1_{22} |
Rectified 1_{22} |
Birectified 1_{22} |
2_{21} |
Rectified 2_{21} | |
orthogonal projections in E_{6} Coxeter plane |
---|
In 6-dimensional geometry, the 1_{22} polytope is a uniform polytope, constructed from the E_{6} group. It was first published in E. L. Elte's 1912 listing of semiregular polytopes, named as V_{72} (for its 72 vertices).^{[1]}
Its Coxeter symbol is 1_{22}, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequence. There are two rectifications of the 1_{22}, construcated by positions points on the elements of 1_{22}. The rectified 1_{22} is constructed by points at the mid-edges of the 1_{22}. The birectified 1_{22} is constructed by points at the triangle face centers of the 1_{22}.
These polytopes are from a family of 39 convex uniform polytopes in 6-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .
Contents
1_22 polytope
1_{22} polytope | |
---|---|
Type | Uniform 6-polytope |
Family | 1_{k2} polytope |
Schläfli symbol | {3,3^{2,2}} |
Coxeter symbol | 1_{22} |
Coxeter-Dynkin diagram | or |
5-faces | 54: 27 1_{21} 27 1_{21} |
4-faces | 702: 270 1_{11} 432 1_{20} |
Cells | 2160: 1080 1_{10} 1080 {3,3} |
Faces | 2160 {3} |
Edges | 720 |
Vertices | 72 |
Vertex figure | Birectified 5-simplex: 0_{22} |
Petrie polygon | Dodecagon |
Coxeter group | E_{6}, [[3,3^{2,2}]], order 103680 |
Properties | convex, isotopic |
The 1_22 polytope contains 72 vertices, and 54 5-demicubic facets. It has a birectified 5-simplex vertex figure. Its 72 vertices represent the root vectors of the simple Lie group E_{6}.
Alternate names
- Pentacontatetra-peton (Acronym Mo) - 54-facetted polypeton (Jonathan Bowers)^{[2]}
Images
E6 [12] |
D5 [8] |
D4 / A2 [6] | |
---|---|---|---|
(1,2) |
(1,3) |
(1,9,12) | |
B6 [12/2] |
A5 [6] |
A4 [[5]] = [10] |
A3 / D3 [4] |
(1,2) |
(2,3,6) |
(1,2) |
(1,6,8,12) |
Construction
It is created by a Wythoff construction upon a set of 6 hyperplane mirrors in 6-dimensional space.
The facet information can be extracted from its Coxeter-Dynkin diagram, .
Removing the node on either of 2-length branches leaves the 5-demicube, 1_{31}, .
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 5-simplex, 0_{22}, .
Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.^{[3]}
E_{6} | k-face | f_{k} | f_{0} | f_{1} | f_{2} | f_{3} | f_{4} | f_{5} | k-figure | notes | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
A_{5} | ( ) | f_{0} | 72 | 20 | 90 | 60 | 60 | 15 | 15 | 30 | 6 | 6 | r{3,3,3} | E_{6}/A_{5} = 72*6!/6! = 72 | |
A_{2}A_{2}A_{1} | { } | f_{1} | 2 | 720 | 9 | 9 | 9 | 3 | 3 | 9 | 3 | 3 | {3}x{3} | E_{6}/A_{2}A_{2}A_{1} = 72*6!/3!/3!/2 = 720 | |
A_{2}A_{1}A_{1} | {3} | f_{2} | 3 | 3 | 2160 | 2 | 2 | 1 | 1 | 4 | 2 | 2 | s{2,4} | E_{6}/A_{2}A_{1}A_{1} = 72*6!/3!/2/2 = 2160 | |
A_{3}A_{1} | {3,3} | f_{3} | 4 | 6 | 4 | 1080 | * | 1 | 0 | 2 | 2 | 1 | { }v( ) | E_{6}/A_{3}A_{1} = 72*6!/4!/2 = 1080 | |
4 | 6 | 4 | * | 1080 | 0 | 1 | 2 | 1 | 2 | ||||||
A_{4}A_{1} | {3,3,3} | f_{4} | 5 | 10 | 10 | 5 | 0 | 216 | * | * | 2 | 0 | { } | E_{6}/A_{4}A_{1} = 72*6!/5!/2 = 216 | |
5 | 10 | 10 | 0 | 5 | * | 216 | * | 0 | 2 | ||||||
D_{4} | h{4,3,3} | 8 | 24 | 32 | 8 | 8 | * | * | 270 | 1 | 1 | E_{6}/D_{4} = 72*6!/8/4! = 270 | |||
D_{5} | h{4,3,3,3} | f_{5} | 16 | 80 | 160 | 80 | 40 | 16 | 0 | 10 | 27 | * | ( ) | E_{6}/D_{5} = 72*6!/16/5! = 27 | |
16 | 80 | 160 | 40 | 80 | 0 | 16 | 10 | * | 27 |
Related complex polyhedron
The regular complex polyhedron _{3}{3}_{3}{4}_{2}, , in has a real representation as the 1_{22} polytope in 4-dimensional space. It has 72 vertices, 216 3-edges, and 54 3{3}3 faces. Its complex reflection group is _{3}[3]_{3}[4]_{2}, order 1296. It has a half-symmetry quasiregular construction as , as a rectification of the Hessian polyhedron, .^{[4]}
Related polytopes and honeycomb
Along with the semiregular polytope, 2_{21}, it is also one of a family of 39 convex uniform polytopes in 6-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .
1_{k2} figures in n dimensions | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Space | Finite | Euclidean | Hyperbolic | ||||||||
n | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |||
Coxeter group |
E_{3}=A_{2}A_{1} | E_{4}=A_{4} | E_{5}=D_{5} | E_{6} | E_{7} | E_{8} | E_{9} = = E_{8}^{+} | E_{10} = = E_{8}^{++} | |||
Coxeter diagram |
|||||||||||
Symmetry (order) |
[3^{−1,2,1}] | [3^{0,2,1}] | [3^{1,2,1}] | [[3^{2,2,1}]] | [3^{3,2,1}] | [3^{4,2,1}] | [3^{5,2,1}] | [3^{6,2,1}] | |||
Order | 12 | 120 | 192 | 103,680 | 2,903,040 | 696,729,600 | ∞ | ||||
Graph | - | - | |||||||||
Name | 1_{−1,2} | 1_{02} | 1_{12} | 1_{22} | 1_{32} | 1_{42} | 1_{52} | 1_{62} |
Geometric folding
The 1_{22} is related to the 24-cell by a geometric folding E6 → F4 of Coxeter-Dynkin diagrams, E6 corresponding to 1_{22} in 6 dimensions, F4 to the 24-cell in 4 dimensions. This can be seen in the Coxeter plane projections. The 24 vertices of the 24-cell are projected in the same two rings as seen in the 1_{22}.
E6/F4 Coxeter planes | |
---|---|
1_{22} |
24-cell |
D4/B4 Coxeter planes | |
1_{22} |
24-cell |
Tessellations
This polytope is the vertex figure for a uniform tessellation of 6-dimensional space, 2_{22}, .
Rectified 1_22 polytope
Rectified 1_{22} | |
---|---|
Type | Uniform 6-polytope |
Schläfli symbol | 2r{3,3,3^{2,1}} r{3,3^{2,2}} |
Coxeter symbol | 0_{221} |
Coxeter-Dynkin diagram | or |
5-faces | 126 |
4-faces | 1566 |
Cells | 6480 |
Faces | 6480 |
Edges | 6480 |
Vertices | 720 |
Vertex figure | 3-3 duoprism prism |
Petrie polygon | Dodecagon |
Coxeter group | E_{6}, [[3,3^{2,2}]], order 103680 |
Properties | convex |
The rectified 1_{22} polytope (also called 0_{221}) can tessellate 6-dimensional space as the Voronoi cell of the E6* honeycomb lattice (dual of E6 lattice).^{[5]}
Alternate names
- Birectified 2_{21} polytope
- Rectified pentacontatetrapeton (acronym Ram) - rectified 54-facetted polypeton (Jonathan Bowers)^{[6]}
Images
Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow.
E6 [12] |
D5 [8] |
D4 / A2 [6] |
B6 [12/2] |
---|---|---|---|
A5 [6] |
A4 [5] |
A3 / D3 [4] | |
Construction
Its construction is based on the E_{6} group and information can be extracted from the ringed Coxeter-Dynkin diagram representing this polytope: .
Removing the ring on the short branch leaves the birectified 5-simplex, .
Removing the ring on the either 2-length branch leaves the birectified 5-orthoplex in its alternated form: t_{2}(2_{11}), .
The vertex figure is determined by removing the ringed node and ringing the neighboring ring. This makes 3-3 duoprism prism, {3}×{3}×{}, .
Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.^{[7]}^{[8]}
E_{6} | k-face | f_{k} | f_{0} | f_{1} | f_{2} | f_{3} | f_{4} | f_{5} | k-figure | notes | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
A_{2}A_{2}A_{1} | ( ) | f_{0} | 720 | 18 | 18 | 18 | 9 | 6 | 18 | 9 | 6 | 9 | 6 | 3 | 6 | 9 | 3 | 2 | 3 | 3 | {3}x{3}x{ } | E_{6}/A_{2}A_{2}A_{1} = 72*6!/3!/3!/2 = 720 | |
A_{1}A_{1}A_{1} | { } | f_{1} | 2 | 6480 | 2 | 2 | 1 | 1 | 4 | 2 | 1 | 2 | 2 | 1 | 2 | 4 | 1 | 1 | 2 | 2 | { }v{ }v( ) | E_{6}/A_{1}A_{1}A_{1} = 72*6!/2/2/2 = 6480 | |
A_{2}A_{1} | {3} | f_{2} | 3 | 3 | 4320 | * | * | 1 | 2 | 1 | 0 | 0 | 2 | 1 | 1 | 2 | 0 | 1 | 2 | 1 | Sphenoid | E_{6}/A_{2}A_{1} = 72*6!/3!/2 = 4320 | |
3 | 3 | * | 4320 | * | 0 | 2 | 0 | 1 | 1 | 1 | 0 | 2 | 2 | 1 | 1 | 1 | 2 | ||||||
A_{2}A_{1}A_{1} | 3 | 3 | * | * | 2160 | 0 | 0 | 2 | 0 | 2 | 0 | 1 | 0 | 4 | 1 | 0 | 2 | 2 | { }∨{ } | E_{6}/A_{2}A_{1}A_{1} = 72*6!/3!/2/2 = 2160 | |||
A_{2}A_{1} | {3,3} | f_{3} | 4 | 6 | 4 | 0 | 0 | 1080 | * | * | * | * | 2 | 1 | 0 | 0 | 0 | 1 | 2 | 0 | { }v( ) | E_{6}/A_{2}A_{1} = 72*6!/3!/2 = 1080 | |
A_{3} | r{3,3} | 6 | 12 | 4 | 4 | 0 | * | 2160 | * | * | * | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | {3} | E_{6}/A_{3} = 72*6!/4! = 2160 | ||
A_{3}A_{1} | 6 | 12 | 4 | 0 | 4 | * | * | 1080 | * | * | 0 | 1 | 0 | 2 | 0 | 0 | 2 | 1 | { }v( ) | E_{6}/A_{3}A_{1} = 72*6!/4!/2 = 1080 | |||
{3,3} | 4 | 6 | 0 | 4 | 0 | * | * | * | 1080 | * | 0 | 0 | 2 | 0 | 1 | 1 | 0 | 2 | |||||
r{3,3} | 6 | 12 | 0 | 4 | 4 | * | * | * | * | 1080 | 0 | 0 | 0 | 2 | 1 | 0 | 1 | 2 | |||||
A_{4} | r{3,3,3} | f_{4} | 10 | 30 | 20 | 10 | 0 | 5 | 5 | 0 | 0 | 0 | 432 | * | * | * | * | 1 | 1 | 0 | { } | E_{6}/A_{4} = 72*6!/5! = 432 | |
A_{4}A_{1} | 10 | 30 | 20 | 0 | 10 | 5 | 0 | 5 | 0 | 0 | * | 216 | * | * | * | 0 | 2 | 0 | E_{6}/A_{4}A_{1} = 72*6!/5!/2 = 216 | ||||
A_{4} | 10 | 30 | 10 | 20 | 0 | 0 | 5 | 0 | 5 | 0 | * | * | 432 | * | * | 1 | 0 | 1 | E_{6}/A_{4} = 72*6!/5! = 432 | ||||
D_{4} | h{4,3,3} | 24 | 96 | 32 | 32 | 32 | 0 | 8 | 8 | 0 | 8 | * | * | * | 270 | * | 0 | 1 | 1 | E_{6}/D_{4} = 72*6!/8/4! = 270 | |||
A_{4}A_{1} | r{3,3,3} | 10 | 30 | 0 | 20 | 10 | 0 | 0 | 0 | 5 | 5 | * | * | * | * | 216 | 0 | 0 | 2 | E_{6}/A_{4}A_{1} = 72*6!/5!/2 = 216 | |||
A_{5} | 2r{3,3,3,3} | f_{5} | 20 | 90 | 60 | 60 | 0 | 15 | 30 | 0 | 15 | 0 | 6 | 0 | 6 | 0 | 0 | 72 | * | * | ( ) | E_{6}/A_{5} = 72*6!/6! = 72 | |
D_{5} | rh{4,3,3,3} | 80 | 480 | 320 | 160 | 160 | 80 | 80 | 80 | 0 | 40 | 16 | 16 | 0 | 10 | 0 | * | 27 | * | E_{6}/D_{5} = 72*6!/16/5! = 27 | |||
80 | 480 | 160 | 320 | 160 | 0 | 80 | 40 | 80 | 80 | 0 | 0 | 16 | 10 | 16 | * | * | 27 |
Truncated 1_22 polytope
Truncated 1_{22} | |
---|---|
Type | Uniform 6-polytope |
Schläfli symbol | t{3,3^{2,2}} |
Coxeter symbol | t(1_{22}) |
Coxeter-Dynkin diagram | or |
5-faces | 72+27+27 |
4-faces | 32+216+432+270+216 |
Cells | 1080+2160+1080+1080+1080 |
Faces | 4320+4320+2160 |
Edges | 6480+720 |
Vertices | 1440 |
Vertex figure | ( )v{3}x{3} |
Petrie polygon | Dodecagon |
Coxeter group | E_{6}, [[3,3^{2,2}]], order 103680 |
Properties | convex |
Alternate names
- Truncated 1_{22} polytope
Construction
Its construction is based on the E_{6} group and information can be extracted from the ringed Coxeter-Dynkin diagram representing this polytope: .
Images
Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow.
E6 [12] |
D5 [8] |
D4 / A2 [6] |
B6 [12/2] |
---|---|---|---|
A5 [6] |
A4 [5] |
A3 / D3 [4] | |
Birectified 1_22 polytope
Birectified 1_{22} polytope | |
---|---|
Type | Uniform 6-polytope |
Schläfli symbol | 2r{3,3^{2,2}} |
Coxeter symbol | 2r(1_{22}) |
Coxeter-Dynkin diagram | or |
5-faces | 126 |
4-faces | 2286 |
Cells | 10800 |
Faces | 19440 |
Edges | 12960 |
Vertices | 2160 |
Vertex figure | |
Coxeter group | E_{6}, [[3,3^{2,2}]], order 103680 |
Properties | convex |
Alternate names
- Bicantellated 2_{21}
- Birectified pentacontitetrapeton (barm) (Jonathan Bowers)^{[9]}
Images
Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow.
E6 [12] |
D5 [8] |
D4 / A2 [6] |
B6 [12/2] |
---|---|---|---|
A5 [6] |
A4 [5] |
A3 / D3 [4] | |
Trirectified 1_22 polytope
Trirectified 1_{22} polytope | |
---|---|
Type | Uniform 6-polytope |
Schläfli symbol | 3r{3,3^{2,2}} |
Coxeter symbol | 3r(1_{22}) |
Coxeter-Dynkin diagram | or |
5-faces | 558 |
4-faces | 4608 |
Cells | 8640 |
Faces | 6480 |
Edges | 2160 |
Vertices | 270 |
Vertex figure | |
Coxeter group | E_{6}, [[3,3^{2,2}]], order 103680 |
Properties | convex |
Alternate names
- Tricantellated 2_{21}
- Trirectified pentacontitetrapeton (trim) (Jonathan Bowers)^{[10]}
See also
Notes
- ^ Elte, 1912
- ^ Klitzing, (o3o3o3o3o *c3x - mo)
- ^ Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203
- ^ Coxeter, H. S. M., Regular Complex Polytopes, second edition, Cambridge University Press, (1991). p.30 and p.47
- ^ The Voronoi Cells of the E6* and E7* Lattices, Edward Pervin
- ^ Klitzing, (o3o3x3o3o *c3o - ram)
- ^ Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203
- ^ Klitzing, Richard. "6D convex uniform polypeta o3o3x3o3o *c3o - ram".
- ^ Klitzing, (o3x3o3x3o *c3o - barm)
- ^ Klitzing, (x3o3o3o3x *c3o - trim^{[permanent dead link]})
References
- Elte, E. L. (1912), The Semiregular Polytopes of the Hyperspaces, Groningen: University of Groningen
- H. S. M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
- Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] See p334 (figure 3.6a) by Peter mcMullen: (12-gonal node-edge graph of 1_{22})
- Klitzing, Richard. "6D uniform polytopes (polypeta)". o3o3o3o3o *c3x - mo, o3o3x3o3o *c3o - ram, o3x3o3x3o *c3o - barm