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Uniform 2 k1 polytope

In geometry, 2<sub>k1</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 as 2<sub>k1</sub> 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 {3,3,3<sup>k,1</sup>}.

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<sub>k−1,1</sub> (n − 1)-polytope facets, each having a vertex figure as an (n − 1)-demicube, {3<sup>1,n−2,1</sup>}.

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

The complete family of 2<sub>k1</sub> polytopes are:

  1. 5-cell: 2<sub>01</sub>, (5 tetrahedra cells)
  2. Pentacross: 2<sub>11</sub>, (32 5-cell (2<sub>01</sub>) facets)
  3. 2<sub>21</sub>, (72 5-simplex and 27 5-orthoplex (2<sub>11</sub>) facets)
  4. 2<sub>31</sub>, (576 6-simplex and 56 2<sub>21</sub> facets)
  5. 2<sub>41</sub>, (17280 7-simplex and 240 2<sub>31</sub> facets)
  6. 2<sub>51</sub>, tessellates Euclidean 8-space (∞ 8-simplex and ∞ 2<sub>41</sub> facets)
  7. 2<sub>61</sub>, tessellates hyperbolic 9-space (∞ 9-simplex and ∞ 2<sub>51</sub> 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