Surprisingly elegant expressions for fundamental constants and other applied numerology

A few years ago, when I was still at Harvard, Ariel Amir knocked on my door. Somehow, he bumped into my old blog post on the (probably) shortest physics article ever, and was wondering whether such a numerological coincidence is truly random or not.

I've seen this paper by Friedrich Lenz for the first time in the Fall of 2006, visiting the Fritz Haber Institute to interview for a PhD position (I cannot believe it was almost ten years ago). There was a talk by an experimentalist who was aiming to measure the change in the proton-to-electron mass ratio using a molecular fountain (I think that was someone from the Rick Bethlem's, but I'm not absolutely sure). There, the "shortest paper ever" was shown as a joke to spice up the introduction.

Since then I was somewhat puzzled whether Lenz actually meant what he wrote, or was basically trolling the scientific community.

Thus, Ariel and I chatted and decided that the only way to answer this was to do an actual calculation. It turned out that Tadashi Tokieda was on sabbatical at the Radcliffe Institute – basically around the corner – and we decided to discuss the idea with him first.

To prove the point, we considered all possible combinations of the 0-9 digits and standard mathematical constants (such as pi or e) and calculated the probability that a combination of three of them would reproduce a 5-digit number of the form xxxx.x.

The probability turned out to be as low as 1.2%!

In other words, the Lenz observation was indeed quite intriguing, in the sense that such an elegant expression for the proton-to-electron mass ratio might have signaled for some underlying physical theory. While, as far as I know, this theory has not ever been revealed (and later measurements actually deviate from 6pi^5), paying attention to such surprises might pay off and deepen our understanding of physics.

Tadashi, Ariel, and I wrote a little popular piece about such surprising coincidences, which will appear in the June issue of the American Mathematical Monthly.

I hope you will enjoy reading it!

Take care,

Misha