I have the following equation:
$$f(x,y)+3=0$$
Where $$f(x,y):=2 \cos(\sqrt{3}y)+4\cos(\frac{3}{2}x)\cos(\frac{\sqrt{3}}{2}y)$$
My goal is to solve the first equation, i.e. find the points (x,y) such that $f(x,y)=-3$.
I already know that there are infinite solutions periodically repeated. In particular the points in the xy plane that satisfy the equation are located on the vertices of a honeycomb structure (many hexagons repeated, like in graphene). For example the following points are possible solutions: $$(x,y)=(\frac{2\pi}{3},\frac{2\pi}{3 \sqrt{3}})$$ or
$$(x,y)=(\frac{2\pi}{3},-\frac{2\pi}{3 \sqrt{3}})$$
or also
$$(x,y)=(0,\frac{4\pi}{3 \sqrt{3}})$$
However when I try to obtain this in Mathematica by
Solve[f[x, y] + 3 == 0, {x, y}]
I get a very complicated expression, where y is expressed in terms of x. So how can I obtain with Mathematica the nice solutions I wrote above?