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, the 3_{21} polytope is a uniform 7-polytope, constructed within the symmetry of the E_{7} group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 7-ic semi-regular figure.^{[1]}
Its Coxeter symbol is 3_{21}, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 3-node sequences.
The rectified 3_{21} is constructed by points at the mid-edges of the 3_{21}. The birectified 3_{21} is constructed by points at the triangle face centers of the 3_{21}. The trirectified 3_{21} is constructed by points at the tetrahedral centers of the 3_{21}, and is the same as the rectified 1_{32}.
These polytopes are part of a family of 127 (2^{7}-1) convex uniform polytopes in 7-dimensions, made of uniform 6-polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .
3_{21} polytope | |
---|---|
Type | Uniform 7-polytope |
Family | k_{21} polytope |
Schläfli symbol | {3,3,3,3^{2,1}} |
Coxeter symbol | 3_{21} |
Coxeter diagram | |
6-faces | 702 total: 126 3_{11} 576 {3^{5}} |
5-faces | 6048: 4032 {3^{4}} 2016 {3^{4}} |
4-faces | 12096 {3^{3}} |
Cells | 10080 {3,3} |
Faces | 4032 {3} |
Edges | 756 |
Vertices | 56 |
Vertex figure | 2_{21} polytope |
Petrie polygon | octadecagon |
Coxeter group | E_{7}, [3^{3,2,1}], order 2903040 |
Properties | convex |
In 7-dimensional geometry, the 3_{21} polytope is a uniform polytope. It has 56 vertices, and 702 facets: 126 3_{11} and 576 6-simplexes.
For visualization this 7-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 56 vertices within an 18-gonal regular polygon (called a Petrie polygon). Its 756 edges are drawn between 3 rings of 18 vertices, and 2 vertices in the center. Specific higher elements (faces, cells, etc.) can also be extracted and drawn on this projection.
The 1-skeleton of the 3_{21} polytope is the Gosset graph.
This polytope, along with the 7-simplex, can tessellate 7-dimensional space, represented by 3_{31} and Coxeter-Dynkin diagram: .
The 56 vertices can be most simply represented in 8-dimensional space, obtained by the 28 permutations of the coordinates and their opposite:
Its construction is based on the E7 group. Coxeter named it as 3_{21} by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 3-node sequence.
The facet information can be extracted from its Coxeter-Dynkin diagram, .
Removing the node on the short branch leaves the 6-simplex, .
Removing the node on the end of the 2-length branch leaves the 6-orthoplex in its alternated form: 3_{11}, .
Every simplex facet touches a 6-orthoplex facet, while alternate facets of the orthoplex touch either a simplex or another orthoplex.
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 2_{21} polytope, .
Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.^{[4]}
E_{7} | k-face | f_{k} | f_{0} | f_{1} | f_{2} | f_{3} | f_{4} | f_{5} | f_{6} | k-figures | notes | |||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
E_{6} | ( ) | f_{0} | 56 | 27 | 216 | 720 | 1080 | 432 | 216 | 72 | 27 | 2_{21} | E_{7}/E_{6} = 72x8!/72x6! = 56 | |
D_{5}A_{1} | { } | f_{1} | 2 | 756 | 16 | 80 | 160 | 80 | 40 | 16 | 10 | 5-demicube | E_{7}/D_{5}A_{1} = 72x8!/16/5!/2 = 756 | |
A_{4}A_{2} | {3} | f_{2} | 3 | 3 | 4032 | 10 | 30 | 20 | 10 | 5 | 5 | rectified 5-cell | E_{7}/A_{4}A_{2} = 72x8!/5!/2 = 4032 | |
A_{3}A_{2}A_{1} | {3,3} | f_{3} | 4 | 6 | 4 | 10080 | 6 | 6 | 3 | 2 | 3 | triangular prism | E_{7}/A_{3}A_{2}A_{1} = 72x8!/4!/3!/2 = 10080 | |
A_{4}A_{1} | {3,3,3} | f_{4} | 5 | 10 | 10 | 5 | 12096 | 2 | 1 | 1 | 2 | isosceles triangle | E_{7}/A_{4}A_{1} = 72x8!/5!/2 = 12096 | |
A_{5}A_{1} | {3,3,3,3} | f_{5} | 6 | 15 | 20 | 15 | 6 | 4032 | * | 1 | 1 | { } | E_{7}/A_{5}A_{1} = 72x8!/6!/2 = 4032 | |
A_{5} | 6 | 15 | 20 | 15 | 6 | * | 2016 | 0 | 2 | E_{7}/A_{5} = 72x8!/6! = 2016 | ||||
A_{6} | {3,3,3,3,3} | f_{6} | 7 | 21 | 35 | 35 | 21 | 10 | 0 | 576 | * | ( ) | E_{7}/A_{6} = 72x8!/7! = 576 | |
D_{6} | {3,3,3,3,4} | 12 | 60 | 160 | 240 | 192 | 32 | 32 | * | 126 | E_{7}/D_{6} = 72x8!/32/6! = 126 |
E7 | E6 / F4 | B7 / 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] |
The 3_{21} is fifth in a dimensional series of semiregular polytopes. Each progressive uniform polytope is constructed vertex figure of the previous polytope. Thorold Gosset identified this series in 1900 as containing all regular polytope facets, containing all simplexes and orthoplexes.
k_{21} figures in n dimensions | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Space | Finite | Euclidean | Hyperbolic | ||||||||
E_{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 | 1,920 | 51,840 | 2,903,040 | 696,729,600 | ∞ | ||||
Graph | - | - | |||||||||
Name | −1_{21} | 0_{21} | 1_{21} | 2_{21} | 3_{21} | 4_{21} | 5_{21} | 6_{21} |
It is in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3_{k1} series. (A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron.)
Space | Finite | Euclidean | Hyperbolic | |||
---|---|---|---|---|---|---|
n | 4 | 5 | 6 | 7 | 8 | 9 |
Coxeter group |
A_{3}A_{1} | A_{5} | D_{6} | E_{7} | =E_{7}^{+} | =E_{7}^{++} |
Coxeter diagram |
||||||
Symmetry | [3^{−1,3,1}] | [3^{0,3,1}] | [[3^{1,3,1}]] = [4,3,3,3,3] |
[3^{2,3,1}] | [3^{3,3,1}] | [3^{4,3,1}] |
Order | 48 | 720 | 46,080 | 2,903,040 | ∞ | |
Graph | - | - | ||||
Name | 3_{1,-1} | 3_{10} | 3_{11} | 3_{21} | 3_{31} | 3_{41} |
Rectified 3_{21} polytope | |
---|---|
Type | Uniform 7-polytope |
Schläfli symbol | t_{1}{3,3,3,3^{2,1}} |
Coxeter symbol | t_{1}(3_{21}) |
Coxeter diagram | |
6-faces | 758 |
5-faces | 44352 |
4-faces | 70560 |
Cells | 48384 |
Faces | 11592 |
Edges | 12096 |
Vertices | 756 |
Vertex figure | 5-demicube prism |
Petrie polygon | octadecagon |
Coxeter group | E_{7}, [3^{3,2,1}], order 2903040 |
Properties | convex |
Its construction is based on the E7 group. Coxeter named it as 3_{21} by its bifurcating Coxeter-Dynkin diagram, with a single node on the end of the 3-node sequence.
The facet information can be extracted from its Coxeter-Dynkin diagram, .
Removing the node on the short branch leaves the 6-simplex, .
Removing the node on the end of the 2-length branch leaves the rectified 6-orthoplex in its alternated form: t_{1}3_{11}, .
Removing the node on the end of the 3-length branch leaves the 2_{21}, .
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 5-demicube prism, .
E7 | E6 / F4 | B7 / 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] |
Birectified 3_{21} polytope | |
---|---|
Type | Uniform 7-polytope |
Schläfli symbol | t_{2}{3,3,3,3^{2,1}} |
Coxeter symbol | t_{2}(3_{21}) |
Coxeter diagram | |
6-faces | 758 |
5-faces | 12348 |
4-faces | 68040 |
Cells | 161280 |
Faces | 161280 |
Edges | 60480 |
Vertices | 4032 |
Vertex figure | 5-cell-triangle duoprism |
Petrie polygon | octadecagon |
Coxeter group | E_{7}, [3^{3,2,1}], order 2903040 |
Properties | convex |
Its construction is based on the E7 group. Coxeter named it as 3_{21} by its bifurcating Coxeter-Dynkin diagram, with a single node on the end of the 3-node sequence.
The facet information can be extracted from its Coxeter-Dynkin diagram, .
Removing the node on the short branch leaves the birectified 6-simplex, .
Removing the node on the end of the 2-length branch leaves the birectified 6-orthoplex in its alternated form: t_{2}(3_{11}), .
Removing the node on the end of the 3-length branch leaves the rectified 2_{21} polytope in its alternated form: t_{1}(2_{21}), .
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes rectified 5-cell-triangle duoprism, .
E7 | E6 / F4 | B7 / 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] |