. By definition of f , the columns of B are axes within the initial cell; moreover, since ∀x ∈ , xf (x) ∈ , each axis of B can be seen as the corresponding axis of A plus an offset o ∈ 3 . But since I is a repeating unit, such an offset is a valid translation, so each axis of B is a valid translation. Finally, since det(A) / ∈ , not all coefficients of A can be integers, so A has at least one coefficient x such that f (x) ̸ = 0, so at least one axis of B is not 0 3 .