= 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 $1/\sqrt2<1$ 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 $1/\sqrt3<1$, 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.
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