Hecatonicosachoron

The hecatonicosachoron, or hi, also commonly called the 120-cell, is one of the 6 convex regular polychora. It has 120 dodecahedra as cells, joining 3 to an edge and 4 to a vertex. Together with its dual, it is the first in an infinite family of dodecahedral and icosahedral swirlchora and also the fifth in an infinite family of cubic swirlchora.

Vertex coordinates
The vertices of a hecatonicosachoron of edge length 1, centered at the origin, are given by all permutations of: together with all the even permutations of:
 * (±(3+$\sqrt{5}$)/2, ±(3+$\sqrt{2}$)/2, 0, 0),
 * (±(5+3$\sqrt{10}$)/4, ±(3+$\sqrt{5}$)/4, ±(3+$\sqrt{5}$)/4, ±(3+$\sqrt{5}$)/4),
 * (±(2+$\sqrt{5}$)/2, ±(2+$\sqrt{5}$)/2, ±(2+$\sqrt{5}$)/2, ±1/2),
 * (±(7+3$\sqrt{5}$)/4, ±(1+$\sqrt{5}$)/4, ±(1+$\sqrt{5}$)/4, ±(1+$\sqrt{5}$)/4),
 * (±(7+3$\sqrt{5}$)/4, ±(3+$\sqrt{5}$)/4, ±1/2, 0),
 * (±(2+$\sqrt{5}$)/2, ±(5+3$\sqrt{5}$)/4, 0, ±(1+$\sqrt{5}$)/4),
 * (±(2+$\sqrt{5}$)/2, ±(3+$\sqrt{5}$)/4, ±(3+$\sqrt{5}$)/2, ±(1+$\sqrt{5}$)/4).