Classifying crystal structures: geometric and arithmetic approach

While the classification of cristals made up by just one atom per cell is well-known and understood (Bravais lattices), that for more complex structures is not. We present a geometric way classifying these crystals and an arithmetic one, the latter introduced in solid mechanics only recently. The two approaches are then compared. Our main result states that they are actually equivalent; this way a geometric interpretation of the arithmetic criterion in given.