</ref> and so on in higher dimensions (such as a [[polychoron]] in four dimensions). Some theories further generalize the idea to include such objects as unbounded polytopes ([[apeirotope]]s and [[tessellation]]s), and [[abstract polytope]]s.
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| When referring to an ''n''-dimensional generalization, the term '''''n''-polytope''' is used. For example, a polygon is a 2-polytope, a polyhedron is a 3-polytope, and a polychoron is a 4-polytope.
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| The term was coined by the mathematician Hoppe, writing in [[German language|German]], and was later introduced to English mathematicians by [[Alicia Boole Stott]], the daughter of logician [[George Boole]].<ref>A. Boole Stott: ''Geometrical deduction of semiregular from regular polytopes and space fillings'', Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910</ref>
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| ==Different approaches to definition==
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| The term ''polytope'' is a broad term that covers a wide class of objects, and different definitions are attested in mathematical literature. Many of these definitions are not equivalent, resulting in different sets of objects being called ''polytopes''. They represent different approaches of generalizing the [[convex polytope]]s to include other objects with similar properties and aesthetic beauty.
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| The original approach broadly followed by Schläfli, Gossett and others begins with the 0-dimensional point as a 0-polytope ([[Vertex (geometry)|vertex]]). A 1-dimensional [[1-polytope]] ([[Edge (geometry)|edge]]) is constructed by bounding a line segment with two 0-polytopes. Then [[2-polytope]]s (polygons) are defined as plane objects whose bounding facets ([[Edge (geometry)|edges]]) are 1-polytopes, [[3-polytope]]s (polyhedra) are defined as solids whose facets ([[Face (geometry)|faces]]) are 2-polytopes, and so forth.
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| A polytope may also be regarded as a [[tessellation]] of some given [[manifold]]. Convex polytopes are equivalent to [[spherical tiling|tilings of the sphere]], while others may be tilings of other [[elliptic space|elliptic]], flat or [[toroid]]al surfaces – see [[elliptic tiling]] and [[toroidal polyhedron]]. Under this definition, [[Tessellation|plane tilings]] and space tilings ([[Honeycomb (geometry)|honeycombs]]) are considered to be polytopes, and are sometimes classed as [[apeirotope]]s because they have infinitely many cells; [[hyperbolic tiling|tilings of hyperbolic spaces]] are also included under this definition.
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| An alternative approach defines a polytope as a set of points that admits a [[simplicial complex|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. However this definition does not allow [[star polytope]]s with interior structures, and so is restricted to certain areas of mathematics.
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| The theory of [[abstract polytope]]s 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 clearly a ''natural underlying space'', such as the [[11-cell]].
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| ==Elements==
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| The elements of a polytope are its vertices, edges, faces, cells and so on. The terminology for these is not entirely consistent across different authors. To give just a few examples: Some authors use ''face'' to refer to an (''n'' − 1)-dimensional element while others use ''face'' to denote a 2-face specifically, and others use ''j''-face or ''k''-face to indicate an element of ''j'' or ''k'' dimensions. Some sources use ''edge'' to refer to a ridge, while [[H. S. M. Coxeter]] uses ''cell'' to denote an (''n'' − 1)-dimensional element.
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| An ''n''-dimensional polytope is bounded by a number of (''n'' − 1)-dimensional ''[[facet (mathematics)|facets]]''. These facets are themselves polytopes, whose facets are (''n'' − 2)-dimensional ''[[Ridge (geometry)|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 [[Face (geometry)|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]], and a 3-dimensional face, sometimes called a ''[[Cell (mathematics)|cell]]'', consists of a [[polyhedron]].
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| {|class="wikitable"
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| !Dimension<br>of element
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| !Element name<br>(in an ''n''-polytope)
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| |-
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| |align=center|−1
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| |Null polytope (necessary in [[Abstract polytope|abstract]] theory)
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| |-
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| |align=center|0
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| |[[Vertex (geometry)|Vertex]]
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| |-
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| |align=center|1
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| |[[Edge (geometry)|Edge]]
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| |-
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| |align=center|2
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| |[[Face (geometry)|Face]]
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| |-
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| |align=center|3
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| |[[Cell (geometry)|Cell]]
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| |-
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| |align=center|4
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| |[[Hypercell]]
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| |-
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| |align=center|<math>\vdots</math>
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| | <math>\vdots</math>
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| |-
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| |align=center|''j''
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| |''j''-face – element of rank ''j'' = −1, 0, 1, 2, 3, ..., ''n''
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| |-
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| |align=center|<math>\vdots</math>
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| | <math>\vdots</math>
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| |-
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| |align=center|''n'' − 3
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| |[[Peak (geometry)|Peak]] – (''n'' − 3)-face
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| |-
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| |align=center|''n'' − 2
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| |[[Ridge (geometry)|Ridge]] or subfacet – (''n'' − 2)-face
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| |-
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| |align=center|''n'' − 1
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| |[[Facet (mathematics)|Facet]] – (''n'' − 1)-face
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| |-
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| |align=center|''n''
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| |Body – ''n''-face
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| |}
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| ==Special classes of polytope==
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| ===Regular polytopes===
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| {{Main|Regular polytope}}
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| A polytope may be ''[[Regular polytope|regular]]''. The [[regular polytope]]s are a class of highly symmetrical and aesthetically pleasing polytopes, including the [[Platonic solid]]s, which have been studied extensively since ancient times.
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| ===Convex polytopes===
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| {{Main|Convex polytope}}
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| A polytope may be ''convex''. The convex polytopes are the simplest kind of polytopes, and form the basis for different generalizations of the concept of polytopes. A convex polytope is sometimes defined as the intersection of a set of [[half-space (geometry)|half-space]]s. This definition allows a polytope to be neither bounded nor finite. Polytopes are defined in this way, e.g., in [[linear programming]]. 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 <math>\{(x,y) \in \mathbb{R}^2 \mid x \geq 0\}</math>. 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.
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| ===Star polytopes===
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| {{Main|Star polytope}}
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| A non-convex polytope may be self-intersecting; this class of polytopes include the [[star polytope]]s.
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| ===Abstract polytopes===
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| {{Main|Abstract polytope}}
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| An abstract polytope is a [[partially ordered set]] 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 of some associated abstract polytope.
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| ===Self-dual polytopes===
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| [[Image:Schlegel wireframe 5-cell.png|120px|thumb|The [[5-cell]] (4-simplex) is self-dual with 5 vertices and 5 tetrahedral cells.]]
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| In 2 dimensions, all [[regular polygon]]s (regular 2-polytopes) are self-[[Dual polyhedron|dual]].
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| In 3 dimensions, the [[tetrahedron]] is self-dual, as well as canonical polygonal pyramids and elongated pyramids.
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| In higher dimensions, every regular ''n''-[[simplex]], with [[Schlafli symbol]] {3<sup>''n''</sup>}, is self-dual.
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| In addition, the [[24-cell]] in 4 dimensions, with [[Schlafli symbol]] {3,4,3}, is self-dual.
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| ==History==
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| {{Main|polygon|polyhedron}}
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| The concept of a polytope originally began with polygons and polyhedra, both of which have been known since ancient times.
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| It was not until the 19th century that higher dimensions were discovered and geometers learned to construct analogues of polygons and polyhedra in them. The first hint of higher dimensions seems to have come in 1827, with [[August Ferdinand Möbius|Möbius]]' discovery that two mirror-image solids can be superimposed by rotating one of them through a fourth dimension. By the 1850s, a handful of other mathematicians such as Cayley and Grassman had considered higher dimensions. [[Ludwig Schläfli]] was the first of these to consider analogues of polygons and polyhedra in such higher spaces. In 1852 he described the six [[convex regular 4-polytope]]s, but his work was not published until 1901, six years after his death. By 1854, [[Bernhard Riemann]]'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.
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| In 1882 Hoppe, writing in German, coined the word ''[[:de:Polytop (Geometrie)|polytop]]'' to refer to this more general concept of polygons and polyhedra. In due course, [[Alicia Boole Stott]] introduced ''polytope'' into the English language.
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| In 1895, [[Thorold Gosset]] not only rediscovered Schläfli's regular polytopes, but also investigated the ideas of [[semiregular polytope]]s and space-filling [[tessellation]]s in higher dimensions. Polytopes were also studied in non-Euclidean spaces such as hyperbolic space.
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| During the early part of the 20th century, higher-dimensional spaces became fashionable, and together with the idea of higher polytopes, inspired artists such as [[Picasso]] to create the movement known as [[cubism]].
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| An important milestone was reached in 1948 with [[H. S. M. Coxeter]]'s book ''[[Regular Polytopes (book)|Regular Polytopes]]'', summarizing work to date and adding findings of his own. [[Branko Grünbaum]] published his influential work on ''Convex Polytopes'' in 1967.
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| More recently, the concept of a polytope has been further generalized. In 1952 Shephard developed the idea of [[complex polytope]]s in complex space, where each real dimension has an imaginary one associated with it. Coxeter went on to publish his book, ''Regular Complex Polytopes'', in 1974. Complex polytopes do not have closed surfaces in the usual way, and are better understood as [[configuration (geometry)|configurations]]. This kind of conceptual issue led to the more general idea of incidence complexes and the study of abstract combinatorial properties relating vertices, edges, faces and so on. This in turn led to the theory of [[abstract polytope]]s as partially ordered sets, or posets, of such elements. McMullen and Schulte published their book ''Abstract Regular Polytopes'' in 2002.
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| Enumerating the [[uniform polytope]]s, convex and nonconvex, in four or more dimensions remains an outstanding problem.
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| In modern times, polytopes and related concepts have found many important applications in fields as diverse as [[computer graphics]], [[Optimization (mathematics)|optimization]], [[Search engine (computing)|search engine]]s, [[cosmology]] and numerous other fields.
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| ==Uses==
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| In the study of [[Optimization (mathematics)|optimization]], [[linear programming]] studies the [[maxima and minima]] of [[linear]] functions constricted to the [[boundary (topology)|boundary]] of an ''n''-dimensional polytope.
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| In [[linear programming]], polytopes occur in the use of [[Generalized barycentric coordinates]] and [[Slack variable]]s.
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| ==See also==
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| <div style="-moz-column-count:2; column-count:2;"> | |
| *[[List of regular polytopes]]
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| *[[Convex polytope]]
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| *[[Regular polytope]]
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| *[[Semiregular polytope]]
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| *[[Uniform polytope]]
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| *[[Abstract polytope]]
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| *[[Bounding volume]]-Discrete oriented polytope
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| *Regular forms
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| *#[[Simplex]]
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| *#[[hypercube]]
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| *#[[Cross-polytope]]
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| *[[Intersection of a polyhedron with a line]]
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| *[[Extension of a polyhedron]]
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| *[[Coxeter group]]
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| *By dimension:
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| *#2-polytope or [[polygon]]
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| *#3-polytope or [[polyhedron]]
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| *#4-polytope or [[polychoron]]
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| *#[[5-polytope]]
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| *#[[6-polytope]]
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| *#[[7-polytope]]
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| *#[[8-polytope]]
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| *#[[9-polytope]]
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| *#[[10-polytope]]
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| *[[Polyform]]
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| *[[Polytope de Montréal]]
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| *[[Schläfli symbol]]
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| *[[Honeycomb (geometry)]]
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| </div> | |
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| ==References==
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| {{reflist}}
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| *{{Citation |last=Coxeter |first=Harold Scott MacDonald |authorlink=Harold Scott MacDonald Coxeter |title=[[Regular Polytopes (book)|Regular Polytopes]] |publisher=[[Dover Publications]] |location=New York |isbn=978-0-486-61480-9 |year=1973}}.
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| *{{Citation |last=Grünbaum |first=Branko |authorlink=Branko Grünbaum |title=Convex polytopes |location=New York & London |publisher=[[Springer-Verlag]] |year=2003 |isbn=0-387-00424-6 |edition=2nd |editor1-first=Volker |editor1-last=Kaibel |editor2-first=Victor |editor2-last=Klee |editor2-link=Victor Klee |editor3-first=Günter M. |editor3-last=Ziegler |editor3-link=Günter M. Ziegler}}.
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| *{{Citation |last=Ziegler |first=Günter M. |authorlink=Günter M. Ziegler |title=Lectures on Polytopes |publisher=[[Springer-Verlag]] |location=Berlin, New York |series=Graduate Texts in Mathematics |year=1995 |volume=152}}.
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| ==External links==
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| {{Wiktionary|polytope}}
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| *{{mathworld |urlname=Polytope |title=Polytope}}
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| *[http://www.businessweek.com/magazine/content/06_04/b3968001.htm "Math will rock your world"] – application of polytopes to a database of articles used to support custom news feeds via the [[Internet]] – (''Business Week Online'')
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| *[http://presh.com/hovinga/regularandsemiregularconvexpolytopesashorthistoricaloverview.html Regular and semi-regular convex polytopes a short historical overview:]
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| {{Dimension topics}}
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| {{Polytopes}}
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| [[Category:Polytopes| ]]
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| [[Category:Real algebraic geometry]]
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