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
[edit]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
[edit]n | 2k1 | Petrie polygon projection |
Name Coxeter-Dynkin diagram |
Facets | Elements | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2k-1,1 polytope | (n-1)-simplex | Vertices | Edges | Faces | Cells | 4-faces | 5-faces | 6-faces | 7-faces | ||||
4 | 201 | 5-cell {32,0,1} |
-- | 5 {33} |
5 | 10 | 10 |
5 | |||||
5 | 211 | pentacross {32,1,1} |
16 {32,0,1} |
16 {34} |
10 | 40 | 80 |
80 |
32 |
||||
6 | 221 | 2 21 polytope {32,2,1} |
27 {32,1,1} |
72 {35} |
27 | 216 | 720 |
1080 |
648 |
99 |
|||
7 | 231 | 2 31 polytope {32,3,1} |
56 {32,2,1} |
576 {36} |
126 | 2016 | 10080 |
20160 |
16128 |
4788 |
632 |
||
8 | 241 | 2 41 polytope {32,4,1} |
240 {32,3,1} |
17280 {37} |
2160 | 69120 | 483840 |
1209600 |
1209600 |
544320 |
144960 |
17520 | |
9 | 251 | 2 51 honeycomb (8-space tessellation) {32,5,1} |
∞ {32,4,1} |
∞ {38} |
∞ | ||||||||
10 | 261 | 2 61 honeycomb (9-space tessellation) {32,6,1} |
∞ {32,5,1} |
∞ {39} |
∞ |
See also
[edit]- k21 polytope family
- 1k2 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 | 0[3] | δ3 | hδ3 | qδ3 | Hexagonal |
E3 | Uniform convex honeycomb | 0[4] | δ4 | hδ4 | qδ4 | |
E4 | Uniform 4-honeycomb | 0[5] | δ5 | hδ5 | qδ5 | 24-cell honeycomb |
E5 | Uniform 5-honeycomb | 0[6] | δ6 | hδ6 | qδ6 | |
E6 | Uniform 6-honeycomb | 0[7] | δ7 | hδ7 | qδ7 | 222 |
E7 | Uniform 7-honeycomb | 0[8] | δ8 | hδ8 | qδ8 | 133 • 331 |
E8 | Uniform 8-honeycomb | 0[9] | δ9 | hδ9 | qδ9 | 152 • 251 • 521 |
E9 | Uniform 9-honeycomb | 0[10] | δ10 | hδ10 | qδ10 | |
E10 | Uniform 10-honeycomb | 0[11] | δ11 | hδ11 | qδ11 | |
En-1 | Uniform (n-1)-honeycomb | 0[n] | δn | hδn | qδn | 1k2 • 2k1 • k21 |