- Uniform 2 k1 polytope
In

geometry ,**2**or {3_{k1}polytope^{2,k,1}} is auniform polytope in n-dimensions (n = k+4) constructed from the E_{n}Coxeter group . The family was named byCoxeter as**2**by its bifurcating_{k1}Coxeter-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**2**(n-1)-polytope facets, each has a_{k-1,1}vertex figure as an (n-1)-demicube, "{3^{1,n-2,1}}".The sequence ends with k=5 (n=9), as an infinite tessellation of 8-space.

The complete family of

**2**polytopes are:_{k1}polytope

#5-cell :**2**, (5 tetrahedra cells)_{01}

#Pentacross :**2**, (32_{11}5-cell (**2**) facets)_{01}

#Gosset 2 21 polytope :**2**, (72 5-_{21}simplex and 27 5-orthoplex (**2**) facets)_{11}

#Gosset 2 31 polytope :**2**, (576 6-_{31}simplex and 56**2**facets)_{21}

#Gosset 2 41 polytope :**2**, (17280 7-_{41}simplex and 240**2**facets)_{31}

#Gosset 2_51 lattice :**2**, tessellates Euclidean 8-space (∞ 8-_{51}simplex and ∞**2**facets)_{41}**Elements****See also*** k

_{21}polytope family

* 1_{k2}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**External links*** [

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

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