List of F4 polytopes
# List of F4 polytopes

* 24-cell

In 4-dimensional geometry, there are 9 uniform 4-polytopes with $F_4$ symmetry, and one chiral half symmetry, the snub 24-cell. There is one self-dual regular form, the 24-cell with 24 vertices.

## Visualization

Each can be visualized as symmetric orthographic projections in Coxeter planes of the $F_4$ Coxeter group, and other subgroups.

The 3D picture are drawn as Schlegel diagram projections, centered on the cell at pos. 3, with a consistent orientation, and the 5 cells at position 0 are shown solid.

F4, [3,4,3] symmetry polytopes :

| # | Name Coxeter diagram Schläfli symbol | Graph | Graph | Graph | Graph | Schlegel diagram | Schlegel diagram | Net |
| - | - | - | - | - | - | - | - | - |
| # | Name Coxeter diagram Schläfli symbol | $F_4$ [12] | $B_4$ [8] | $B_3$ [6] | $B_3$ [6] | $B_2$ [4] | Octahedron centered | Net | Dual octahedron centered |
| 1 | 24-cell (rectified 16-cell) = {3,4,3} = r{3,3,4} | | | | | | | | |
| 2 | rectified 24-cell (cantellated 16-cell) = r{3,4,3} = rr{3,3,4} | | | | | | | | |
| 3 | truncated 24-cell (cantitruncated 16-cell) = t{3,4,3} = tr{3,3,4} | | | | | | | | |
| 4 | cantellated 24-cell rr{3,4,3} | | | | | | | | |
| 5 | cantitruncated 24-cell tr{3,4,3} | | | | | | | | |
| 6 | runcitruncated 24-cell $t_{0,1,3}{3,4,3}$ | | | | | | | | |

[[3,3,3]] extended symmetries of F4 :

| # | Name Coxeter diagram Schläfli symbol | Graph | Graph | Graph | Graph | Schlegel diagram | Net |
| - | - | - | - | - | - | - | - |
| # | Name Coxeter diagram Schläfli symbol | $F_4$ [[12]] = [24] | $B_4$ [8] | $B_3$ [6] | $B_2$ [[4]] = [8] | Octahedron centered | Net |
| 7 | *runcinated 24-cell $t_{0,3}{3,4,3}$ | | | | | | |
| 8 | *bitruncated 24-cell 2t{3,4,3} | | | | | | |
| 9 | *omnitruncated 24-cell $t_{0,1,2,3}{3,4,3}$ | | | | | | |

$[3^+,4,3]$ half symmetries of F4 :

| # | Name Coxeter diagram Schläfli symbol | Graph | Graph | Graph | Graph | Schlegel diagram | Schlegel diagram | Orthogonal Projection | Net |
| - | - | - | - | - | - | - | - | - | - |
| # | Name Coxeter diagram Schläfli symbol | $F_4$ $[12]^+$ | $B_4$ [8] | $B_3$ $[6]^+$ | $B_2$ [4] | Octahedron centered | Dual octahedron centered | Octahedron centered | Net |
| 10 | snub 24-cell s{3,4,3} | | | | | | | | |
| 11 Nonuniform | runcic snub 24-cell $s_3{3,4,3}$ | | | | | | | | |

## Coordinates

Vertex coordinates for all 15 forms are given below, including dual configurations from the two regular 24-cells. (The dual configurations are named in bold.) Active rings in the first and second nodes generate points in the first column. Active rings in the third and fourth nodes generate the points in the second column. The sum of each of these points are then permutated by coordinate positions, and sign combinations. This generates all vertex coordinates. Edge lengths are 2.

The only exception is the snub 24-cell, which is generated by half of the coordinate permutations, only an even number of coordinate swaps. φ=(√5+1)/2.

24-cell family coordinates :

| # | Base point(s) t(0,1) | Base point(s) t(2,3) | Schläfli symbol | Name |
| - | - | - | - | - |
| 1 | | (0,0,1,1)√2 | {3,4,3} | 24-cell |
| 2 | | (0,1,1,2)√2 | r{3,4,3} | rectified 24-cell |
| 3 | | (0,1,2,3)√2 | t{3,4,3} | truncated 24-cell |
| 10 | | (0,1,φ,φ+1)√2 | s{3,4,3} | snub 24-cell |
| 2 | (0,2,2,2) (1,1,1,3) | | r{3,4,3} | rectified 24-cell |
| 4 | (0,2,2,2) + (1,1,1,3) + | (0,0,1,1)√2 " | rr{3,4,3} | cantellated 24-cell |
| 8 | (0,2,2,2) + (1,1,1,3) + | (0,1,1,2)√2 " | 2t{3,4,3} | bitruncated 24-cell |
| 5 | (0,2,2,2) + (1,1,1,3) + | (0,1,2,3)√2 " | tr{3,4,3} | cantitruncated 24-cell |
| 1 | (0,0,0,2) (1,1,1,1) | | {3,4,3} | 24-cell |
| 7 | (0,0,0,2) + (1,1,1,1) + | (0,0,1,1)√2 " | $t_{0,3}{3,4,3}$ | runcinated 24-cell |
| 4 | (0,0,0,2) + (1,1,1,1) + | (0,1,1,2)√2 " | $t_{1,3}{3,4,3}$ | cantellated 24-cell |
| 6 | (0,0,0,2) + (1,1,1,1) + | (0,1,2,3)√2 " | $t_{0,1,3}{3,4,3}$ | runcitruncated 24-cell |
| 3 | (1,1,1,5) (1,3,3,3) (2,2,2,4) | | t{3,4,3} | truncated 24-cell |
| 6 | (1,1,1,5) + (1,3,3,3) + (2,2,2,4) + | (0,0,1,1)√2 " " | $t_{0,2,3}{3,4,3}$ | runcitruncated 24-cell |
| 5 | (1,1,1,5) + (1,3,3,3) + (2,2,2,4) + | (0,1,1,2)√2 " " | tr{3,4,3} | cantitruncated 24-cell |
| 9 | (1,1,1,5) + (1,3,3,3) + (2,2,2,4) + | (0,1,2,3)√2 " " | $t_{0,1,2,3}{3,4,3}$ | Omnitruncated 24-cell |
