H4 polytope



120-cell

600-cell

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 and. Coxeter expressed them as 5-dimensional coordinates.
n120-cell600-cell
4D
The 600 vertices of the 120-cell include all permutations of
and all even permutations of
The vertices of a 600-cell centered at the origin of 4-space, with edges of length 1/φ, can be given as follows: 16 vertices of the form
and 8 vertices obtained from
The remaining 96 vertices are obtained by taking even permutations of
Zero-sum permutation:
Zero-sum permutation: