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Uniform 1 k2 polytope

In geometry, 1<sub>k2</sub> polytope is a uniform polytope in n dimensions (n = k + 4) constructed from the E<sub>n</sub> Coxeter group. The family was named by their Coxeter symbol 1<sub>k2</sub> 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<sup>k,2</sup>}.

Family members

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<sub>k−1,2</sub> and (n−1)-demicube facets. Each has a vertex figure of a {3<sup>1,n−2,2</sup>} polytope, is a birectified n-simplex, t<sub>2</sub>{3<sup>n</sup>}.

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

The complete family of 1<sub>k2</sub> polytopes are:

  1. 5-cell: 1<sub>02</sub>, (5 tetrahedral cells)
  2. 1<sub>12</sub> polytope, (16 5-cell, and 10 16-cell facets)
  3. 1<sub>22</sub> polytope, (54 demipenteract facets)
  4. 1<sub>32</sub> polytope, (56 1<sub>22</sub> and 126 demihexeract facets)
  5. 1<sub>42</sub> polytope, (240 1<sub>32</sub> and 2160 demihepteract facets)
  6. 1<sub>52</sub> honeycomb, tessellates Euclidean 8-space (∞ 1<sub>42</sub> and ∞ demiocteract facets)
  7. 1<sub>62</sub> honeycomb, tessellates hyperbolic 9-space (∞ 1<sub>52</sub> and ∞ demienneract facets)

Elements

See also

References

  • 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