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And as you keep raising the number to higher and higher powers, the number will spiral inward or outward, depending on whether the number is on the inside or on the outside of the unit circle, a circle centered at the origin with radius 1.

In topology, his proof of the Poincare´ Conjecture in dimension 1, showing that the unit circle is the only simply connected compact 1-manifold without boundary, sent topology into a decade-long tailspin.

The distinction doesn’t really matter—they could multiply things by π as needed to ship their work over to the standard unit circle—but this fact can help us understand part of how they did their work.

After hearing the sad news, I found myself browsing his apers on the arXiv and decided to click on “The densest sequence in the unit circle,” which he and Jon cowrote a decade ago.

You can just read the sine off of the picture! After a few weeks of struggling to remember which trig function was which for which angle in a triangle, learning about and then internalizing the unit circle was huge for me.