Carnegie Mellon University
March 14, 2019

Do Perfect Circles Exist? Maybe.

By Emily Payne

Jocelyn Duffy
  • Associate Dean for Communications, MCS
  • 412-268-9982

Can a perfect circle exist? Mathematically speaking, of course. A circle is a collection of points equidistant from a fixed center point, and a simple equation can tell us whenever a shape meets this definition. But in the physical world, things get a bit murkier. It's hard to say with certainty whether a perfect circle or a sphere, a circle's three-dimensional counterpart, exist outside of mathematical abstraction.

Why is that? To the human eye, circles and spheres are abundant in nature and in our universe. They can occur naturally — in planets, stars, celestial bodies, tree rings, rain drops — or they can be man-made — such as traffic roundabouts, buttons, volleyballs, pizza. But there is a nuance to what our eyes see as a circle and what math would tell us about their true shape.

"How do you know something in nature is a perfect circle? You might know if you found one, but if you haven't found one, you haven't proved that they don't exist." — David Kinderlehrer


Perhaps nothing appears more perfectly spherical than the gaseous ball of fire we see in the sky every day. Gravitational forces pull matter toward the center of mass, making most of the objects in the solar system, like the sun, settle on a spherical plane. As stars, planets and moons spin on their axes, centrifugal force causes these objects to bulge at their equators, making them wider than they are tall. The faster an object spins, the more oblate than truly spherical it becomes. The sun, for example, bulges 10 kilometers at its equator; but when scaled down, this difference is infinitesimal.

This doesn't mean that a perfect circle or sphere does not exist somewhere.

"How do you know something in nature is a perfect circle? You might know if you found one, but if you haven't found one, you haven't proved that they don't exist," said David Kinderlehrer, Alumni Professor of Mathematical Sciences.

While nature might be out of our control, shouldn't it at least be possible to draw or make a perfect circle? For a circle to be perfect, we would need to measure an infinite number of points around the circle's circumference to know for sure. Each point would need to be precise from the particle level to the molecular level, whether the circle is stationary or in motion, which makes determining perfection a tricky feat.

Much like Schrodinger's cat's suspended existence, the answer is not clear cut — there are all kinds of possibilities.

"There are certainly circles people can draw that you can't tell that the set of points are not equidistant from that fixed point because you don't have the equipment to tell that," Kinderlehrer continued.

In this vein, maybe a circle in nature is perfect, maybe it isn't, but our ways of knowing are limited by the constraints of our physical senses. What we do know is that perfect circles abound in mathematics where lines and points are safe from the finite restrictions and forces of the material world.

Circles Around CMU

Man-made circles — that may or may not be perfect — abound around CMU's Pittsburgh campus. Have you seen all of the items and locations highlighted below?