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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:
- 5-cell: 102, (5 tetrahedral cells)
- 112 polytope, (16 5-cell, and 10 16-cell facets)
- 122 polytope, (54 demipenteract facets)
- 132 polytope, (56 122 and 126 demihexeract facets)
- 142 polytope, (240 132 and 2160 demihepteract facets)
- 152 honeycomb, tessellates Euclidean 8-space (∞ 142 and ∞ demiocteract facets)
- 162 honeycomb, tessellates hyperbolic 9-space (∞ 152 and ∞ demienneract facets)
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See also
- k21 polytope family
- 2k1 polytope family
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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