# Analytic derivation of honeycomb lattice Brillouin zone

My goal is to analytically derive the first Brillouin zone of the honeycomb lattice. Geometrically it's clear how to do this by just finding the space on the lattice nearest to a particular point of the lattice; this is just another hexagon, so the first Brillouin zone should be a hexagon in $$k$$-space.

Here's my attempt so far. Let the nearest neighbor spacing be $$a$$, and choose lattice vectors $$\vec{a}_{1} = a\sqrt{3}(-1/2,\sqrt{3}/2)$$ and $$\vec{a}_{2} = a\sqrt{3}(1/2, \sqrt{3}/2)$$. I assume that a translation by $$N_{1}\vec{a}_{1}$$ or $$N_{2}\vec{a}_{2}$$ is a symmetry of the lattice, i.e., we have periodic boundary conditions and these translations take us around the torus. This seems to suggest the constraints

$$\exp\left[iN_{1}\vec{a}_{1}\cdot\vec{k}\right] = \exp\left[iN_{2}\vec{a}_{2}\cdot\vec{k}\right] = 1$$

After some massaging, this gives the equations

$$k_{x} = \frac{2\pi}{a\sqrt{3}}\left(\frac{q}{N_{2}}-\frac{m}{N_{1}}\right), k_{y}=\frac{2\pi}{3a}\left(\frac{q}{N_{2}}+\frac{m}{N_{1}}\right)$$

for integers $$q,m$$ that have to be chosen to restrict the momenta to the first Brillouin zone. However I can't see how some choice of these integers traces out a hexagon. Is this reasoning somehow flawed, or is there perhaps a way to extract the Brillouin zone from this result?