On the symmetry classification of the four-dimensional parallelohedra

A complete description of the 52 combinatorial types of parallelohedra in Euclidean space E 4 is presented, and for each type a realization through a Dirichlet parallelohedron having maximal symmetry is given. The cone of four-dimensional positive-definite quadratic forms is subdivided into existence fields of symmetry types of parallelohedra. Complete results are given for metric tensors with one, two, and three free parameters.