List of A6 polytopes
Encyclopedia
6-simplex |
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 35 uniform polytopes with A6 symmetry. There is one self-dual regular form, the 6-simplex with 7 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 A6 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 35 polytopes can be made in the A6, A5, A4, A3, A2 Coxeter planes. Ak graphs have [k+1] symmetry. For even k and symmetrically nodea_1ed-diagrams, symmetry doubles to [2(k+1)].
These 63 polytopes are each shown in these 5 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.
| # | A6 [7] |
A5 [6] |
A4 [5] |
A3 [4] |
A2 [3] |
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 Name |
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| 1 | t0{3,3,3,3,3} 6-simplex Heptapeton (hop) |
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| 2 | t1{3,3,3,3,3} Rectified 6-simplex Rectified 6-simplex In six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex.There are three unique degrees of rectifications, including the zeroth, the 6-simplex itself. Vertices of the rectified 6-simplex are located at the edge-centers of the... Rectified heptapeton (ril) |
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| 3 | t0,1{3,3,3,3,3} Truncated 6-simplex Truncated 6-simplex In six-dimensional geometry, a truncated 6-simplex is a convex uniform 6-polytope, being a truncation of the regular 6-simplex.There are unique 3 degrees of truncation. Vertices of the truncation 6-simplex are located as pairs on the edge of the 6-simplex. Vertices of the bitruncated 6-simplex are... Truncated heptapeton (til) |
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| 4 | t2{3,3,3,3,3} Birectified 6-simplex Birectified heptapeton (bril) |
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| 5 | t0,2{3,3,3,3,3} Cantellated 6-simplex Cantellated 6-simplex In six-dimensional geometry, a cantellated 6-simplex is a convex uniform 6-polytope, being a cantellation of the regular 6-simplex.There are unique 4 degrees of cantellation for the 6-simplex, including truncations.- Cantellated 6-simplex:... Small rhombated heptapeton (sril) |
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| 6 | t1,2{3,3,3,3,3} Bitruncated 6-simplex Bitruncated heptapeton (batal) |
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| 7 | t0,1,2{3,3,3,3,3} Cantitruncated 6-simplex Great rhombated heptapeton (gril) |
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| 8 | t0,3{3,3,3,3,3} Runcinated 6-simplex Runcinated 6-simplex In six-dimensional geometry, a runcinated 6-simplex is a convex uniform 6-polytope constructed as a runcination of the regular 6-simplex.... Small prismated heptapeton (spil) |
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| 9 | t1,3{3,3,3,3,3} Bicantellated 6-simplex Small birhombated heptapeton (sabril) |
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| 10 | t0,1,3{3,3,3,3,3} Runcitruncated 6-simplex Prismatotruncated heptapeton (patal) |
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| 11 | t2,3{3,3,3,3,3} Tritruncated 6-simplex Tetradecapeton (fe) |
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| 12 | t0,2,3{3,3,3,3,3} Runcicantellated 6-simplex Prismatorhombated heptapeton (pril) |
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| 13 | t1,2,3{3,3,3,3,3} Bicantitruncated 6-simplex Great birhombated heptapeton (gabril) |
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| 14 | t0,1,2,3{3,3,3,3,3} Runcicantitruncated 6-simplex Great prismated heptapeton (gapil) |
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| 15 | t0,4{3,3,3,3,3} Stericated 6-simplex Stericated 6-simplex In six-dimensional geometry, a stericated 6-simplex is a convex uniform 6-polytope with 4th order truncations of the regular 6-simplex.... Small cellated heptapeton (scal) |
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| 16 | t1,4{3,3,3,3,3} Biruncinated 6-simplex Small biprismato-tetradecapeton (sibpof) |
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| 17 | t0,1,4{3,3,3,3,3} Steritruncated 6-simplex cellitruncated heptapeton (catal) |
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| 18 | t0,2,4{3,3,3,3,3} Stericantellated 6-simplex Cellirhombated heptapeton (cral) |
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| 19 | t1,2,4{3,3,3,3,3} Biruncitruncated 6-simplex Biprismatorhombated heptapeton (bapril) |
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| 20 | t0,1,2,4{3,3,3,3,3} Stericantitruncated 6-simplex Celligreatorhombated heptapeton (cagral) |
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| 21 | t0,3,4{3,3,3,3,3} Steriruncinated 6-simplex Celliprismated heptapeton (copal) |
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| 22 | t0,1,3,4{3,3,3,3,3} Steriruncitruncated 6-simplex celliprismatotruncated heptapeton (captal) |
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| 23 | t0,2,3,4{3,3,3,3,3} Steriruncicantellated 6-simplex celliprismatorhombated heptapeton (copril) |
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| 24 | t1,2,3,4{3,3,3,3,3} Biruncicantitruncated 6-simplex Great biprismato-tetradecapeton (gibpof) |
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| 25 | t0,1,2,3,4{3,3,3,3,3} Steriruncicantitruncated 6-simplex Great cellated heptapeton (gacal) |
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| 26 | t0,5{3,3,3,3,3} Pentellated 6-simplex Pentellated 6-simplex In six-dimensional geometry, a pentellated 6-simplex is a convex uniform 6-polytope with 5th order truncations of the regular 6-simplex.There are unique 10 degrees of pentellations of the 6-simplex with permutations of truncations, cantellations, runcinations, and sterications... Small teri-tetradecapeton (staf) |
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| 27 | t0,1,5{3,3,3,3,3} Pentitruncated 6-simplex Tericellated heptapeton (tocal) |
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| 28 | t0,2,5{3,3,3,3,3} Penticantellated 6-simplex Teriprismated heptapeton (tapal) |
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| 29 | t0,1,2,5{3,3,3,3,3} Penticantitruncated 6-simplex Terigreatorhombated heptapeton (togral) |
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| 30 | t0,1,3,5{3,3,3,3,3} Pentiruncitruncated 6-simplex Tericellirhombated heptapeton (tocral) |
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| 31 | t0,2,3,5{3,3,3,3,3} Pentiruncicantellated 6-simplex Teriprismatorhombi-tetradecapeton (taporf) |
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| 32 | t0,1,2,3,5{3,3,3,3,3} Pentiruncicantitruncated 6-simplex Terigreatoprismated heptapeton (tagopal) |
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| 33 | t0,1,4,5{3,3,3,3,3} Pentisteritruncated 6-simplex tericellitrunki-tetradecapeton (tactaf) |
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| 34 | t0,1,2,4,5{3,3,3,3,3} Pentistericantitruncated 6-simplex tericelligreatorhombated heptapeton (tacogral) |
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| 35 | t0,1,2,3,4,5{3,3,3,3,3} Omnitruncated 6-simplex Great teri-tetradecapeton (gotaf) |