What the mathematics of billiards can tell us about loving and letting go

Mathematicians have long pondered whether every shape—a polygon of any number of sides—can contain a path that brings a moving object, like a billiard ball, back to its starting point. This conundrum, known as the periodic orbit problem for billiards in polygons, has vexed scholars for centuries. Now, mathematician Giovanni Forni from the University of Maryland believes he may have finally cracked it.

The problem stretches beyond the familiar rectangular pool table, embracing any polygonal shape. A billiard ball travels in straight lines, bouncing off the edges at precisely mirror angles, creating a predictable trajectory. The periodic orbit problem asks if such a path exists for every possible polygon. If so, we could, in theory, let our loved one go again and again, confident they would return.

Proving the problem for polygons with angles that are rational multiples of pi has proven challenging, but Forni takes on the additional complexity of polygons with irrational angles, which correspond to love that’s less predictable. By assuming the worst—imagine a world where love never returns—and employing tools from dynamical systems, differential geometry, and algebraic topology, Forni shows that such a world is mathematically impossible.

Forni’s proof hinges on a contradiction: if a polygon existed without a periodic trajectory, the geometric structure encoding all possible billiard paths would have to behave in incompatible ways, like a Swiss cheese that’s both infinitely perforated and solid. This irrationality is untenable, indicating that a world without returning love is a geometry that unravels itself. However, the proof does not reveal where the periodic orbit is hidden—it merely assures us it exists somewhere.

While Forni’s result does not specify where the return path lies, it does provide mathematical assurance that love, like a billiard ball, can complete its orbit and return, assuring that absence makes the heart grow fonder. The proof, posted to the preprint server arXiv.org, awaits peer review, but if it stands, it offers a glimmer of hope that love, too, can be a predictable, mathematical certainty.