List of E7 polytopes
Encyclopedia
321 |
231 |
132 |
In 7-dimensional geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, there are 127 uniform polytopes with E7 symmetry. The three simplest forms are the 321, 231, and 132 polytopes, composed of 56, 126, and 576 vertices
Vertex (geometry)
In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...
respectively.
They can be visualized as symmetric orthographic projection
Orthographic projection
Orthographic projection is a means of representing a three-dimensional object in two dimensions. It is a form of parallel projection, where all the projection lines are orthogonal to the projection plane, resulting in every plane of the scene appearing in affine transformation on the viewing surface...
s in Coxeter planes of the E7 Coxeter group, and other subgroups.
Graphs
Symmetric orthographic projectionOrthographic projection
Orthographic projection is a means of representing a three-dimensional object in two dimensions. It is a form of parallel projection, where all the projection lines are orthogonal to the projection plane, resulting in every plane of the scene appearing in affine transformation on the viewing surface...
s of these 127 polytopes can be made in the E7, E6, D6, D5, D4, D3, D2, A7, A6, A5, A4, A3, A2 Coxeter planes. Ak has k+1 symmetry, Dk has 2(k-1) symmetry, and E6 and E7 have 12, 18 symmetry respectively.
For 10 of 127 polytopes (7-one nodea_1, and 3 truncations), they are shown in these 9 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.
| # | Coxeter plane graphs | Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors... Schläfli symbol Names |
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| E7 [18] |
E6 / F4 [12] |
A6 / B7 [7x2] |
A5 [6] |
D7 [12/2] |
A4 / D6 [10] |
D5 [8] |
A2 / D4 [6] |
A3 / D3 [4] |
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| 1 | 231 Gosset 2 31 polytope In 7-dimensional geometry, 231 is a uniform polytope, constructed from the E7 group.Coxeter named it 231 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 2-node sequences.... (laq) |
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| 2 | Rectified 231 (rolaq) |
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| 3 | Rectified 132 (rolin) |
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| 4 | 132 (lin) |
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| 5 | Birectified 321 (branq) |
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| 6 | Rectified 321 (ranq) |
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| 7 | 321 Gosset 3 21 polytope In 7-dimensional geometry, the 321 polytope is a uniform 6-polytope, constructed within the symmetry of the E7 group. It was discovered by Thorold Gosset, published in his 1900 paper... (naq) |
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| 8 | Truncated 231 (talq) |
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| 9 | Truncated 132 (tilin) |
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| 10 | Truncated 321 (tanq) |
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