View source for Polytope

{{Short description|Geometric object with flat sides}}
{{Distinguish|Polytrope}}
{| class="wikitable" style="margin-left:1em" align="right"
|-
|[[File:First stellation of octahedron.svg|50px]]
|[[File:First stellation of dodecahedron.svg|50px]]
|[[File:Second stellation of dodecahedron.svg|50px]]
|[[File:Third stellation of dodecahedron.svg|50px]]
|[[File:Sixteenth stellation of icosahedron.svg|50px]]
|[[File:First stellation of icosahedron.svg|50px]]
|-
| colspan="6" | A [[polyhedron]] is a 3-dimensional polytope
|}
[[File:Assorted polygons.svg|thumb|400px|right|A [[polygon]] is a 2-dimensional polytope. Polygons can be characterised according to various criteria. Some examples are: open (excluding its boundary), bounding circuit only (ignoring its interior), closed (including both its boundary and its interior), and self-intersecting with varying densities of different regions.]]

In elementary [[geometry]], a '''polytope''' is a geometric object with [[Flat (geometry)|flat]] sides (''[[Face (geometry)|faces]]''). Polytopes are the generalization of three-dimensional [[polyhedron|polyhedra]] to any number of dimensions. Polytopes may exist in any general number of dimensions {{mvar|n}} as an {{mvar|n}}-dimensional polytope or '''{{mvar|n}}-polytope'''. For example, a two-dimensional [[polygon]] is a 2-polytope and a three-dimensional [[polyhedron]] is a 3-polytope. In this context, "flat sides" means that the sides of a {{math|(''k'' + 1)}}-polytope consist of {{mvar|k}}-polytopes that may have {{math|(''k'' – 1)}}-polytopes in common.

Some theories further generalize the idea to include such objects as unbounded [[apeirotope]]s and [[tessellation]]s, decompositions or tilings of curved [[manifold]]s including [[spherical polyhedra]], and set-theoretic [[abstract polytope]]s.

Polytopes of more than three dimensions were first discovered by [[Ludwig Schläfli]] before 1853, who called such a figure a '''polyschem'''.{{Sfn|Coxeter|1973|pp=141-144|loc=§7-x. Historical remarks}} The [[German language|German]] term ''polytop'' was coined by the mathematician [[Reinhold Hoppe]], and was introduced to English mathematicians as ''polytope'' by [[Alicia Boole Stott]].

==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]]s to include other objects with similar properties.

The original approach broadly followed by [[Ludwig Schläfli]], [[Thorold Gosset]] 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.<ref name="coxeter1973">Coxeter (1973)</ref>

Attempts to generalise the [[Euler characteristic]] of polyhedra to higher-dimensional polytopes led to the development of [[topology]] and the treatment of a decomposition or [[CW-complex]] as analogous to a polytope.<ref>{{cite book|author-link=David Richeson|last=Richeson|first=D.|title=Euler's Gem: The Polyhedron Formula and the Birth of Topology|title-link= Euler's Gem|publisher=Princeton University Press|year=2008}}</ref> In this approach, a polytope may be regarded as a [[tessellation]] or decomposition of some given [[manifold]]. An example of this 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.<ref name="Grünbaum2003">Grünbaum (2003)</ref> However this definition does not allow [[star polytope]]s with interior structures, and so is restricted to certain areas of mathematics.

The discovery of [[star polyhedron|star polyhedra]] and other unusual constructions led to the idea of a polyhedron as a bounding surface, ignoring its interior.<ref>Cromwell, P.; ''Polyhedra'', CUP (ppbk 1999) pp 205 ff.</ref> In this light convex polytopes in ''p''-space are equivalent to [[spherical tiling|tilings of the (''p''−1)-sphere]], while others may be tilings of other [[elliptic space|elliptic]], flat or [[toroid]]al (''p''−1)-surfaces – see [[elliptic tiling]] and [[toroidal polyhedron]]. A [[polyhedron]] is understood as a surface whose [[Face (geometry)|faces]] are [[polygons]], a [[4-polytope]] as a hypersurface whose facets ([[Face (geometry)|cells]]) 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 (geometry)|edge]]) seen as a [[1-polytope]] bounded by a point pair, and a point or [[Vertex (geometry)|vertex]] as a 0-polytope. This approach is used for example in the theory of [[abstract polytope]]s.

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 set|bounded]] polyhedron.<ref>Nemhauser and Wolsey, "Integer and Combinatorial Optimization," 1999, {{isbn|978-0471359432}}, Definition 2.2.</ref> This terminology is typically confined to polytopes and polyhedra that are [[Convex body|convex]]. With this terminology, a convex polyhedron is the intersection of a finite number of [[Half-space (geometry)|halfspaces]] and is defined by its sides while a convex polytope is the [[convex hull]] of a finite number of points and is defined by its vertices.

Polytopes in lower numbers of dimensions have standard names:
{|class="wikitable"
!Dimension<br>of polytope
!Description<ref name="johnson224"/>
|-
|align=center|−1
|[[Nullitope]]
|-
|align=center|0
|[[Hypercube#Elements|Monon]]
|-
|align=center|1
|Dion
|-
|align=center|2
|[[Polygon]]
|-
|align=center|3
|[[Polyhedron]]
|-
|align=center|4
|[[Polychoron]]<ref name=johnson224/>
|}

==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''&nbsp;−&nbsp;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]] uses ''cell'' to denote an (''n''&nbsp;−&nbsp;1)-dimensional element.<ref>Regular polytopes, p. 127 ''The part of the polytope that lies in one of the hyperplanes is called a cell''</ref>{{citation needed|date=February 2015|reason=need to cite each definition claimed}} <!-- Note that "each definition claimed" means "each definition claimed" and this tag should remain until each definition claimed has been cited -->

The terms adopted in this article are given in the table below:

{|class="wikitable"
!Dimension<br>of element
!Term<br>(in an ''n''-polytope)
|-
|align=center|−1
|Nullity (necessary in [[Abstract polytope|abstract]] theory)<ref name="johnson224">Johnson, Norman W.; ''Geometries and Transformations'', Cambridge University Press, 2018, p.224.</ref>
|-
|align=center|0
|[[Vertex (geometry)|Vertex]]
|-
|align=center|1
|[[Edge (geometry)|Edge]]
|-
|align=center|2
|[[Face (geometry)|Face]]
|-
|align=center|3
|[[Cell (geometry)|Cell]]
|-
|align=center|<math>\vdots</math>
|&nbsp;<math>\vdots</math>
|-
|align=center|''j''
|''j''-face – element of rank ''j'' = −1, 0, 1, 2, 3, ..., ''n''
|-
|align=center|<math>\vdots</math>
|&nbsp;<math>\vdots</math>
|-
|align=center|''n'' − 3
|[[Peak (geometry)|Peak]] – (''n'' − 3)-face
|-
|align=center|''n'' − 2
|[[Ridge (geometry)|Ridge]] or subfacet – (''n'' − 2)-face
|-
|align=center|''n'' − 1
|[[Facet (mathematics)|Facet]] – (''n'' − 1)-face
|-
|align=center|''n''
|The polytope itself
|}

An ''n''-dimensional polytope is bounded by a number of (''n''&nbsp;−&nbsp;1)-dimensional ''[[facet (mathematics)|facets]]''. These facets are themselves polytopes, whose facets are (''n''&nbsp;−&nbsp;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''&nbsp;−&nbsp;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]].

==Important classes of polytopes==

===Convex polytopes===
{{Main|Convex polytope}}
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-space (geometry)|half-spaces]]. 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.
It is an [[integral polytope]] if all of its vertices have integer coordinates.

A certain class of convex polytopes are ''reflexive'' polytopes. An integral {{nobr|<math>d</math>-polytope}} <math>\mathcal{P}</math> is reflexive if for some [[integer matrix|integral matrix]] <math>\mathbf{A}</math>, <math>\mathcal{P} = \{\mathbf{x} \in \mathbb{R}^d : \mathbf{Ax} \leq \mathbf{1}\}</math>, where <math>\mathbf{1}</math> denotes a vector of all ones, and the inequality is component-wise. It follows from this definition that <math>\mathcal{P}</math> is reflexive if and only if <math>(t+1)\mathcal{P}^\circ \cap \mathbb{Z}^d = t\mathcal{P} \cap \mathbb{Z}^d</math> for all <math>t \in \mathbb{Z}_{\geq 0}</math>. In other words, a {{nobr|<math>(t + 1)</math>-dilate}} of <math>\mathcal{P}</math> differs, in terms of integer lattice points, from a {{nobr|<math>t</math>-dilate}} of <math>\mathcal{P}</math> only by lattice points gained on the boundary. Equivalently, <math>\mathcal{P}</math> is reflexive if and only if its [[dual polyhedron|dual polytope]] <math>\mathcal{P}^*</math> is an integral polytope.<ref>Beck, Matthias; Robins, Sinai (2007), ''[[Computing the Continuous Discretely|Computing the Continuous Discretely: Integer-point enumeration in polyhedra]]'', Undergraduate Texts in Mathematics, New York: Springer-Verlag, {{ISBN|978-0-387-29139-0}}, MR 2271992</ref>

===Regular polytopes===
{{Main|Regular polytope}}
[[Regular polytope]]s have the highest degree of symmetry of all polytopes. The symmetry group of a regular polytope acts transitively on its [[flag (geometry)|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:
*[[Simplex|Simplices]], including the [[equilateral triangle]] and the [[regular tetrahedron]].
*[[Hypercube]]s or measure polytopes, including the [[square]] and the [[cube]].
*[[Orthoplex]]es or cross polytopes, including the [[square]] and [[regular octahedron]].

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 polygon]]s of ''n''-fold symmetry, both convex and (for ''n'' ≥ 5) star. But in higher dimensions there are no other regular polytopes.<ref name="coxeter1973"/>

In three dimensions the convex [[Platonic solid]]s include the fivefold-symmetric [[dodecahedron]] and [[icosahedron]], 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-polytope]]s include one additional convex solid with fourfold symmetry and two with fivefold symmetry. There are ten star [[Schläfli-Hess 4-polytope]]s, all with fivefold symmetry, giving in all sixteen regular 4-polytopes.

===Star polytopes===
{{Main|Star polytope}}
A non-convex polytope may be self-intersecting; this class of polytopes include the [[star polytope]]s. Some regular polytopes are stars.<ref name="coxeter1973"/>

== Properties ==

=== Euler characteristic ===
Since a (filled) convex polytope ''P'' in <math>d</math> dimensions is [[Contractible space|contractible]] to a point, the [[Euler characteristic]] <math>\chi</math> of its boundary ∂P is given by the alternating sum:
:<math>\chi = n_0 - n_1 + n_2 - \cdots \plusmn n_{d-1} = 1 + (-1)^{d-1}</math>, where <math>n_j</math> is the number of <math>j</math>-dimensional faces.

This generalizes [[Euler's formula for polyhedra]].<ref name="pands"/>

=== Internal angles ===
The [[Gram–Euler theorem]] similarly generalizes the alternating sum of [[Internal and external angles|internal angle]]s <math display="inline"> \sum \varphi</math> for convex polyhedra to higher-dimensional polytopes:<ref name="pands">M. A. Perles and G. C. Shephard. 1967. "Angle sums of convex polytopes". ''Math. Scandinavica'', Vol 21, No 2. March 1967. pp. 199–218.</ref>

: <math>\sum \varphi = (-1)^{d-1}</math>

==Generalisations of a polytope==

===Infinite polytopes===
{{Main|Apeirotope}}
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. [[Tessellation|plane tilings]], space-filling ([[Honeycomb (geometry)|honeycombs]]) and [[hyperbolic tiling]]s are in this sense polytopes, and are sometimes called [[apeirotope]]s because they have infinitely many cells.

Among these, there are regular forms including the [[regular skew polyhedron|regular skew polyhedra]] and the infinite series of tilings represented by the regular [[apeirogon]], square tiling, cubic honeycomb, and so on.

===Abstract polytopes===
{{Main|Abstract polytope}}

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 an intuitive underlying space, such as the [[11-cell]].

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 in some real space of the associated abstract polytope.<ref>{{citation | last1 = McMullen | first1 = Peter | author1-link = Peter McMullen | first2 = Egon | last2 = Schulte | title = Abstract Regular Polytopes | edition = 1st | publisher = [[Cambridge University Press]] | isbn = 0-521-81496-0 | date = December 2002 | url-access = registration | url = https://archive.org/details/abstractregularp0000mcmu }}</ref>

===Complex polytopes===
{{Main|Complex polytope}}

Structures analogous to polytopes exist in complex [[Hilbert space]]s <math> \Complex^n</math> where ''n'' real dimensions are accompanied by ''n'' [[imaginary number|imaginary]] ones. [[Regular complex polytope]]s are more appropriately treated as [[configuration (polytope)|configurations]].<ref>Coxeter, H.S.M.; ''Regular Complex Polytopes'', 1974</ref>

==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''&nbsp;−&nbsp;1)-dimensional elements for (''n''&nbsp;−&nbsp;''j'')-dimensional elements (for ''j''&nbsp;=&nbsp;1 to ''n''&nbsp;−&nbsp;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]]s for regular polytopes, where the symbol for the dual polytope is simply the reverse of the original. For example, {4,&nbsp;3,&nbsp;3} is dual to&nbsp;{3,&nbsp;3,&nbsp;4}.

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.<ref>Wenninger, M.; ''Dual Models'', CUP (1983).</ref>

If the dual is reversed, then the original polytope is recovered. Thus, polytopes exist in dual pairs.

===Self-dual polytopes===
[[File:Schlegel wireframe 5-cell.png|120px|thumb|The [[5-cell]] (4-simplex) is self-dual with 5 vertices and 5 tetrahedral cells.]]
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 any number of dimensions, with [[Schläfli symbol]] {3<sup>''n''</sup>}. These include the [[equilateral triangle]] {3}, [[regular tetrahedron]] {3,3}, and [[5-cell]]  {3,3,3}.
*Every [[hypercubic honeycomb]], in any number of dimensions. These include the [[apeirogon]] {∞}, [[square tiling]] {4,4} and [[cubic honeycomb]] {4,3,4}.
*Numerous compact, paracompact and noncompact hyperbolic tilings, such as the [[icosahedral honeycomb]] {3,5,3}, and [[order-5 pentagonal tiling]] {5,5}.
*In 2 dimensions, all [[regular polygon]]s (regular 2-polytopes)
*In 3 dimensions, the [[canonical form|canonical]] [[polygonal pyramid]]s and [[elongated pyramid]]s, and [[tetrahedrally diminished dodecahedron]].
*In 4 dimensions, the [[24-cell]], with [[Schläfli symbol]] {3,4,3}. Also the [[great 120-cell]] {5,5/2,5} and [[grand stellated 120-cell]] {5/2,5,5/2}.

==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]] and [[Hermann Grassmann]] 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-polytope]]s in 1852 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.

In 1882 [[Reinhold 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]], daughter of logician [[George Boole]], introduced the anglicised ''polytope'' into the English language.<ref name="coxeter1973"/>{{rp|vi}}

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 also began to be studied in non-Euclidean spaces such as hyperbolic space.

An important milestone was reached in 1948 with [[Harold Scott MacDonald Coxeter|H. S. M. Coxeter]]'s book ''[[Regular Polytopes (book)|Regular Polytopes]]'', summarizing work to date and adding new findings of his own.

Meanwhile, the French mathematician [[Henri Poincaré]] had developed the [[topology|topological]] idea of a polytope as the piecewise decomposition (e.g. [[CW-complex]]) of a [[manifold (topology)|manifold]]. [[Branko Grünbaum]] published his influential work on ''[[Convex Polytopes]]'' in 1967.

In 1952 [[Geoffrey Colin Shephard]] generalised the idea as [[complex polytope]]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 polytope]]s as partially ordered sets, or posets, of such elements. [[Peter McMullen]] and Egon Schulte published their book ''Abstract Regular Polytopes'' in 2002.

Enumerating the [[uniform polytope]]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;<ref>[http://math.fau.edu/Yiu/Oldwebsites/RM2003/cmjConway825.pdf John Horton Conway: Mathematical Magus] - Richard K. Guy</ref><ref>{{cite journal | doi=10.1098/rsbm.2021.0034 | title=John Horton Conway. 26 December 1937—11 April 2020 | journal=Biographical Memoirs of Fellows of the Royal Society | date=June 2022 | volume=72 | pages=117–138 | last1=Curtis | first1=Robert Turner | doi-access=free }}</ref> in higher dimensions this problem was still open as of 1997.<ref>[http://mathserver.neu.edu/~schulte/symchapter.pdf Symmetry of Polytopes and Polyhedra], Egon Schulte. p. 12: "However, there are many more uniform polytopes but a complete list is known only for d = 4 [Joh]."</ref> The full enumeration for nonconvex uniform polytopes is not known in dimensions four and higher as of 2008.<ref>[[John Horton Conway]], Heidi Burgiel, and [[Chaim Goodman-Strauss]]: ''[[The Symmetries of Things]]'', p. 408. "There are also starry analogs of the Archimedean polyhedra...So far as we know, nobody has yet enumerated the analogs in four or higher dimensions."</ref>

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]], [[quantum mechanics]] and numerous other fields. In 2013 the [[amplituhedron]] was discovered as a simplifying construct in certain calculations of theoretical physics.

==Applications==
In the field of [[Optimization (mathematics)|optimization]], [[linear programming]] studies the [[maxima and minima]] of [[linear]] functions; these maxima and minima occur on the [[boundary (topology)|boundary]] of an ''n''-dimensional polytope. In linear programming, polytopes occur in the use of [[generalized barycentric coordinates]] and [[slack variable]]s.

In  [[twistor theory]], a branch of [[theoretical physics]], a polytope called the [[amplituhedron]] 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.<ref>{{cite journal|last1=Arkani-Hamed |first1=Nima |last2=Trnka |first2=Jaroslav |year=2013 |arxiv=1312.2007 |title=The Amplituhedron |doi=10.1007/JHEP10(2014)030 |volume=2014 |journal=Journal of High Energy Physics|issue=10 |page=30 |bibcode=2014JHEP...10..030A }}</ref>

==See also==
{{div col|colwidth=30em}}
*[[List of regular polytopes]]
*[[Bounding volume]]-discrete oriented polytope
*[[Intersection of a polyhedron with a line]]
*[[Extension of a polyhedron]]
*[[Polytope de Montréal]]
*[[Honeycomb (geometry)]]
*[[Opetope]]
{{div col end}}

==References==
===Citations===
{{Reflist}}
===Bibliography===
{{Refbegin}}
*{{Citation |last=Coxeter |first=Harold Scott MacDonald |author-link=Harold Scott MacDonald Coxeter |title=[[Regular Polytopes (book)|Regular Polytopes]] |publisher=[[Dover Publications]] |location=New York |isbn=978-0-486-61480-9 |year=1973}}.
*{{Citation |last=Grünbaum |first=Branko |author-link=Branko Grünbaum |title=Convex polytopes|title-link=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}}.
*{{Citation |last=Ziegler |first=Günter M. |author-link=Günter M. Ziegler |title=Lectures on Polytopes |publisher=[[Springer-Verlag]] |location=Berlin, New York |series=Graduate Texts in Mathematics |year=1995 |volume=152}}.
{{Refend}}

==External links==
{{Wiktionary|polytope}}
*{{MathWorld |urlname=Polytope |title=Polytope}}
*[https://web.archive.org/web/20060115040615/http://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'')
*[https://web.archive.org/web/20030818121038/http://presh.com/hovinga/regularandsemiregularconvexpolytopesashorthistoricaloverview.html Regular and semi-regular convex polytopes a short historical overview:]

{{Dimension topics}}
{{Polytopes}}
{{Authority control}}

[[Category:Polytopes| ]]
[[Category:Real algebraic geometry]]