It is not difficult to show that the points of an integer lattice do not span any equilateral triangles (for example, see here).

Is it also true that the points of a triangular lattice do not span any squares?

That is, a lattice that is composed of the vertices of a tiling of equilateral triangles, or equivalently: $\left\{a\cdot (1,0) + b\cdot (1/2, \sqrt{3}/2) \ \mid \mbox{ for any } a,b\in{\mathbb Z} \right\}$.




With $\rho=\frac{1+i\sqrt 3}2$, we have $\rho\notin\mathbb Q[i]$ and $i\notin\mathbb Q[\rho]$


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