List of E6 polytopes
Encyclopedia
221 |
122 |
In 6-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 39 uniform polytopes with E6 symmetry. The two simplest forms are the 221 and 122 polytopes, composed of 27 and 72 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 E6 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 39 polytopes can be made in the E6, D5, D4, D2, A5, A4, A3 Coxeter planes. Ak has k+1 symmetry, Dk has 2(k-1) symmetry, and E6 has 12 symmetry.
Six symmetry planes graphs are shown for 9 of the 39 polytopes in the E6 symmetry. The vertices and edges drawn with 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... Names |
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| E6 / F4 [12] |
D5 / B4 [8] |
D4 / B3 / G2 / A2 [6] |
A5 [6] |
B6 [12/2] |
D6 / B5 / A4 [10/2] |
D3 / B2 / A3 [4] |
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| 1 | 221 Icosihepta-heptacontidipeton (jak) |
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| 2 | Rectified 221 Rectified icosihepta-heptacontidipeton (rojak) |
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| 3 | Rectified 122 / Birectified 221 Rectified pentacontatetrapeton (ram) |
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| 4 | Trirectified 221 Trirectified icosihepta-heptacontidipeton (harjak) |
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| 5 | 122 Pentacontatetrapeton (mo) |
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| 6 | Truncated 221 Truncated icosihepta-heptacontidipeton (tojak) |
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| 7 | Cantellated 221 Cantellated icosihepta-heptacontidipeton |
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| 8 | Bicantellated 221 / Birectified 122 Birectified pentacontatetrapeton (barm) |
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| 9 | Truncated pentacontatetrapeton (tim) |
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