Studying crystals for solid state physics I figured that we must be able to define a crystal as an at most countable subset $C\subset M$ where $M$ is an affine space modeled after a vector space $V$ such that there exist a vector $v\in V$ such that $C+v=C$. Is it true that such a definition permits the theorem (which is what is commonly used in solid state physics as the definition of a crystal): there exists vectors $a_1,\dots,a_{dim V},b_1,\dots,b_m\in V$ for some $m\in\mathbb{N}^*$ such that if $p\in C$ then $C = \{p+\sum_{i=1}^{dimV}n_ia_i + b_j|n_1,\dots n_{dimV}\in \mathbb{Z}, j\in \{1,\dots,m\}\}$?
I hope there aren't any stupid counter examples. If this is true it would be nice.