In the book Eigenvalues in Riemannian Geometry of Isaac Chavel page $28$ - $29$, they talk about the lattice $\Gamma$ and it is defined as $$\Gamma = \left\{\sum_{j=1}^n \alpha^j v_j : \alpha^j \in \mathbb{Z}, j=1,\dots, n\right\}.$$ If I take the canonical basis (simplifying the problem) $v_1=(1,0)$, $v_2=(0,\sqrt{2})$, then I obtain $\Gamma=\begin{pmatrix} 1 & 0 \\ 0 & \sqrt{2} \end{pmatrix}\mathbb{Z}^2$. Now, it is possible to associate to the lattice $\Gamma,$ the dual lattice, $\Gamma^*$, given by $$\Gamma^*=\{y \in \mathbb{R}^n : \langle x,y\rangle \in \mathbb{Z} \text{ for all } x \in \Gamma\}.$$
Questions : Does it exist a way to interpret the dual lattice $\Gamma^*$ graphically, a sketch? How should we interpret this concept in the resolution of the spectrum on the torus?
Thanks for your help!