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Uniform 1 k2 polytope

Uniform polytope From Wikipedia, the free encyclopedia

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In geometry, 1k2 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 1k2 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 {3,3k,2}.

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 1k−1,2 and (n−1)-demicube facets. Each has a vertex figure of a {31,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 1k2 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)
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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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