Uniform 1 k2 polytope
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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[edit]
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[edit]
n | 1k2 | Petrie polygon projection | Name Coxeter-Dynkin diagram | Facets | Elements | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1k-1,2 | (n-1)-demicube | Vertices | Edges | Faces | Cells | 4-faces | 5-faces | 6-faces | 7-faces | ||||
4 | 102 | 120 | -- | 5 110 | 5 | 10 | 10 | 5 | |||||
5 | 112 | 121 | 16 120 | 10 111 | 16 | 80 | 160 | 120 | 26 | ||||
6 | 122 | 122 | 27 112 | 27 121 | 72 | 720 | 2160 | 2160 | 702 | 54 | |||
7 | 132 | 132 | 56 122 | 126 131 | 576 | 10080 | 40320 | 50400 | 23688 | 4284 | 182 | ||
8 | 142 | 142 | 240 132 | 2160 141 | 17280 | 483840 | 2419200 | 3628800 | 2298240 | 725760 | 106080 | 2400 | |
9 | 152 | 152 (8-space tessellation) | ∞ 142 | ∞ 151 | ∞ | ||||||||
10 | 162 | 162 (9-space hyperbolic tessellation) | ∞ 152 | ∞ 161 | ∞ |
See also[edit]
- k21 polytope family
- 2k1 polytope family
References[edit]
- 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[edit]
Space | Family | / / | ||||
---|---|---|---|---|---|---|
E2 | Uniform tiling | {3[3]} | δ3 | hδ3 | qδ3 | Hexagonal |
E3 | Uniform convex honeycomb | {3[4]} | δ4 | hδ4 | qδ4 | |
E4 | Uniform 4-honeycomb | {3[5]} | δ5 | hδ5 | qδ5 | 24-cell honeycomb |
E5 | Uniform 5-honeycomb | {3[6]} | δ6 | hδ6 | qδ6 | |
E6 | Uniform 6-honeycomb | {3[7]} | δ7 | hδ7 | qδ7 | 222 |
E7 | Uniform 7-honeycomb | {3[8]} | δ8 | hδ8 | qδ8 | 133 • 331 |
E8 | Uniform 8-honeycomb | {3[9]} | δ9 | hδ9 | qδ9 | 152 • 251 • 521 |
E9 | Uniform 9-honeycomb | {3[10]} | δ10 | hδ10 | qδ10 | |
E10 | Uniform 10-honeycomb | {3[11]} | δ11 | hδ11 | qδ11 | |
En-1 | Uniform (n-1)-honeycomb | {3[n]} | δn | hδn | qδn | 1k2 • 2k1 • k21 |