List of F4 polytopes

24-cell

In 4-dimensional geometry, there are 9 uniform 4-polytopes with F₄ 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 F₄ 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.

Coordinates

Vertex coordinates for all 15 forms are given below, including dual configurations from the two regular 24-cells. 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=/2.