Uniform 2 k1 polytope
{{Short description|Uniform polytope}}
{{DISPLAYTITLE:Uniform 2 k1 polytope}}
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 has 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 polytope polytopes are:
- 5-cell: 201, (5 tetrahedra cells)
- Pentacross: 211, (32 5-cell (201) facets)
- 221, (72 5-simplex and 27 5-orthoplex (211) facets)
- 231, (576 6-simplex and 56 221 facets)
- 241, (17280 7-simplex and 240 231 facets)
- 251, tessellates Euclidean 8-space (∞ 8-simplex and ∞ 241 facets)
- 261, tessellates hyperbolic 9-space (∞ 9-simplex and ∞ 251 facets)
Elements
| class="wikitable"
|+ Gosset 2k1 figures |
| rowspan=2|n
!rowspan=2|2k1 !rowspan=2| Petrie !rowspan=2| Name !colspan=2|Facets !colspan=8|Elements |
|---|
| 2k-1,1 polytope
!(n-1)-simplex ! Vertices ! Edges ! Faces ! Cells ! 4-faces ! 5-faces ! 6-faces ! 7-faces |
| align=center
|4 |201 |80px |5-cell | -- | 5 | 10 | 10 | 5 | | | | |
| align=center
|5 |211 |80px |pentacross | 10 | 40 | 80 | 80 | 32 | | | |
| align=center
|6 |221 |80px |2 21 polytope |27 |216 |720 |1080 |648 | | |
| align=center
|7 |231 |80px |2 31 polytope |126 |2016 |10080 |20160 |16128 | |
| align=center
|8 |241 |80px |2 41 polytope |2160 |69120 |483840 |1209600 |1209600 |
| align=center
|9 |251 | |2 51 honeycomb |colspan=8|∞ |
| align=center
|10 |261 | |2 61 honeycomb |∞ |colspan=8|∞ |
See also
- k21 polytope family
- 1k2 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 (Gossetoctotope)]
{{Polytopes}}
{{Honeycombs}}