Conventional physics quickly gets messy in an irregular world, according to Harvard scientists who found a few new surprises while tackling an old problem: what happens when a ball rolls downhill.
L. Mahadevan, a professor at the Harvard John A. Paulson School of Engineering and Applied Sciences (SEAS), led the team as they peered into the complex physics of imperfect objects. Their work moved from simulation to real-world work, with surprising results that replayed some mathematical oldies but provided new insights into fundamental physics.
“We go about the world seeing just about what everyone else sees,” Mahadevan said. “But if we choose to pause and wonder even as we wander, we learn about the world, and perhaps even about ourselves. Drawing connections between different fields of mathematics and physics by exploring this simple problem was fun – who knows, it might even turn out to be useful one day.”
Simulating the Plane
The Harvard team began with computer simulations in which they rolled slightly irregular spheres and cylinders down ramps set at differing degrees of incline. From their first simulations, the team saw that while perfectly round objects continually roll along, irregular ones stammer as they descend. Steeper ramps exhibited a higher propensity for the object to roll, with objects more likely to falter on a flatter plane. Mahadevan’s team found the most intriguing physics at the liminal point, the angle at which motion ceased.
“Indeed the behavior of the object near the transition angle, or a critical point, has the features of a phase transition, or bifurcation, which separates two qualitatively distinct states – rolling and not rolling,” first author Daoyuan Qian said.
The team measured entropy to better understand this behavior by observing each object’s terminal rolling speed just before it stopped. Their analysis revealed that inertia and object dimensions significantly influenced rolling behavior. They found that as an object resolves into a stable rolling motion, the time it can roll without stopping theoretically diverges to infinity. Interestingly, spheres behaved differently from cylinders due to their ability to roll across their entire surface in multiple directions.
While the simulations revealed fascinating behavior, the team used real-world experiments to validate their data. Using irregular cylinders and spheres on ramps of varying inclines, their laboratory results largely aligned with their simulations, though not without a few surprises.
At first glance, the lurching motion of a rolling sphere appeared random. But as the team mapped out its movement, a pattern emerged. Once a steady state was reached, the sphere repeated periodic motions, rolling over twice during each period before returning to its original orientation.
Explaining the Physics
“This was something we did not see coming at all,” Qian said.
Despite their surprise, the team found mathematical theorems that explained the phenomena. One was the Hairy Ball Theorem, which Mahadevan described using the metaphor, “You cannot comb the hair on a sphere without a cowlick.” Another was Dirac’s Plate Trick, which states that an object with attached strings must rotate twice to return to its original state—a concept mirrored in the sphere’s jerky, doubled rolling motion.
Phase transitions can be either continuous—like a magnet slowly demagnetizing with heat—or discontinuous, such as the abrupt shift from water to ice. There is no in-between.
“In the rolling study, we see both – when there is no inertia, the rolling speed can be very very small close to the critical ramp angle (average speed continuously increasing from 0), but under inertia effects the object accelerates once it starts rolling, so the final speed changes from 0 to some large number discontinuously at the transition,” Qian explained to The Debrief.
Next Steps
“It’s quite interesting how we can see these kinds of abstract mathematics made visible with this simple experiment,” said co-author and postdoctoral fellow Yeonsu Jung. “And then the question could be, ‘What else can we do?’… Maybe we could explore something that hasn’t been studied by mathematicians yet.”
The work has practical applications in nanoscale cellular transport and robotics, and Qian explained to The Debrief how the knowledge could be applied to many real-life scenarios, including bowling.
“One of the most surprising things we found is that the rolling motion of an irregular object is actually regular, so if you know the shape really well, then you can indeed predict how it rolls, Qian said. “This contrasts with what is commonly known as a random walk, where one would not be able to perform this prediction.”
“The most interesting part of the research (to me) is in demonstrating the possibility of simplifying a very complex problem by thinking about it in a different way,” Qian concluded.
The paper “Phase Transitions in the Rolling of Irregular Cylinders and Spheres” appeared on March 25, 2025, in Proceedings of the National Academy of Sciences.
Ryan Whalen covers science and technology for The Debrief. He holds an MA in History and a Master of Library and Information Science with a certificate in Data Science. He can be contacted at ryan@thedebrief.org, and follow him on Twitter @mdntwvlf.