# Third Brillouin zone for a quadratic 2D lattice

As far as I understand, the construction of Brillouin zones stems from the relation$$2 \vec{k}\cdot \vec{G} +G^2 = 0 \,,$$where $\vec{k}$ is the wave vector and and $\vec{G}$ is the reciprocal lattice vector. This condition is supposed to be fulfilled when $\vec{k}$ terminates on a line normal to $\vec{G}$ at half of the length of $\vec{G}.$

If so, why does the third Brillouin zone take the form of Figure 1, rather than Figure 2, for a quadratic lattice? The second figure shows the area enclosed by lines situated half of $2b\hat{x}$ and $2b\hat{y}$ from the origin, perpendicular to the $x$- and $y$- directions respectively.

$\hspace{50px} {\def\place#1#2#3{\smash{\rlap{\hskip{#1px}\raise{#2px}{#3}}}}} \place{0}{220}{\begin{array}{c}\textbf{Figure 1} \\ \hspace{250px} \end{array}}$ $\hspace{50px}{\def\place#1#2#3{\smash{\rlap{\hskip{#1px}\raise{#2px}{#3}}}}} \place{0}{220}{\begin{array}{c}\textbf{Figure 2} \\ \hspace{250px} \end{array}}$ To see it another way, notice that the black dot on the far right (the middle of the three vertical dots) is a BZ center, a $\Gamma$ point. The black dot directly above that one (far upper right corner) is also a $\Gamma$ point, but it must be in the next BZ. Half-way between these two $\Gamma$ points must be a zone boundary, exactly as shown in your Figure 1.