Classification of closed minimal networks on tetrahedra

A complete description is obtained of the closed locally minimal networks on quasiregular tetrahedra. It is shown that on quasiregular tetrahedra the sets of closed locally minimal networks have the same structure as on flat 2-dimensional tori.
The paper describes the set of all quasiregular tetrahedra on which there exist closed locally minimal networks of a specified type. In the other direction, it describes the set of all closed minimal networks on a given quasiregular tetrahedron.