H4 polytope

In 4-dimensional geometry, there are 15 uniform polytopes with H4 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 H4 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
The coordinates of uniform polytopes from the H4 family are complicated. The regular ones can be expressed in terms of the golden ratio φ = (1 + $\sqrt{5}$)/2 and σ = (3$\sqrt{5}$ + 1)/2. Coxeter expressed them as 5-dimensional coordinates.