3_{21} |
2_{31} |
1_{32} | |||
Rectified 3_{21} |
birectified 3_{21} | ||||
Rectified 2_{31} |
Rectified 1_{32} | ||||
Orthogonal projections in E_{7} Coxeter plane |
---|
In 7-dimensional geometry, 2_{31} is a uniform polytope, constructed from the E7 group.
Its Coxeter symbol is 2_{31}, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node branch.
The rectified 2_{31} is constructed by points at the mid-edges of the 2_{31}.
These polytopes are part of a family of 127 (or 2^{7}−1) convex uniform polytopes in 7-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .
Gosset 2_{31} polytope | |
---|---|
Type | Uniform 7-polytope |
Family | 2_{k1} polytope |
Schläfli symbol | {3,3,3^{3,1}} |
Coxeter symbol | 2_{31} |
Coxeter diagram | |
6-faces | 632: 56 2_{21} 576 {3^{5}} |
5-faces | 4788: 756 2_{11} 4032 {3^{4}} |
4-faces | 16128: 4032 2_{01} 12096 {3^{3}} |
Cells | 20160 {3^{2}} |
Faces | 10080 {3} |
Edges | 2016 |
Vertices | 126 |
Vertex figure | 1_{31} |
Petrie polygon | Octadecagon |
Coxeter group | E_{7}, [3^{3,2,1}] |
Properties | convex |
The 2_{31} is composed of 126 vertices, 2016 edges, 10080 faces (Triangles), 20160 cells (tetrahedra), 16128 4-faces (3-simplexes), 4788 5-faces (756 pentacrosses, and 4032 5-simplexes), 632 6-faces (576 6-simplexes and 56 2_{21}). Its vertex figure is a 6-demicube. Its 126 vertices represent the root vectors of the simple Lie group E_{7}.
This polytope is the vertex figure for a uniform tessellation of 7-dimensional space, 3_{31}.
It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 7-dimensional space.
The facet information can be extracted from its Coxeter-Dynkin diagram, .
Removing the node on the short branch leaves the 6-simplex. There are 576 of these facets. These facets are centered on the locations of the vertices of the 3_{21} polytope, .
Removing the node on the end of the 3-length branch leaves the 2_{21}. There are 56 of these facets. These facets are centered on the locations of the vertices of the 1_{32} polytope, .
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 6-demicube, 1_{31}, .
Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.^{[3]}
E_{7} | k-face | f_{k} | f_{0} | f_{1} | f_{2} | f_{3} | f_{4} | f_{5} | f_{6} | k-figures | notes | ||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
D_{6} | ( ) | f_{0} | 126 | 32 | 240 | 640 | 160 | 480 | 60 | 192 | 12 | 32 | 6-demicube | E_{7}/D_{6} = 72x8!/32/6! = 126 | |
A_{5}A_{1} | { } | f_{1} | 2 | 2016 | 15 | 60 | 20 | 60 | 15 | 30 | 6 | 6 | rectified 5-simplex | E_{7}/A_{5}A_{1} = 72x8!/6!/2 = 2016 | |
A_{3}A_{2}A_{1} | {3} | f_{2} | 3 | 3 | 10080 | 8 | 4 | 12 | 6 | 8 | 4 | 2 | tetrahedral prism | E_{7}/A_{3}A_{2}A_{1} = 72x8!/4!/3!/2 = 10080 | |
A_{3}A_{2} | {3,3} | f_{3} | 4 | 6 | 4 | 20160 | 1 | 3 | 3 | 3 | 3 | 1 | tetrahedron | E_{7}/A_{3}A_{2} = 72x8!/4!/3! = 20160 | |
A_{4}A_{2} | {3,3,3} | f_{4} | 5 | 10 | 10 | 5 | 4032 | * | 3 | 0 | 3 | 0 | {3} | E_{7}/A_{4}A_{2} = 72x8!/5!/3! = 4032 | |
A_{4}A_{1} | 5 | 10 | 10 | 5 | * | 12096 | 1 | 2 | 2 | 1 | Isosceles triangle | E_{7}/A_{4}A_{1} = 72x8!/5!/2 = 12096 | |||
D_{5}A_{1} | {3,3,3,4} | f_{5} | 10 | 40 | 80 | 80 | 16 | 16 | 756 | * | 2 | 0 | { } | E_{7}/D_{5}A_{1} = 72x8!/32/5! = 756 | |
A_{5} | {3,3,3,3} | 6 | 15 | 20 | 15 | 0 | 6 | * | 4032 | 1 | 1 | E_{7}/A_{5} = 72x8!/6! = 72*8*7 = 4032 | |||
E_{6} | {3,3,3^{2,1}} | f_{6} | 27 | 216 | 720 | 1080 | 216 | 432 | 27 | 72 | 56 | * | ( ) | E_{7}/E_{6} = 72x8!/72x6! = 8*7 = 56 | |
A_{6} | {3,3,3,3,3} | 7 | 21 | 35 | 35 | 0 | 21 | 0 | 7 | * | 576 | E_{7}/A_{6} = 72x8!/7! = 72×8 = 576 |
E7 | E6 / F4 | B6 / A6 |
---|---|---|
[18] |
[12] |
[7x2] |
A5 | D7 / B6 | D6 / B5 |
[6] |
[12/2] |
[10] |
D5 / B4 / A4 | D4 / B3 / A2 / G2 | D3 / B2 / A3 |
[8] |
[6] |
[4] |
2_{k1} 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 | [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 | 384 | 51,840 | 2,903,040 | 696,729,600 | ∞ | ||||
Graph | - | - | |||||||||
Name | 2_{−1,1} | 2_{01} | 2_{11} | 2_{21} | 2_{31} | 2_{41} | 2_{51} | 2_{61} |
Rectified 2_{31} polytope | |
---|---|
Type | Uniform 7-polytope |
Family | 2_{k1} polytope |
Schläfli symbol | {3,3,3^{3,1}} |
Coxeter symbol | t_{1}(2_{31}) |
Coxeter diagram | |
6-faces | 758 |
5-faces | 10332 |
4-faces | 47880 |
Cells | 100800 |
Faces | 90720 |
Edges | 30240 |
Vertices | 2016 |
Vertex figure | 6-demicube |
Petrie polygon | Octadecagon |
Coxeter group | E_{7}, [3^{3,2,1}] |
Properties | convex |
The rectified 2_{31} is a rectification of the 2_{31} polytope, creating new vertices on the center of edge of the 2_{31}.
It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 7-dimensional space.
The facet information can be extracted from its Coxeter-Dynkin diagram, .
Removing the node on the short branch leaves the rectified 6-simplex, .
Removing the node on the end of the 2-length branch leaves the, 6-demicube, .
Removing the node on the end of the 3-length branch leaves the rectified 2_{21}, .
The vertex figure is determined by removing the ringed node and ringing the neighboring node.
E7 | E6 / F4 | B6 / A6 |
---|---|---|
[18] |
[12] |
[7x2] |
A5 | D7 / B6 | D6 / B5 |
[6] |
[12/2] |
[10] |
D5 / B4 / A4 | D4 / B3 / A2 / G2 | D3 / B2 / A3 |
[8] |
[6] |
[4] |