Editing Regular polytope (section)

{{Short description|Polytope with highest degree of symmetry}}
{{more footnotes|date=July 2014}}
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|+ Regular polytope examples
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|[[File:Regular pentagon.svg|160px]]<BR>A regular [[pentagon]] is a [[polygon]], a two-dimensional polytope with 5 [[Edge (geometry)|edges]], represented by [[Schläfli symbol]] {{math|{5}.}}
|[[Image:POV-Ray-Dodecahedron.svg|160px]]<BR>A regular [[dodecahedron]] is a [[polyhedron]], a three-dimensional polytope, with 12 pentagonal [[Face (geometry)|faces]], represented by Schläfli symbol {{math|{5,3}.}}
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|[[File:Schlegel wireframe 120-cell.png|160px]]<BR>A regular [[120-cell]] is a [[polychoron]], a four-dimensional polytope, with 120 dodecahedral [[Cell (geometry)|cells]], represented by Schläfli symbol {{math|{5,3,3}.}} (shown here as a [[Schlegel diagram]])
|[[File:Cubic honeycomb.png|160px]]<BR>A regular [[cubic honeycomb]] is a [[tessellation]], an infinite polytope, represented by Schläfli symbol {{math|{4,3,4}.}}
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|colspan=2|[[File:Octeract Petrie polygon.svg|320px]]<BR>The 256 vertices and 1024 edges of an [[8-cube]] can be shown in this orthogonal projection ([[Petrie polygon]])
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In [[mathematics]], a '''regular polytope''' is a [[polytope]] whose [[symmetry group]] acts [[transitive group action|transitively]] on its [[flag (geometry)|flags]], thus giving it the highest degree of symmetry. In particular, all its elements or {{mvar|j}}-faces (for all {{math|0 ≤ ''j'' ≤ ''n''}}, where {{mvar|n}} is the [[dimension]] of the polytope) &mdash; cells, faces and so on &mdash; are also transitive on the symmetries of the polytope, and are themselves regular polytopes of dimension {{math|'' j''≤ ''n''}}.

Regular polytopes are the generalised analog in any number of dimensions of [[regular polygon]]s (for example, the [[square (geometry)|square]] or the regular pentagon) and [[regular polyhedra]] (for example, the [[cube]]). The strong symmetry of the regular polytopes gives them an [[aesthetics|aesthetic]] quality that interests both mathematicians and non-mathematicians.

Classically, a regular polytope in {{mvar|n}} dimensions may be defined as having regular [[Facet (geometry)|facets]] ({{math|[''n''–1]}}-faces) and regular [[vertex figure]]s. These two conditions are sufficient to ensure that all faces are alike and all [[Vertex (geometry)|vertices]] are alike. Note, however, that this definition does not work for [[abstract polytope]]s.

A regular polytope can be represented by a [[Schläfli symbol]] of the form {{math|{a, b, c, ..., y, z},}} with regular facets as {{math|{a, b, c, ..., y},}} and regular vertex figures as {{math|{b, c, ..., y, z}.}}