As the saying goes, “If you love something, set it free. If it comes back, it is yours. If it doesn’t, it never was.”
What most people don’t tell you is that if you let that something go from exactly the right place at exactly the right angle, it will return to you with mathematical certainty.
At least, that’s what mathematicians have been trying to prove for decades. Now, to the delight of star-crossed lovers everywhere, mathematician Giovanni Forni of the University of Maryland believes he has finally done it.
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Formally, the problem is known as the periodic orbit problem for billiards in polygons—not quite as romantic, but stay with me.
Imagine a pool table.
You probably imagined a rectangular table. Think again. This is a mathematical pool table, which means it can have any (finite—no apeirogons allowed) number of sides. On this table, a ball travels in straight lines and reflects perfectly off the walls. The periodic orbit problem asks: Does every such table have at least one trajectory that eventually returns the ball to its starting point, repeating the same sequence of bounces forever?
If the answer is yes, then we can let our beloved go again and again, secure in the knowledge that they will complete their orbit and eventually return to us. Absence, after all, makes the heart grow fonder.
Versions of the periodic orbit problem have circulated—pun intended—since the 18th century. In 2004 it earned a spot on a list of the five most resistant problems in dynamical systems, a field of mathematics that studies how motion and change unfold over time.
As University of Chicago mathematician Howard Masur, who was not involved in Forni’s new result, puts it, “Billiards are typically related to lots of other subjects in mathematics; it makes them a very attractive thing to study.”
Mathematicians, to their credit, made substantial progress roughly 50 years ago when they proved these paths existed for tables whose angles are rational multiples of pi (aka pi multiplied by a fraction). But the question remained unsolved for polygons with irrational angles.
And, if we’re honest with ourselves, when has love ever been rational?
To solve the problem, Forni asks us to assume the worst: Imagine a world in which our love never comes back. Then, using tools drawn from dynamical systems, differential geometry and algebraic topology, he proves that such a world simply cannot exist.
Forni’s argument hinges on a contradiction. If a polygon existed with no periodic trajectory—that is, no way to return to its starting point—then the geometric object encoding all possible billiard paths would be forced to behave in two incompatible ways. Such a structure must necessarily have an infinite amount of “holes” or loops but also finitely many at the same time. It makes no sense, like a slice of infinitely perforated Swiss cheese that is also solid cheddar.
In a sense, the proof shows that a world in which love never returns is a world whose geometry tears itself apart—assuming, of course, the proof holds up.
Forni’s argument has been posted to the preprint server arXiv.org but has yet to undergo peer review.
If the paper does bear mathematical fruit, there is one important caveat: Forni’s proof shows that for every polygon there exists a starting point and direction that produce a periodic trajectory. What it does not do is tell us where that point is or how to find it. The orbit is guaranteed to be out there somewhere, but its location remains a mystery.
So while mathematics may soon assure us that love can always come back, it still cannot tell us where to stand or which way to let it go.
For now, at least, love remains a leap of faith.
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