English: Demonstration of the reciprocal lattice. A real space 2D lattice with primitive vectors ${\displaystyle a_{1}}$ and ${\displaystyle a_{2}}$, shown blue and green arrows respectively. ${\displaystyle R}$ is a general lattice vector, shown red. ${\displaystyle r}$ is a vector from the origin to any point. Plotting plane waves of the form ${\displaystyle e^{iG\cdot r}}$, we see that for any integer combination of reciprocal lattice vector basis ${\displaystyle g_{1}}$ and ${\displaystyle g_{2}}$, the resulting plane waves have the same periodicity of the lattice - that is that any translation from point ${\displaystyle r}$ to a point ${\displaystyle R+r}$, the value of the plane wave is the same. These plane waves can be added together and the above relation will still apply.