= Square-tiled surface
A square-tiled surface is a connected <translation surface> built by gluing finitely many equal Euclidean squares along paired parallel edges by <translations>. After normalizing side length to one, its area is the number of squares. Vertex identifications can create cone singularities with angle $2\pi k$ for a positive integer $k$.
Taking local square coordinates modulo $\mathbb Z^2$ gives a finite map to the flat <torus> $\mathbb R^2/\mathbb Z^2$. It is a translation covering away from the images of square vertices, with possible branching over the torus origin. Its degree equals the number of squares.
One square with opposite edges identified gives the ordinary flat torus. Two adjacent unit squares forming a $2$-by-$1$ rectangle, with opposite rectangle edges identified, give another flat torus and a degree-two unbranched map to the unit torus. More elaborate vertex gluings produce higher-genus surfaces. This gives a finite combinatorial encoding of examples used in flat geometry and dynamics.
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