= Solution
Every primitive basis of the <Bravais lattice> has the form
$$
\boxed{
a_1'=p a_1+q a_2,
\qquad
a_2'=r a_1+s a_2
},
$$
where $p,q,r,s\in\mathbb Z$ and
$$
\boxed{ps-qr=\pm1}.
$$
The unimodular determinant condition is exactly what makes the integer change of basis invertible over $\mathbb Z$.
Let $b_i$ be the primitive vectors of the <reciprocal lattice>, with
$$
a_i\cdot b_j=2\pi\delta_{ij}.
$$
Solving these four equations gives
$$
\boxed{
b_1=\left(2\pi,\frac{2\pi}{\sqrt3}\right),
\qquad
b_2=\left(0,\frac{4\pi}{\sqrt3}\right)
}.
$$
The six reciprocal-lattice points nearest the origin are
$$
\pm b_1,\qquad
\pm b_2,\qquad
\pm(b_2-b_1),
$$
all at distance $4\pi/\sqrt3$.
The <Wigner-Seitz cell> is bounded by the perpendicular bisectors of the segments joining the origin to those six points. It is the regular hexagon with vertices
$$
\boxed{
\left(\frac{4\pi}{3},0\right),
\left(\frac{2\pi}{3},\frac{2\pi}{\sqrt3}\right),
\left(-\frac{2\pi}{3},\frac{2\pi}{\sqrt3}\right),
\left(-\frac{4\pi}{3},0\right),
\left(-\frac{2\pi}{3},-\frac{2\pi}{\sqrt3}\right),
\left(\frac{2\pi}{3},-\frac{2\pi}{\sqrt3}\right)
}.
$$
Its area equals the reciprocal primitive-cell area:
$$
\boxed{
|b_1\times b_2|=\frac{8\pi^2}{\sqrt3}
}.
$$
This is the <reciprocal lattice and Wigner-Seitz cell of the unit triangular lattice>.
For a periodic potential, <Bloch theorem> identifies wavevectors differing by a reciprocal-lattice vector. The first <Brillouin zone> is the Wigner--Seitz cell just found. More generally, the $n$th Brillouin zone consists of wavevectors reached from the origin after crossing exactly $n-1$ reciprocal-lattice Bragg planes; its boundaries are the perpendicular bisectors
$$
k\cdot G=\frac{|G|^2}{2},
\qquad G\in\Lambda^*\setminus\{0\}.
$$
For this triangular reciprocal lattice, the first zone is the central regular hexagon. The second is the sixfold-symmetric collection of regions immediately outside its six sides, bounded next by the bisectors associated with the next reciprocal points. These zones organize free-particle states into bands: Bragg coupling is strongest at their boundaries and opens energy gaps when the periodic potential is introduced.
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