Imagine if our skies were always filled with a thick layer of opaque clouds. With no way to see the stars, or to view our planet from above, would we have ever discovered that the Earth is round?
The answer is yes. By measuring particular distances and angles on the ground, we can determine that the Earth is a sphere and not, say, flat or doughnut-shaped — even without a satellite picture.
Mathematicians have found that this is often true of two-dimensional surfaces more generally: A relatively small amount of local information about the surface is all you need to figure out its overall form. The part uniquely defines the whole.
But in some exceptional cases, this limited local information might describe more than one surface. Mathematicians have spent the past 150 years cataloging these exceptions: instances in which local measurements that usually define just one surface in fact describe more than one. But the only exceptions they managed to find weren’t nice, closed-up surfaces like orbs or doughnuts — instead, they stretched on forever in some direction, or had edges you could fall off of.
Nobody could find a closed-up surface that broke the rule. It began to seem as though there simply weren’t any. Perhaps such surfaces could always be uniquely defined by the usual local information.
Now, mathematicians have finally uncovered one of those long-sought exceptions. In a paper published in October, three researchers — Alexander Bobenko of the Technical University of Berlin, Tim Hoffmann of the Technical University of Munich, and Andrew Sageman-Furnas of North Carolina State University — describe a pair of very twisty, closed-up surfaces that, despite having the same local information, have completely different global structures.
Finding them took years of toil, a few very overheated laptops, and an unexpected clue from a seemingly unrelated corner of geometry.
Geometric Misfits
Mathematicians have all sorts of ways to describe a surface locally, but two are especially useful.
One captures information about the surface’s “extrinsic” curvature. Choose a point on your surface. At that point, there are infinitely many directions in which you can calculate how quickly the surface bends in space — what’s known as its curvature. Focus only on the directions where you get the biggest and smallest curvature values, then take the average of the two. The number you get is called the mean curvature. You can compute the mean curvature for any given point on the surface to gain a better understanding of how it’s situated in the space surrounding it.
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