Semiregular polytope: Difference between revisions
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In [[geometry]], a '''semiregular [[polytope]]''' is usually taken to be a [[polytope]] that is [[vertex-uniform]] and has all its [[facet (geometry)|facets]] being [[regular polytope]]s. |
In [[geometry]], by [[Thorold Gosset]]'s definition a '''semiregular [[polytope]]''' is usually taken to be a [[polytope]] that is [[vertex-uniform]] and has all its [[facet (geometry)|facets]] being [[regular polytope]]s. |
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In [[three-dimensional space]] and below, the terms ''semiregular polytope'' and ''[[uniform polytope]]'' have identical meanings, because all uniform [[polygon]]s must be [[regular polygon|regular]]. However, since not all [[uniform polyhedra]] are [[regular polyhedra|regular]], the number of semiregular polytopes in dimensions higher than three is much smaller than the number of uniform polytopes in the same number of dimensions. |
In [[three-dimensional space]] and below, the terms ''semiregular polytope'' and ''[[uniform polytope]]'' have identical meanings, because all uniform [[polygon]]s must be [[regular polygon|regular]]. However, since not all [[uniform polyhedra]] are [[regular polyhedra|regular]], the number of semiregular polytopes in dimensions higher than three is much smaller than the number of uniform polytopes in the same number of dimensions. |
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Revision as of 04:49, 7 February 2013
 Rectified 5-cell |
 Snub 24-cell |
 Rectified 600-cell |
In geometry, by Thorold Gosset's definition a semiregular polytope is usually taken to be a polytope that is vertex-uniform and has all its facets being regular polytopes.
In three-dimensional space and below, the terms semiregular polytope and uniform polytope have identical meanings, because all uniform polygons must be regular. However, since not all uniform polyhedra are regular, the number of semiregular polytopes in dimensions higher than three is much smaller than the number of uniform polytopes in the same number of dimensions.
The three convex semiregular polychora (4-polytopes) are the rectified 5-cell, snub 24-cell and rectified 600-cell. The only semiregular polytopes in higher dimensions are the k21 polytopes, where the rectified 5-cell is the special case of k = 0.
Semiregular polytopes can be extended to semiregular honeycombs. The semiregular Euclidean honeycombs are the tetrahedral-octahedral honeycomb (3D), gyrated alternated cubic honeycomb (3D) and the 521 honeycomb (8D).
See also
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