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} 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 dimensional | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
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<sup>1,3,1</sup>]] = [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] |
In six-dimensional geometry, a uniform polypeton is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes.
In geometry, the Gosset–Elte figures, named by Coxeter after Thorold Gosset and E. L. Elte, are a group of uniform polytopes which are not regular, generated by a Wythoff construction with mirrors all related by order-2 and order-3 dihedral angles. They can be seen as one-end-ringed Coxeter–Dynkin diagrams.
In geometry, a uniform k_{21} polytope is a polytope in k + 4 dimensions constructed from the E_{n} Coxeter group, and having only regular polytope facets. The family was named by their Coxeter symbol k_{21} by its bifurcating Coxeter–Dynkin diagram, with a single ring on the end of the k-node sequence.
In five-dimensional geometry, a rectified 5-simplex is a convex uniform 5-polytope, being a rectification of the regular 5-simplex.
In 7-dimensional geometry, 2_{31} is a uniform polytope, constructed from the E7 group.
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).
In 7-dimensional geometry, 1_{32} is a uniform polytope, constructed from the E7 group.
In 8-dimensional geometry, the 1_{42} is a uniform 8-polytope, constructed within the symmetry of the E_{8} group.
In 8-dimensional geometry, the 2_{41} is a uniform 8-polytope, constructed within the symmetry of the E_{8} group.
In 6-dimensional geometry, the 2_{21} polytope is a uniform 6-polytope, constructed within the symmetry of the E_{6} group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 6-ic semi-regular figure. It is also called the Schläfli polytope.
In 8-dimensional geometry, the 4_{21} is a semiregular uniform 8-polytope, constructed within the symmetry of the E_{8} group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 8-ic semi-regular figure.
In geometry, the 2_{22} honeycomb is a uniform tessellation of the six-dimensional Euclidean space. It can be represented by the Schläfli symbol {3,3,3^{2,2}}. It is constructed from 2_{21} facets and has a 1_{22} vertex figure, with 54 2_{21} polytopes around every vertex.
In 7-dimensional geometry, the 3_{31} honeycomb is a uniform honeycomb, also given by Schläfli symbol {3,3,3,3^{3,1}} and is composed of 3_{21} and 7-simplex facets, with 56 and 576 of them respectively around each vertex.
In 7-dimensional geometry, 1_{33} is a uniform honeycomb, also given by Schläfli symbol {3,3^{3,3}}, and is composed of 1_{32} facets.
In six-dimensional geometry, a pentellated 6-simplex is a convex uniform 6-polytope with 5th order truncations of the regular 6-simplex.
In six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex.
In geometry, the 5_{21} honeycomb is a uniform tessellation of 8-dimensional Euclidean space. The symbol 5_{21} is from Coxeter, named for the length of the 3 branches of its Coxeter-Dynkin diagram.
In seven-dimensional geometry, a rectified 7-simplex is a convex uniform 7-polytope, being a rectification of the regular 7-simplex.
In eight-dimensional geometry, a rectified 8-simplex is a convex uniform 8-polytope, being a rectification of the regular 8-simplex.
In geometry, an E_{9} honeycomb is a tessellation of uniform polytopes in hyperbolic 9-dimensional space. , also (E_{10}) is a paracompact hyperbolic group, so either facets or vertex figures will not be bounded.