Uniform 1 k2 polytope

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Uniform 1 k2 polytope

In geometry, 1k2 polytope or {31,k,2} is a uniform polytope in n-dimensions (n = k+4) constructed from the En Coxeter group. The family was named by Coxeter as 1k2 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequence.

The family starts uniquely as 6-polytopes, but can be extended backwards to include the 5-demicube (demipenteract) in 5-dimensions, and the 4-simplex (5-cell) in 4-dimensions.

Each polytope is constructed from 1k-1,2 and (n-1)-demicube facets. Each has a vertex figure of a "{31,n-2,2}" polytope is a birectified n-simplex, "t2{3n}".

The sequence ends with k=5 (n=9), as an infinite tessellation of 8-space.

The complete family of 1k2 polytope polytopes are:
# 5-cell: 102, (5 tetrahedral cells)
# Demipenteract: 112, (16 5-cell, and 10 16-cell facets)
# Gosset 1 22 polytope: 122, (54 demipenteract facets)
# Gosset 1 32 polytope: 132, (56 122 and 126 demihexeract facets)
# Gosset 1 42 polytope: 142, (240 132 and 2160 demihepteract facets)
# Gosset 1_52 lattice: 152, tessellates Euclidean 8-space (∞ 142 and ∞ demiocteract facets)

Elements

* k21 polytope family
* 2k1 polytope family

References

* 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, 1988

* [http://www.geocities.com/os2fan2/gloss.htm#gossetfig PolyGloss v0.05: Gosset figures (Gossetododecatope)]

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