List of A5 polytopes
Encyclopedia
5-simplex |
In 5-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 19 uniform polytopes with A5 symmetry. There is one self-dual regular form, the 5-simplex with 6 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 A5 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 19 polytopes can be made in the 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 19 polytopes are each shown in these 4 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 Name |
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|---|---|---|---|---|---|
| [6] | [5] | [4] | [3] | ||
| A5 | A4 | A3 | A2 | ||
| 1 | t0{3,3,3,3} 5-simplex (hix) |
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| 2 | t1{3,3,3,3} Rectified 5-simplex (rix) |
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| 3 | t2{3,3,3,3} Birectified 5-simplex (dot) |
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| 4 | t0,1{3,3,3,3} Truncated 5-simplex (tix) |
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| 5 | t1,2{3,3,3,3} Bitruncated 5-simplex (bittix) |
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| 6 | t0,2{3,3,3,3} Cantellated 5-simplex (sarx) |
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| 7 | t1,3{3,3,3,3} Bicantellated 5-simplex (sibrid) |
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| 8 | t0,3{3,3,3,3} Runcinated 5-simplex Runcinated 5-simplex In six-dimensional geometry, a runcinated 5-simplex is a convex uniform 5-polytope with 3rd order truncations of the regular 5-simplex.There are 4 unique runcinations of the 5-simplex with permutations of truncations, and cantellations.... (spix) |
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| 9 | t0,4{3,3,3,3} Stericated 5-simplex (scad) |
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| 10 | t0,1,2{3,3,3,3} Cantitruncated 5-simplex (garx) |
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| 11 | t1,2,3{3,3,3,3} Bicantitruncated 5-simplex (gibrid) |
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| 12 | t0,1,3{3,3,3,3} Runcitruncated 5-simplex (pattix) |
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| 13 | t0,2,3{3,3,3,3} Runcicantellated 5-simplex (pirx) |
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| 14 | t0,1,4{3,3,3,3} Steritruncated 5-simplex (cappix) |
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| 15 | t0,2,4{3,3,3,3} Stericantellated 5-simplex (card) |
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| 16 | t0,1,2,3{3,3,3,3} Runcicantitruncated 5-simplex (gippix) |
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| 17 | t0,1,2,4{3,3,3,3} Stericantitruncated 5-simplex (cograx) |
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| 18 | t0,1,3,4{3,3,3,3} Steriruncitruncated 5-simplex (captid) |
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| 19 | t0,1,2,3,4{3,3,3,3} Omnitruncated 5-simplex (gocad) |