# H4 polytope

| 120-cell | 600-cell |
In 4-dimensional geometry, there are 15 uniform polytopes with H$_4$ symmetry. Two of these, the 120-cell and 600-cell, are regular.

## Visualizations

Each can be visualized as symmetric orthographic projections in Coxeter planes of the H$_4$ 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.
| # | Name | Coxeter plane projections | Coxeter plane projections | Coxeter plane projections | Coxeter plane projections | Coxeter plane projections | Coxeter plane projections | Schlegel diagrams | Schlegel diagrams | Net |
| # | Name | F4 [12] | [20] | H4 [30] | H3 [10] | A3 [4] | A2 [3] | Dodecahedron centered | Tetrahedron centered | Net |
| 1 | 120-cell {5,3,3} | | | | | | | | | |
| 2 | rectified 120-cell r{5,3,3} | | | | | | | | | |
| 3 | rectified 600-cell r{3,3,5} | | | | | | | | | |
| 4 | 600-cell {3,3,5} | | | | | | | | | |
| 5 | truncated 120-cell t{5,3,3} | | | | | | | | | |
| 6 | cantellated 120-cell rr{5,3,3} | | | | | | | | | |
| 7 | runcinated 120-cell (also runcinated 600-cell) t$_{0,3}${5,3,3} | | | | | | | | | |
| 8 | bitruncated 120-cell (also bitruncated 600-cell) t$_{1,2}${5,3,3} | | | | | | | | | |
| 9 | cantellated 600-cell t$_{0,2}${3,3,5} | | | | | | | | | |
| 10 | truncated 600-cell t{3,3,5} | | | | | | | | | |
| 11 | cantitruncated 120-cell tr{5,3,3} | | | | | | | | | |
| 12 | runcitruncated 120-cell t$_{0,1,3}${5,3,3} | | | | | | | | | |
| 13 | runcitruncated 600-cell t$_{0,1,3}${3,3,5} | | | | | | | | | |
| 14 | cantitruncated 600-cell tr{3,3,5} | | | | | | | | | |
| 15 | omnitruncated 120-cell (also omnitruncated 600-cell) t$_{0,1,2,3}${5,3,3} | | | | | | | | | |
Diminished forms :
| # | Name | Coxeter plane projections | Coxeter plane projections | Coxeter plane projections | Coxeter plane projections | Coxeter plane projections | Coxeter plane projections | Schlegel diagrams | Schlegel diagrams | Net |
| # | Name | F4 [12] | [20] | H4 [30] | H3 [10] | A3 [4] | A2 [3] | Dodecahedron centered | Tetrahedron centered | Net |
| 16 | 20-diminished 600-cell (grand antiprism) | | | | | | | | | |
| 17 | 24-diminished 600-cell (snub 24-cell) | | | | | | | | | |
| 18 Nonuniform | Bi-24-diminished 600-cell | | | | | | | | | |
| 19 Nonuniform | 120-diminished rectified 600-cell | | | | | | | | | |

## Coordinates

The coordinates of uniform polytopes from the H$_4$ family are complicated. The regular ones can be expressed in terms of the golden ratio φ = (1 + √5)/2 and σ = (3√5 + 1)/2. Coxeter expressed them as 5-dimensional coordinates.
| n | 120-cell | 600-cell |
| 4D | The 600 vertices of the 120-cell include all permutations of (0, 0, ±2, ±2) (±1, ±1, ±1, ±√5) (±φ$^{−2}$, ±φ, ±φ, ±φ) (±φ$^{−1}$, ±φ$^{−1}$, ±φ$^{−1}$, ±φ$^2$) and all even permutations of (0, ±φ$^{−2}$, ±1, ±φ$^2$) (0, ±φ$^{−1}$, ±φ, ±√5) (±φ$^{−1}$, ±1, ±φ, ±2) | The vertices of a 600-cell centered at the origin of 4-space, with edges of length 1/φ (where φ = (1+√5)/2 is the golden ratio), can be given as follows: 16 vertices of the form ⁠1/2⁠ (±1, ±1, ±1, ±1), and 8 vertices obtained from (0, 0, 0, ±1) by permuting coordinates. The remaining 96 vertices are obtained by taking even permutations of ⁠1/2⁠ (±φ, ±1, ±1/φ, 0). |
| 5D | Zero-sum permutation: (30): √5 (1, 1, 0, −1, −1) (10): ±(4, −1, −1, −1, −1) (40): ±(φ$^{−1}$, φ$^{−1}$, φ$^{−1}$, 2, −σ) (40): ±(φ, φ, φ, −2, −(σ−1)) (120): ±√5 (φ, 0, 0, φ$^{−1}$, −1) (120): ±(2, 2, φ$^{−1}$√5, −φ, −3) (240): ±(φ$^2$, 2φ$^{−1}$, φ$^{−2}$, −1, −2φ) | Zero-sum permutation: (20): √5 (1, 0, 0, 0, −1) (40): ±(φ$^2$, φ$^{−2}$, −1, −1, −1) (60): ±(2, φ$^{−1}$, φ$^{−1}$, −φ, −φ) |