600-cell

Description

The 600-cell is the convex regular polytope in $\R^4$ with Schläfli symbol $\set{3, 3, 5}$. Its cells are $600$ regular tetrahedra, five around each edge, and its vertex figure is a regular icosahedron. It has $120$ vertices, $720$ edges, $1200$ triangular faces and $600$ cells.

A model: let $\varphi = (1 + \sqrt5)/2$ be the golden ratio. Take the $8$ points obtained from $(\pm1, 0, 0, 0)$ by permuting coordinates, and the $16$ points $\frac12(\pm1, \pm1, \pm1, \pm1)$. Add the $96$ points obtained from $\frac12(\pm\varphi, \pm1, \pm\varphi^{-1}, 0)$ by even permutations of the coordinates. These $8 + 16 + 96 = 120$ points lie on the unit sphere, and their convex hull is a 600-cell with edge length $\varphi^{-1}$.

Check of the counts: the icosahedral vertex figure has $12$ vertices, so each vertex lies on $12$ edges and there are $120 \cdot 12 / 2 = 720$ edges. Each tetrahedron has $4$ triangles, each lying in $2$ cells, so $600 \cdot 4 / 2 = 1200$ faces. Then $120 - 720 + 1200 - 600 = 0$.