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We study the repeated folding of a two-parameter family of quadrilaterals about their successively transformed diagonals by examining the evolution of the diagonal lengths. Successively mapped pairs of squared lengths lie on an elliptic curve on which folding acts as translation under the group law. We prove the rotation number attains all possible values and any value determines a unique curve in parameter space. For rational parameters we give an algorithm to determine if the folding map is periodic. This gives a partial explanation for the diversity and intricacy of the curves traced out by the paths of the vertices of the transformed quadrilaterals.

The screensaver map: Dynamics on elliptic curves arising from polygonal folding