120-cell

Description

The 120-cell is the convex regular polytope in $\R^4$ with Schläfli symbol $\set{5, 3, 3}$. Its cells are $120$ regular dodecahedra, three around each edge, and four cells meet at each vertex. It has $600$ vertices, $1200$ edges, $720$ pentagonal faces and $120$ cells.

Check of the counts: each dodecahedron has $12$ pentagons, and each pentagon lies in exactly $2$ cells, so there are $120 \cdot 12 / 2 = 720$ faces. Each pentagon has $5$ edges, and each edge lies in $3$ pentagons, so there are $720 \cdot 5 / 3 = 1200$ edges. The vertex figure is a regular tetrahedron $\set{3,3}$, so each vertex lies on $4$ edges, and there are $1200 \cdot 2 / 4 = 600$ vertices. Then $600 - 1200 + 720 - 120 = 0$, as Euler's relation for convex $4$-polytopes requires.