Uniform 2 k1 polytope
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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
| 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
- 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
Fundamental convex regular and uniform polytopes in dimensions 2–10
| ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Family | An | Bn | I2(p) / Dn | E6 / E7 / E8 / F4 / G2 | Hn | |||||||
| Regular polygon | Triangle | Square | p-gon | Hexagon | Pentagon | |||||||
| Uniform polyhedron | Tetrahedron | Octahedron • Cube | Demicube | Dodecahedron • Icosahedron | ||||||||
| Uniform 4-polytope | 5-cell | 16-cell • Tesseract | Demitesseract | 24-cell | 120-cell • 600-cell | |||||||
| Uniform 5-polytope | 5-simplex | 5-orthoplex • 5-cube | 5-demicube | |||||||||
| Uniform 6-polytope | 6-simplex | 6-orthoplex • 6-cube | 6-demicube | 122 • 221 | ||||||||
| Uniform 7-polytope | 7-simplex | 7-orthoplex • 7-cube | 7-demicube | 132 • 231 • 321 | ||||||||
| Uniform 8-polytope | 8-simplex | 8-orthoplex • 8-cube | 8-demicube | 142 • 241 • 421 | ||||||||
| Uniform 9-polytope | 9-simplex | 9-orthoplex • 9-cube | 9-demicube | |||||||||
| Uniform 10-polytope | 10-simplex | 10-orthoplex • 10-cube | 10-demicube | |||||||||
| Uniform n-polytope | n-simplex | n-orthoplex • n-cube | n-demicube | 1k2 • 2k1 • k21 | n-pentagonal polytope | |||||||
| Topics: Polytope families • Regular polytope • List of regular polytopes and compounds | ||||||||||||
Fundamental convex regular and uniform honeycombs in dimensions 2-9
| ||||||
|---|---|---|---|---|---|---|
| Space | Family | [math]\displaystyle{ {\tilde{A}}_{n-1} }[/math] | [math]\displaystyle{ {\tilde{C}}_{n-1} }[/math] | [math]\displaystyle{ {\tilde{B}}_{n-1} }[/math] | [math]\displaystyle{ {\tilde{D}}_{n-1} }[/math] | [math]\displaystyle{ {\tilde{G}}_2 }[/math] / [math]\displaystyle{ {\tilde{F}}_4 }[/math] / [math]\displaystyle{ {\tilde{E}}_{n-1} }[/math] |
| 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 | |
| En-1 | Uniform (n-1)-honeycomb | {3[n]} | δn | hδn | qδn | 1k2 • 2k1 • k21 |
 |  |
