This illustrates the homeomorphism between the quotient space $\mathbb{R}^2/E$ and the torus $T$, where $xEy$ iff $x-y\in\mathbb{Z}^2$. The equivalence classes for $E$ are cosets of the subgroup $\mathbb{Z}^2<\mathbb{R}^2$. Each coset is a regular array of dots, as shown on the left. The point in $T$ corresponding to a given coset is shown on the right. You can drag the coset (or use the arrow keys) and watch the motion of the point on $T$. Note that if you drag the dots one space to the right then the new grid is the same as the old one and the point on $T$ travels all the way around and returns to its starting point.