This short post describes an interesting phenomenon I’ve been puzzled about recently; I’m not a mathematician and perhaps there is an obvious explanation for this that I’m not aware of, but I’ve asked a few professionals and no one has had an answer yet.
Before you read further, take a guess. The fundamental constants of mathematics – what range of values would you expect them to span? After all, the number line is infinite in both directions – tons of room for them spread out in, from the absurdly small to the inconceivably large. If you were to sprinkle them randomly across available number line real estate, they should be spread far and wide. Now, let’s look.
First, consider the fundamental constants of physics. I’m talking about dimensionless constants, such that their magnitude is not dependent on our definition of any units. On a sym-log plot, their distribution looks like this:
It’s hard to visualize their distribution because they range from the extremely small cosmological constant, on the order of 10^-123, to the very large 10^+36 for the ratio of the electrostatic and the gravitational forces between two protons, and many others between those extremes. In other words, they span a huge range – ~160 orders of magnitude across the number line, just as one would expect if their values were not constrained in any particular way. They use a massive range of possible values.
Now, consider the mathematical constants (like Pi, e, etc.) Of course we can argue about which ones belong in this set (e.g., what about the monster group?) and we can arbitrarily define new ones with larger values, but let’s take a look at a moderately canonical list. What you might notice is that they are all ridiculously tightly clustered – roughly between about +0.5 and +5! I was shocked to notice this; out of the whole enormous number line, why are all the important mathematical constants crowded into a tiny segment, and by the way, with broken symmetry (they are all positive)?
No need for a log plot for this one – a nice linear plot shows them all off clearly enough. What is going on here?
One might be tempted to advance a selectionist explanation: the truly enormous ones have been out of the range of human mathematicians’ ability to discover them; fine, but surely we could have handled ones closer to say, 20 – or 50? Why aren’t there any such known? What explains the enrichment for mathematical constants in such an infinitesimally small region?
Another thing to point out is that given that the number line is infinite, the range of 10^-123 to 10^+36 is also technically speaking extremely tight in the grand scheme of things. But statistically, the distributions of values among mathematical vs. physical constants are very different, and I know of no a priori reason why anyone would guess in advance that the former would be constrained to such a small segment. Why don’t important mathematical constants exceed 5 or occur below 0? It’s really bugging me. I think there’s something important here, and such unexpected enrichment (clustering) calls for an explanation. What would an “explanation” of such a thing even look like? Perhaps not like the explanation of the distribution of spectral lines of elements, or maybe somewhat like it, with a more fundamental underlying phenomenon. Perhaps it’s related to the structure of regions of the Platonic Space.
If you’re a number theorist and can tell me why this clustering occurs, please drop the answer here!
Featured image by MidJourney.
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