Regular (InstanceAttribute, 4)
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In two dimensions, there are infinitely many regular polytopes, each one having a different number of sides. | In two dimensions, there are infinitely many regular polytopes, each one having a different number of sides. | ||
| - | In three dimensions and above, there are | + | In three dimensions and above, there are three distinct sets of regular polytopes: the [[simplex|simplices]], which is self-dual, and the [[hypercube]]s and [[cross polytope]]s which are dual to each other. In three or four dimensions only, there are two more regular polytopes: the [[dodecahedron]] and the [[icosahedron]], which are dual to each other. In four dimensions and no other dimension, there is also a sixth regular polytope, with several unique properties: the [[icositetrachoron]], which is self-dual. |
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| - | + | For the three main aforementioned sets, the simplices are all [[rotope]]s, the cross polytopes are all [[bracketope]]s and the hypercubes are all both rotopes and bracketopes. | |
| - | Note that it does not make sense to speak of regularity in dimensions less than two. | + | Note that it does not make sense to speak of regularity in dimensions less than two. Also, since [[shape]]s can have curved hypercells, there are infinitely many regular ''shapes'' in any dimension, which is why we specify that regularity usually applies only to polytopes. |
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[[Category:Geometric properties]] | [[Category:Geometric properties]] | ||
Revision as of 20:39, 22 September 2007
A regular polytope is a polytope whose hypercells are all congruent.
In two dimensions, there are infinitely many regular polytopes, each one having a different number of sides.
In three dimensions and above, there are three distinct sets of regular polytopes: the simplices, which is self-dual, and the hypercubes and cross polytopes which are dual to each other. In three or four dimensions only, there are two more regular polytopes: the dodecahedron and the icosahedron, which are dual to each other. In four dimensions and no other dimension, there is also a sixth regular polytope, with several unique properties: the icositetrachoron, which is self-dual.
For the three main aforementioned sets, the simplices are all rotopes, the cross polytopes are all bracketopes and the hypercubes are all both rotopes and bracketopes.
Note that it does not make sense to speak of regularity in dimensions less than two. Also, since shapes can have curved hypercells, there are infinitely many regular shapes in any dimension, which is why we specify that regularity usually applies only to polytopes.
