List of A7 polytopes
Encyclopedia
7-simplex |
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 71 uniform polytopes with A7 symmetry. There is one self-dual regular form, the 7-simplex with 8 vertices.
Each 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 A7 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 135 polytopes can be made in the A7, A6, A5, A4, A3, A2 Coxeter planes. Ak has [k+1] symmetry. For even k and symmetrically ringed-diagrams, symmetry doubles to [2(k+1)].
These 63 polytopes are each shown in these 6 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.
| # | 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 Johnson name |
Ak orthogonal projection graphs | |||||
|---|---|---|---|---|---|---|---|
| A7 [8] |
A6 [7] |
A5 [6] |
A4 [5] |
A3 [4] |
A2 [3] |
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| 1 | t0{3,3,3,3,3,3} 7-simplex |
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| 2 | t1{3,3,3,3,3,3} Rectified 7-simplex Rectified 7-simplex In seven-dimensional geometry, a rectified 7-simplex is a convex uniform 7-polytope, being a rectification of the regular 7-simplex.There are four unique degrees of rectifications, including the zeroth, the 7-simplex itself. Vertices of the rectified 7-simplex are located at the edge-centers of the... |
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| 3 | t2{3,3,3,3,3,3} Birectified 7-simplex |
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| 4 | t3{3,3,3,3,3,3} Trirectified 7-simplex |
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| 5 | t0,1{3,3,3,3,3,3} Truncated 7-simplex Truncated 7-simplex In seven-dimensional geometry, a truncated 7-simplex is a convex uniform 7-polytope, being a truncation of the regular 7-simplex.There are unique 3 degrees of truncation. Vertices of the truncation 7-simplex are located as pairs on the edge of the 7-simplex. Vertices of the bitruncated 7-simplex... |
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| 6 | t0,2{3,3,3,3,3,3} Cantellated 7-simplex Cantellated 7-simplex In seven-dimensional geometry, a cantellated 7-simplex is a convex uniform 7-polytope, being a cantellation of the regular 7-simplex.There are unique 6 degrees of cantellation for the 7-simplex, including truncations.- Cantellated 7-simplex:... |
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| 7 | t1,2{3,3,3,3,3,3} Bitruncated 7-simplex |
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| 8 | t0,3{3,3,3,3,3,3} Runcinated 7-simplex Runcinated 7-simplex In seven-dimensional geometry, a runcinated 7-simplex is a convex uniform 7-polytope with 3rd order truncations of the regular 7-simplex.There are 8 unique runcinations of the 7-simplex with permutations of truncations, and cantellations.... |
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| 9 | t1,3{3,3,3,3,3,3} Bicantellated 7-simplex |
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| 10 | t2,3{3,3,3,3,3,3} Tritruncated 7-simplex |
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| 11 | t0,4{3,3,3,3,3,3} Stericated 7-simplex Stericated 7-simplex In seven-dimensional geometry, a stericated 7-simplex is a convex uniform 7-polytope with 4th order truncations of the regular 7-simplex.... |
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| 12 | t1,4{3,3,3,3,3,3} Biruncinated 7-simplex |
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| 13 | t2,4{3,3,3,3,3,3} Tricantellated 7-simplex |
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| 14 | t0,5{3,3,3,3,3,3} Pentellated 7-simplex Pentellated 7-simplex In seven-dimensional geometry, a pentellated 7-simplex is a convex uniform 7-polytope with 5th order truncations of the regular 7-simplex.... |
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| 15 | t1,5{3,3,3,3,3,3} Bistericated 7-simplex |
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| 16 | t0,6{3,3,3,3,3,3} Hexicated 7-simplex Hexicated 7-simplex In seven-dimensional geometry, a hexicated 7-simplex is a convex uniform 7-polytope, including 6th-order truncations from the regular 7-simplex.... |
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| 17 | t0,1,2{3,3,3,3,3,3} Cantitruncated 7-simplex |
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| 18 | t0,1,3{3,3,3,3,3,3} Runcitruncated 7-simplex |
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| 19 | t0,2,3{3,3,3,3,3,3} Runcicantellated 7-simplex |
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| 20 | t1,2,3{3,3,3,3,3,3} Bicantitruncated 7-simplex |
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| 21 | t0,1,4{3,3,3,3,3,3} Steritruncated 7-simplex |
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| 22 | t0,2,4{3,3,3,3,3,3} Stericantellated 7-simplex |
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| 23 | t1,2,4{3,3,3,3,3,3} Biruncitruncated 7-simplex |
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| 24 | t0,3,4{3,3,3,3,3,3} Steriruncinated 7-simplex |
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| 25 | t1,3,4{3,3,3,3,3,3} Biruncicantellated 7-simplex |
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| 26 | t2,3,4{3,3,3,3,3,3} Tricantitruncated 7-simplex |
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| 27 | t0,1,5{3,3,3,3,3,3} Pentitruncated 7-simplex |
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| 28 | t0,2,5{3,3,3,3,3,3} Penticantellated 7-simplex |
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| 29 | t1,2,5{3,3,3,3,3,3} Bisteritruncated 7-simplex |
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| 30 | t0,3,5{3,3,3,3,3,3} Pentiruncinated 7-simplex |
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| 31 | t1,3,5{3,3,3,3,3,3} Bistericantellated 7-simplex |
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| 32 | t0,4,5{3,3,3,3,3,3} Pentistericated 7-simplex |
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| 33 | t0,1,6{3,3,3,3,3,3} Hexitruncated 7-simplex |
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| 34 | t0,2,6{3,3,3,3,3,3} Hexicantellated 7-simplex |
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| 35 | t0,3,6{3,3,3,3,3,3} Hexiruncinated 7-simplex |
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| 36 | t0,1,2,3{3,3,3,3,3,3} Runcicantitruncated 7-simplex |
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| 37 | t0,1,2,4{3,3,3,3,3,3} Stericantitruncated 7-simplex |
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| 38 | t0,1,3,4{3,3,3,3,3,3} Steriruncitruncated 7-simplex |
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| 39 | t0,2,3,4{3,3,3,3,3,3} Steriruncicantellated 7-simplex |
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| 40 | t1,2,3,4{3,3,3,3,3,3} Biruncicantitruncated 7-simplex |
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| 41 | t0,1,2,5{3,3,3,3,3,3} Penticantitruncated 7-simplex |
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| 42 | t0,1,3,5{3,3,3,3,3,3} Pentiruncitruncated 7-simplex |
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| 43 | t0,2,3,5{3,3,3,3,3,3} Pentiruncicantellated 7-simplex |
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| 44 | t1,2,3,5{3,3,3,3,3,3} Bistericantitruncated 7-simplex |
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| 45 | t0,1,4,5{3,3,3,3,3,3} Pentisteritruncated 7-simplex |
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| 46 | t0,2,4,5{3,3,3,3,3,3} Pentistericantellated 7-simplex |
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| 47 | t1,2,4,5{3,3,3,3,3,3} Bisteriruncitruncated 7-simplex |
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| 48 | t0,3,4,5{3,3,3,3,3,3} Pentisteriruncinated 7-simplex |
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| 49 | t0,1,2,6{3,3,3,3,3,3} Hexicantitruncated 7-simplex |
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| 50 | t0,1,3,6{3,3,3,3,3,3} Hexiruncitruncated 7-simplex |
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| 51 | t0,2,3,6{3,3,3,3,3,3} Hexiruncicantellated 7-simplex |
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| 52 | t0,1,4,6{3,3,3,3,3,3} Hexisteritruncated 7-simplex |
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| 53 | t0,2,4,6{3,3,3,3,3,3} Hexistericantellated 7-simplex |
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| 54 | t0,1,5,6{3,3,3,3,3,3} Hexipentitruncated 7-simplex |
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| 55 | t0,1,2,3,4{3,3,3,3,3,3} Steriruncicantitruncated 7-simplex |
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| 56 | t0,1,2,3,5{3,3,3,3,3,3} Pentiruncicantitruncated 7-simplex |
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| 57 | t0,1,2,4,5{3,3,3,3,3,3} Pentistericantitruncated 7-simplex |
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| 58 | t0,1,3,4,5{3,3,3,3,3,3} Pentisteriruncitruncated 7-simplex |
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| 59 | t0,2,3,4,5{3,3,3,3,3,3} Pentisteriruncicantellated 7-simplex |
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| 60 | t1,2,3,4,5{3,3,3,3,3,3} Bisteriruncicantitruncated 7-simplex |
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| 61 | t0,1,2,3,6{3,3,3,3,3,3} Hexiruncicantitruncated 7-simplex |
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| 62 | t0,1,2,4,6{3,3,3,3,3,3} Hexistericantitruncated 7-simplex |
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| 63 | t0,1,3,4,6{3,3,3,3,3,3} Hexisteriruncitruncated 7-simplex |
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| 64 | t0,2,3,4,6{3,3,3,3,3,3} Hexisteriruncicantellated 7-simplex |
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| 65 | t0,1,2,5,6{3,3,3,3,3,3} Hexipenticantitruncated 7-simplex |
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| 66 | t0,1,3,5,6{3,3,3,3,3,3} Hexipentiruncitruncated 7-simplex |
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| 67 | t0,1,2,3,4,5{3,3,3,3,3,3} Pentisteriruncicantitruncated 7-simplex |
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| 68 | t0,1,2,3,4,6{3,3,3,3,3,3} Hexisteriruncicantitruncated 7-simplex |
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| 69 | t0,1,2,3,5,6{3,3,3,3,3,3} Hexipentiruncicantitruncated 7-simplex |
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| 70 | t0,1,2,4,5,6{3,3,3,3,3,3} Hexipentistericantitruncated 7-simplex |
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| 71 | t0,1,2,3,4,5,6{3,3,3,3,3,3} Omnitruncated 7-simplex |
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