- Uniform 2 k1 polytope
In
geometry , 2k1 polytope or {32,k,1} is auniform polytope in n-dimensions (n = k+4) constructed from the EnCoxeter group . The family was named byCoxeter as 2k1 by its bifurcatingCoxeter-Dynkin diagram , with a single ring on the end of the 2-node sequence.The family starts uniquely as
6-polytope s, but can be extended backwards to include the 5-orthoplex (pentacross ) in 5-dimensions, and the 4-simplex (5-cell ) in 4-dimensions.Each polytope is constructed from (n-1)-
simplex and 2k-1,1 (n-1)-polytope facets, each has avertex figure as an (n-1)-demicube, "{31,n-2,1}".The sequence ends with k=5 (n=9), as an infinite tessellation of 8-space.
The complete family of 2k1 polytope polytopes are:
#5-cell : 201, (5 tetrahedra cells)
#Pentacross : 211, (325-cell (201) facets)
#Gosset 2 21 polytope : 221, (72 5-simplex and 27 5-orthoplex (211) facets)
#Gosset 2 31 polytope : 231, (576 6-simplex and 56 221 facets)
#Gosset 2 41 polytope : 241, (17280 7-simplex and 240 231 facets)
#Gosset 2_51 lattice : 251, tessellates Euclidean 8-space (∞ 8-simplex and ∞ 241 facets)Elements
See also
* k21 polytope family
* 1k2 polytope familyReferences
*
Alicia 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
** Stott, A. B. "Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings." Verhandelingen der Koninklijke Akad. Wetenschappen Amsterdam 11, 3-24, 1910.
** Alicia Boole Stott, "Geometrical deduction of semiregular from regular polytopes and space fillings," Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam, (eerste sectie), Vol. 11, No. 1, pp. 1-24 plus 3 plates, 1910.
** Stott, A. B. 1910. "Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings." Verhandelingen der Koninklijke Akad. Wetenschappen Amsterdam
* Schoute, P. H., Analytical treatment of the polytopes regularly derived from the regular polytopes, "Ver. der Koninklijke Akad. van Wetenschappen te Amsterdam" (eerstie sectie), vol 11.5, 1913.
* H.S.M. Coxeter: Regular and Semi-Regular Polytopes, Part I, Mathematische Zeitschrift, Springer, Berlin, 1940
* N.W. Johnson: "The Theory of Uniform Polytopes and Honeycombs", Ph.D. Dissertation, University of Toronto, 1966
* H.S.M. Coxeter: Regular and Semi-Regular Polytopes, Part II, Mathematische Zeitschrift, Springer, Berlin, 1985
* H.S.M. Coxeter: Regular and Semi-Regular Polytopes, Part III, Mathematische Zeitschrift, Springer, Berlin, 1988External links
* [http://www.geocities.com/os2fan2/gloss.htm#gossetfig PolyGloss v0.05: Gosset figures (Gossetooctotope)]
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