Although the frieze pattern considered as a translational series of motifs s.l. involves every point of the two-dimensional plane, and as such is not a proper subset of the plane, but (as point set) equal to the plane, it is a proper subgroup of the continuous group of symmetries of the two-dimensional plane. To see this, we must realize that a symmetry group consists of elements which are not points but transformations. Now, although a discrete symmetry group describing a frieze contains infinitely many symmetry transformations as its elements, as does the continuous group of symmetries of the two-dimensional plane, it nevertheless lacks (infinitely) many symmetry transformations that are present in the continuous group. For example, with respect to translations, the discrete symmetry group describing a frieze, has only translations that are integer multiples of a certain definite translation (including the latter itself), while the continuous group possesses integer multiples of all translations (i.e. of translations of any length).