 Rotational symmetry determines square and hexagonal period lattices

A rank-two period lattice $\Lambda \subset \mathbb{C}$ stable under multiplication by $i$ is, up to nonzero complex scaling, $\mathbb{Z}+i\mathbb{Z}$. Choose a shortest nonzero period $\omega$. Every complex number lies within $1/\sqrt{2}<1$ of a Gaussian integer, so for any $\eta \in \Lambda$, subtraction of a suitable $(m+ni)\omega$ leaves a shorter period and hence zero. This proves $\Lambda =\omega \mathbb{Z}[i]$.

Similarly, stability under a primitive cube root $\zeta$ gives $\Lambda =\omega (\mathbb{Z}+\zeta \mathbb{Z})$. The triangular lattice has covering radius $1/\sqrt{3}<1$, so the same shortest-period argument applies. Stability under a primitive sixth root is equivalent, since $e^{\pi i/3}=1+e^{2\pi i/3}$. These facts identify complex tori with automorphisms fixing their origin of orders four and three or six, respectively.