Uniform 1 k2 polytope
{{Short description|Uniform polytope}}
{{DISPLAYTITLE:Uniform 1 k2 polytope}}
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 polytope 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)
Elements
| class="wikitable"
|+ Gosset 1k2 figures |
| rowspan=2|n
!rowspan=2|1k2 !rowspan=2| Petrie !rowspan=2| Name !colspan=2|Facets !colspan=8|Elements |
|---|
| 1k-1,2
! Vertices ! Edges ! Faces ! Cells ! 4-faces ! 5-faces ! 6-faces ! 7-faces |
| align=center
|4 |102 |80px |120 | -- | 5 | 10 | 10 | 5 | | | | |
| align=center
|5 |112 |80px |121 |16 |80 |160 |120 | | | |
| align=center
|6 |122 |80px |122 |72 |720 |2160 |2160 |54 | | |
| align=center
|7 |132 |80px |132 |576 |10080 |40320 |50400 | |
| align=center
|8 |142 |80px |142 |17280 |483840 |2419200 |3628800 |
| align=center
|9 |152 | |152 |colspan=8|∞ |
| align=center
|10 |162 | |162 |∞ |colspan=8|∞ |
See also
- k21 polytope family
- 2k1 polytope family
References
- Alicia Boole Stott Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910
- Stott, A. B. "Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings." Verhandelingen der Koninklijke Akad. Wetenschappen Amsterdam 11, 3-24, 1910.
- Alicia Boole Stott, "Geometrical deduction of semiregular from regular polytopes and space fillings," Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam, (eerste sectie), Vol. 11, No. 1, pp. 1–24 plus 3 plates, 1910.
- Stott, A. B. 1910. "Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings." Verhandelingen der Koninklijke Akad. Wetenschappen Amsterdam
- Schoute, P. H., Analytical treatment of the polytopes regularly derived from the regular polytopes, Ver. der Koninklijke Akad. van Wetenschappen te Amsterdam (eerstie sectie), vol 11.5, 1913.
- 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
- [http://os2fan2.com/gloss.htm#gossetfig PolyGloss v0.05: Gosset figures (Gossetododecatope)]
{{Polytopes}}
{{Honeycombs}}