his animation explains how one can naturally represent a torus as a flat square. First, a torus is cut vertically along a plane passing through its center of symmetry. After the obtained cylinder is straightened, it is cut along its axis and flattened. ♦ One could reconstruct the original torus by glueing the opposite sides of the square. Glueing the first pair would give back the cylinder, glueing the second (with the square's sides having become circles) - would give back the torus. ♦ Hence a torus can be thought of as a square with suitably identified points on its perimeter. In this representation, the motions of horizontal and vertical bands on a torus become wrapping motions of rows and columns (and diagonals) of the square.
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