Geometry’s near-miss that wasn’t

On June 8, Nautilus published a piece by Evelyn Lamb talking about mathematical near-misses. Imagine a mathematician trying to solve a problem using a specific technique and imagine it allows her to get really, really close to a solution – but not the solution itself. That’s a mathematical near-miss, and the technique becomes of particular interest to mathematicians because they can reveal potential connections between seemingly unconnected areas of mathematics. Lamb starts the piece talking about geometry but further down she’s got the simplest example: the Ramanujan constant. It is enumerated as e^{π(163^0.5)} (in English, you’d be reading this as “e to the power pi-times the square-root of 163”). It’s equal to 262,537,412,640,768,743.99999999999925. According to mathematician John Baez (quoted in the same article), this amazing near-miss is thanks to 163 being a so-called Heegner number. “Exponentials related to these numbers are nearly integers,” Lamb writes. Her piece concludes thus:

Near misses live in the murky boundary between idealistic, unyielding mathematics and our indulgent, practical senses. They invert the logic of approximation. Normally the real world is an imperfect shadow of the Platonic realm. The perfection of the underlying mathematics is lost under realizable conditions. But with near misses, the real world is the perfect shadow of an imperfect realm. An approximation is “a not-right estimate of a right answer,” Kaplan says, whereas “a near-miss is an exact representation of an almost-right answer.”

It was an entirely fun article (not just because I’ve a thing for articles discussing science that has no known paractical applications). However, the minute I read the headline (‘The Impossible Mathematics of the Real World’), one other science story from the past – which turned out to be of immense practical relevance – immediately came to mind: that of the birth of non-Euclidean geometry. In 19th century Europe, the German polymath Carl Friedrich Gauss realised that though people regularly approximated the shapes of real-world objects to those conceived by Euclid in c. 300 BC, there were enough dissimilarities to suspect that some truths of the world could be falling through the cracks. For example, Earth isn’t a perfect sphere; mountains aren’t perfect cones; and perfect cubes and cuboids don’t exist in nature. Yet we seem perfectly okay with ‘solving’ problems by making often unreasonable approximations. Which one is the imperfect shadow here?

A lecture delivered by Bernhard Riemann, a student of Gauss’s at the University of Gottingen, in June 1854 put his teacher’s suspicions to rest and showed that Euclid’s shapes had been the imperfect shadows. He’d done this by inventing the mathematical tools and rules to describe a geometry that existed in more than three dimensions and could deal with curved surfaces. (E.g., the three angles of a Euclidean triangle add up to 180º – but draw a triangle on the surface of a sphere and the sum of the angles is greater than 180º.) In effect, Euclid’s geometry was a lower dimensional variant of Riemannian geometry.

But the extent of Euclidean geometry’s imperfections only really came to light when physicists* used Riemann’s geometry to set up the theories of relativity, which unified space and time and discovered that gravity’s effects could be understood as the experience of moving through the curvature of spacetime. These realisations wouldn’t have been possible without Gauss wondering why Euclid’s shapes made any sense at all in a world filled with jags and bumps. To me, this illustrates a fascinating kind of a near-miss: one where real-world objects were squeezed into mathematical rules so we could make approximate real-world predictions for over 2,300 years without really noticing that most of Euclid’s shapes looked nothing like anything in the natural universe.

*It wasn’t just Albert Einstein. Among others, the list of contributors included Hendrik Lorentz, Henri Poincare, Hermann Minkowski, Marcel Grossmann and Arnold Sommerfeld.

Featured image credit: Pexels/pixabay.