Delaunay and Voronoi polytopes of the root lattice $E_7$ and of the dual lattice $E_7^*$

We give a detailed geometrically clear description of all faces of the Delaunay and Voronoi polytopes of the root lattice $E_7$ and the dual lattice $E_7^*$. Here three uniform polytopes related to the Coxeter–Dynkin diagram of the Lie algebra $E_7$ play a special role. These are the Gosset polytope $P_\mathrm{Gos}=3_{21}$, which is a Delaunay polytope, the contact polytope $2_{31}$ (both for the lattice $E_7$), and the Voronoi polytope $P_\mathrm V(E_7^*)=1_{32}$ of the dual lattice $E_7^*$. This paper can be considered as an illustration of the methods for studying Delaunay and Voronoi polytopes.