Uniform 1 k2 polytope

In geometry, 1_k2 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 1_k2 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequence. It can be named by an extended Schläfli symbol {3,3^k,2}.

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 1_k-1,2 and (n-1)-demicube facets. Each has a vertex figure of a {3^1,n-2,2} polytope is a birectified n-simplex, t₂{3ⁿ}.

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

The complete family of 1_k2 polytope polytopes are:

1. 5-cell: 1₀₂, (5 tetrahedral cells)
2. 1₁₂ polytope, (16 5-cell, and 10 16-cell facets)
3. 1₂₂ polytope, (54 demipenteract facets)
4. 1₃₂ polytope, (56 1₂₂ and 126 demihexeract facets)
5. 1₄₂ polytope, (240 1₃₂ and 2160 demihepteract facets)
6. 1₅₂ honeycomb, tessellates Euclidean 8-space (∞ 1₄₂ and ∞ demiocteract facets)
7. 1₆₂ honeycomb, tessellates hyperbolic 9-space (∞ 1₅₂ and ∞ demienneract facets)

• k₂₁ polytope family
• 2_k1 polytope family

• 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
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  • 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

• PolyGloss v0.05: Gosset figures (Gossetododecatope)