 5-simplex    |
In 5-dimensional geometry, 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 projections in the Coxeter planes of the A5 Coxeter group and other subgroups.
Symmetric orthographic projections 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 Schläfli symbol Name | |||
|---|---|---|---|---|---|
| [6] | [5] | [4] | [3] | ||
| A5 | A4 | A3 | A2 | ||
| 1 |  |  |  |  | {3,3,3,3} 5-simplex (hix) |
| 2 |  |  |  |  | t1{3,3,3,3} or r{3,3,3,3} Rectified 5-simplex (rix) |
| 3 |  |  |  |  | t2{3,3,3,3} or 2r{3,3,3,3} Birectified 5-simplex (dot) |
| 4 |  |  |  |  | t0,1{3,3,3,3} or t{3,3,3,3} Truncated 5-simplex (tix) |
| 5 |  |  |  |  | t1,2{3,3,3,3} or 2t{3,3,3,3} Bitruncated 5-simplex (bittix) |
| 6 |  |  |  |  | t0,2{3,3,3,3} or rr{3,3,3,3} Cantellated 5-simplex (sarx) |
| 7 |  |  |  |  | t1,3{3,3,3,3} or 2rr{3,3,3,3} Bicantellated 5-simplex (sibrid) |
| 8 |  |  |  |  | t0,3{3,3,3,3} Runcinated 5-simplex (spix) |
| 9 |  |  |  |  | t0,4{3,3,3,3} or 2r2r{3,3,3,3} Stericated 5-simplex (scad) |
| 10 |  |  |  |  | t0,1,2{3,3,3,3} or tr{3,3,3,3} Cantitruncated 5-simplex (garx) |
| 11 |  |  |  |  | t1,2,3{3,3,3,3} or 2tr{3,3,3,3} Bicantitruncated 5-simplex (gibrid) |
| 12 |  |  |  |  | t0,1,3{3,3,3,3} Runcitruncated 5-simplex (pattix) |
| 13 |  |  |  |  | t0,2,3{3,3,3,3} Runcicantellated 5-simplex (pirx) |
| 14 |  |  |  |  | t0,1,4{3,3,3,3} Steritruncated 5-simplex (cappix) |
| 15 |  |  |  |  | t0,2,4{3,3,3,3} Stericantellated 5-simplex (card) |
| 16 |  |  |  |  | t0,1,2,3{3,3,3,3} Runcicantitruncated 5-simplex (gippix) |
| 17 |  |  |  |  | t0,1,2,4{3,3,3,3} Stericantitruncated 5-simplex (cograx) |
| 18 |  |  |  |  | t0,1,3,4{3,3,3,3} Steriruncitruncated 5-simplex (captid) |
| 19 |  |  |  |  | t0,1,2,3,4{3,3,3,3} Omnitruncated 5-simplex (gocad) |
| A5 polytopes | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| t0 | t1 | t2 | t0,1 | t0,2 | t1,2 | t0,3 | |||||
| t1,3 | t0,4 | t0,1,2 | t0,1,3 | t0,2,3 | t1,2,3 | t0,1,4 | |||||
| t0,2,4 | t0,1,2,3 | t0,1,2,4 | t0,1,3,4 | t0,1,2,3,4 | |||||||
In 8-dimensional geometry, there are 255 uniform polytopes with E8 symmetry. The three simplest forms are the 421, 241, and 142 polytopes, composed of 240, 2160 and 17280 vertices respectively.
In 7-dimensional geometry, 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 respectively.
In 6-dimensional geometry, there are 39 uniform polytopes with E6 symmetry. The two simplest forms are the 221 and 122 polytopes, composed of 27 and 72 vertices respectively.
In five-dimensional geometry, a truncated 5-simplex is a convex uniform 5-polytope, being a truncation of the regular 5-simplex.
In six-dimensional geometry, a truncated 6-simplex is a convex uniform 6-polytope, being a truncation of the regular 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.
In 8-dimensional geometry, there are 135 uniform polytopes with A8 symmetry. There is one self-dual regular form, the 8-simplex with 9 vertices.
In 7-dimensional geometry, there are 128 uniform polytopes with B7 symmetry. There are two regular forms, the 7-orthoplex, and 8-cube with 14 and 128 vertices respectively. The 7-demicube is added with half of the symmetry.
In 7-dimensional geometry, there are 71 uniform polytopes with A7 symmetry. There is one self-dual regular form, the 7-simplex with 8 vertices.
In 6-dimensional geometry, there are 35 uniform polytopes with A6 symmetry. There is one self-dual regular form, the 6-simplex with 7 vertices.
In 6-dimensional geometry, there are 64 uniform polytopes with B6 symmetry. There are two regular forms, the 6-orthoplex, and 6-cube with 12 and 64 vertices respectively. The 6-demicube is added with half the symmetry.
In 6-dimensional geometry, there are 47 uniform polytopes with D6 symmetry, of which 16 are unique and 31 are shared with the B6 symmetry. There are two regular forms, the 6-orthoplex, and 6-demicube with 12 and 32 vertices respectively.
In 7-dimensional geometry, there are 95 uniform polytopes with D7 symmetry; 32 are unique, and 63 are shared with the B7 symmetry. There are two regular forms, the 7-orthoplex, and 7-demicube with 14 and 64 vertices respectively.
In 5-dimensional geometry, there are 31 uniform polytopes with B5 symmetry. There are two regular forms, the 5-orthoplex, and 5-cube with 10 and 32 vertices respectively. The 5-demicube is added as an alternation of the 5-cube.
In 5-dimensional geometry, there are 23 uniform polytopes with D5 symmetry, 8 are unique, and 15 are shared with the B5 symmetry. There are two special forms, the 5-orthoplex, and 5-demicube with 10 and 16 vertices respectively.
In seven-dimensional geometry, a cantic 7-cube or truncated 7-demicube as a uniform 7-polytope, being a truncation of the 7-demicube.
In 4-dimensional geometry, there are 9 uniform polytopes with A4 symmetry. There is one self-dual regular form, the 5-cell with 5 vertices.
In 4-dimensional geometry, there are 15 uniform 4-polytopes with B4 symmetry. There are two regular forms, the tesseract and 16-cell, with 16 and 8 vertices respectively.
In 4-dimensional geometry, there are 7 uniform 4-polytopes with reflections of D4 symmetry, all are shared with higher symmetry constructions in the B4 or F4 symmetry families. there is also one half symmetry alternation, the snub 24-cell.