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Uniform 2 k1 polytope

Uniform polytope From Wikipedia, the free encyclopedia

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In geometry, 2k1 polytope is a uniform polytope in n dimensions (n = k + 4) constructed from the En Coxeter group. The family was named by their Coxeter symbol as 2k1 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 {3,3,3k,1}.

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 2k−1,1 (n − 1)-polytope facets, each having a vertex figure as an (n − 1)-demicube, {31,n−2,1}.

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

The complete family of 2k1 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)
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Elements

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See also

References

  • A. Boole Stott (1910). "Geometrical deduction of semiregular from regular polytopes and space fillings" (PDF). Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam. XI (1). Amsterdam: Johannes Müller. Archived from the original (PDF) on 29 April 2025.
  • P. H. Schoute (1911). "Analytical treatment of the polytopes regularly derived from the regular polytopes" (PDF). Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam. Section I. XI (3). Amsterdam: Johannes Müller. Archived from the original (PDF) on 22 January 2025.
  • P. H. Schoute (1913). "Analytical treatment of the polytopes regularly derived from the regular polytopes" (PDF). Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam. Sections II, III, IV. XI (5). Amsterdam: Johannes Müller. Archived from the original (PDF) on 22 February 2025.
  • 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
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