4_{21} |
1_{42} |
2_{41} |
Rectified 4_{21} |
Rectified 1_{42} |
Rectified 2_{41} |
Birectified 4_{21} |
Trirectified 4_{21} | |
Orthogonal projections in E_{6} Coxeter plane |
---|
In 8-dimensional geometry, the 2_{41} is a uniform 8-polytope, constructed within the symmetry of the E_{8} group.
Its Coxeter symbol is 2_{41}, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequences.
The rectified 2_{41} is constructed by points at the mid-edges of the 2_{41}. The birectified 2_{41} is constructed by points at the triangle face centers of the 2_{41}, and is the same as the rectified 1_{42}.
These polytopes are part of a family of 255 (2^{8} − 1) convex uniform polytopes in 8-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .
2_{41} polytope | |
---|---|
Type | Uniform 8-polytope |
Family | 2_{k1} polytope |
Schläfli symbol | {3,3,3^{4,1}} |
Coxeter symbol | 2_{41} |
Coxeter diagram | |
7-faces | 17520: 240 2_{31} 17280 {3^{6}} |
6-faces | 144960: 6720 2_{21} 138240 {3^{5}} |
5-faces | 544320: 60480 2_{11} 483840 {3^{4}} |
4-faces | 1209600: 241920 {2_{01} 967680 {3^{3}} |
Cells | 1209600 {3^{2}} |
Faces | 483840 {3} |
Edges | 69120 |
Vertices | 2160 |
Vertex figure | 1_{41} |
Petrie polygon | 30-gon |
Coxeter group | E_{8}, [3^{4,2,1}] |
Properties | convex |
The 2_{41} is composed of 17,520 facets (240 2_{31} polytopes and 17,280 7-simplices), 144,960 6-faces (6,720 2_{21} polytopes and 138,240 6-simplices), 544,320 5-faces (60,480 2_{11} and 483,840 5-simplices), 1,209,600 4-faces (4-simplices), 1,209,600 cells (tetrahedra), 483,840 faces (triangles), 69,120 edges, and 2160 vertices. Its vertex figure is a 7-demicube.
This polytope is a facet in the uniform tessellation, 2_{51} with Coxeter-Dynkin diagram:
The 2160 vertices can be defined as follows:
It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space.
The facet information can be extracted from its Coxeter-Dynkin diagram: .
Removing the node on the short branch leaves the 7-simplex: . There are 17280 of these facets
Removing the node on the end of the 4-length branch leaves the 2_{31}, . There are 240 of these facets. They are centered at the positions of the 240 vertices in the 4_{21} polytope.
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 7-demicube, 1_{41}, .
Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.^{[3]}
E_{8} | k-face | f_{k} | f_{0} | f_{1} | f_{2} | f_{3} | f_{4} | f_{5} | f_{6} | f_{7} | k-figure | notes | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
D_{7} | ( ) | f_{0} | 2160 | 64 | 672 | 2240 | 560 | 2240 | 280 | 1344 | 84 | 448 | 14 | 64 | h{4,3,3,3,3,3} | E_{8}/D_{7} = 192*10!/64/7! = 2160 | |
A_{6}A_{1} | { } | f_{1} | 2 | 69120 | 21 | 105 | 35 | 140 | 35 | 105 | 21 | 42 | 7 | 7 | r{3,3,3,3,3} | E_{8}/A_{6}A_{1} = 192*10!/7!/2 = 69120 | |
A_{4}A_{2}A_{1} | {3} | f_{2} | 3 | 3 | 483840 | 10 | 5 | 20 | 10 | 20 | 10 | 10 | 5 | 2 | {}x{3,3,3} | E_{8}/A_{4}A_{2}A_{1} = 192*10!/5!/3!/2 = 483840 | |
A_{3}A_{3} | {3,3} | f_{3} | 4 | 6 | 4 | 1209600 | 1 | 4 | 4 | 6 | 6 | 4 | 4 | 1 | {3,3}V( ) | E_{8}/A_{3}A_{3} = 192*10!/4!/4! = 1209600 | |
A_{4}A_{3} | {3,3,3} | f_{4} | 5 | 10 | 10 | 5 | 241920 | * | 4 | 0 | 6 | 0 | 4 | 0 | {3,3} | E_{8}/A_{4}A_{3} = 192*10!/5!/4! = 241920 | |
A_{4}A_{2} | 5 | 10 | 10 | 5 | * | 967680 | 1 | 3 | 3 | 3 | 3 | 1 | {3}V( ) | E_{8}/A_{4}A_{2} = 192*10!/5!/3! = 967680 | |||
D_{5}A_{2} | {3,3,3^{1,1}} | f_{5} | 10 | 40 | 80 | 80 | 16 | 16 | 60480 | * | 3 | 0 | 3 | 0 | {3} | E_{8}/D_{5}A_{2} = 192*10!/16/5!/2 = 40480 | |
A_{5}A_{1} | {3,3,3,3} | 6 | 15 | 20 | 15 | 0 | 6 | * | 483840 | 1 | 2 | 2 | 1 | { }V( ) | E_{8}/A_{5}A_{1} = 192*10!/6!/2 = 483840 | ||
E_{6}A_{1} | {3,3,3^{2,1}} | f_{6} | 27 | 216 | 720 | 1080 | 216 | 432 | 27 | 72 | 6720 | * | 2 | 0 | { } | E_{8}/E_{6}A_{1} = 192*10!/72/6! = 6720 | |
A_{6} | {3,3,3,3,3} | 7 | 21 | 35 | 35 | 0 | 21 | 0 | 7 | * | 138240 | 1 | 1 | E_{8}/A_{6} = 192*10!/7! = 138240 | |||
E_{7} | {3,3,3^{3,1}} | f_{7} | 126 | 2016 | 10080 | 20160 | 4032 | 12096 | 756 | 4032 | 56 | 576 | 240 | * | ( ) | E_{8}/E_{7} = 192*10!/72!/8! = 240 | |
A_{7} | {3,3,3,3,3,3} | 8 | 28 | 56 | 70 | 0 | 56 | 0 | 28 | 0 | 8 | * | 17280 | E_{8}/A_{7} = 192*10!/8! = 17280 |
Petrie polygon projections can be 12, 18, or 30-sided based on the E6, E7, and E8 symmetries. The 2160 vertices are all displayed, but lower symmetry forms have projected positions overlapping, shown as different colored vertices. For comparison, a B6 coxeter group is also shown.
E8 [30] |
[20] | [24] |
---|---|---|
(1) |
||
E7 [18] |
E6 [12] |
[6] |
(1,8,24,32) |
D3 / B2 / A3 [4] |
D4 / B3 / A2 [6] |
D5 / B4 [8] |
---|---|---|
D6 / B5 / A4 [10] |
D7 / B6 [12] |
D8 / B7 / A6 [14] |
(1,3,9,12,18,21,36) |
||
B8 [16/2] |
A5 [6] |
A7 [8] |
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_{41} polytope | |
---|---|
Type | Uniform 8-polytope |
Schläfli symbol | t_{1}{3,3,3^{4,1}} |
Coxeter symbol | t_{1}(2_{41}) |
Coxeter diagram | |
7-faces | 19680 total:
240 t_{1}(2_{21}) |
6-faces | 313440 |
5-faces | 1693440 |
4-faces | 4717440 |
Cells | 7257600 |
Faces | 5322240 |
Edges | 19680 |
Vertices | 69120 |
Vertex figure | rectified 6-simplex prism |
Petrie polygon | 30-gon |
Coxeter group | E_{8}, [3^{4,2,1}] |
Properties | convex |
The rectified 2_{41} is a rectification of the 2_{41} polytope, with vertices positioned at the mid-edges of the 2_{41}.
It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space, defined by root vectors of the E_{8} Coxeter group.
The facet information can be extracted from its Coxeter-Dynkin diagram: .
Removing the node on the short branch leaves the rectified 7-simplex: .
Removing the node on the end of the 4-length branch leaves the rectified 2_{31}, .
Removing the node on the end of the 2-length branch leaves the 7-demicube, 1_{41}.
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the rectified 6-simplex prism, .
Petrie polygon projections can be 12, 18, or 30-sided based on the E6, E7, and E8 symmetries. The 2160 vertices are all displayed, but lower symmetry forms have projected positions overlapping, shown as different colored vertices. For comparison, a B6 coxeter group is also shown.
E8 [30] |
[20] | [24] |
---|---|---|
(1) |
||
E7 [18] |
E6 [12] |
[6] |
(1,8,24,32) |
D3 / B2 / A3 [4] |
D4 / B3 / A2 [6] |
D5 / B4 [8] |
---|---|---|
D6 / B5 / A4 [10] |
D7 / B6 [12] |
D8 / B7 / A6 [14] |
(1,3,9,12,18,21,36) |
||
B8 [16/2] |
A5 [6] |
A7 [8] |