Uniform 2 k1 polytope explained

In geometry, 2_k1 polytope is a uniform polytope in n dimensions (n = k+4) constructed from the E_n Coxeter group. The family was named by their Coxeter symbol as 2_k1 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequence. It can be named by an extended Schläfli symbol .

Family members

The family starts uniquely as 6-polytopes, 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_k-1,1 (n-1)-polytope facets, each has a vertex figure as an (n-1)-demicube, .

The sequence ends with k=6 (n=10), as an infinite hyperbolic tessellation of 9-space.

The complete family of 2_k1 polytope polytopes are:

1. 5-cell: 2₀₁, (5 tetrahedra cells)
2. Pentacross: 2₁₁, (32 5-cell (2₀₁) facets)
3. 2₂₁, (72 5-simplex and 27 5-orthoplex (2₁₁) facets)
4. 2₃₁, (576 6-simplex and 56 2₂₁ facets)
5. 2₄₁, (17280 7-simplex and 240 2₃₁ facets)
6. 2₅₁, tessellates Euclidean 8-space (∞ 8-simplex and ∞ 2₄₁ facets)
7. 2₆₁, tessellates hyperbolic 9-space (∞ 9-simplex and ∞ 2₅₁ facets)

See also

• k₂₁ 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

• PolyGloss v0.05: Gosset figures (Gossetoctotope)