The Riemann hypothesis, prime numbers, and other instances of theorems going pear-shaped

When flying back from a big work conference recently, something occurred to me. If you aren’t into math, you’ll want to click “back” right now.

Okay so a while back after I made post #1061, I mentioned that 1,061 was prime. (It failed to fall into neat pieces in my head even with some effort.) Then, when I made post #1063, I realized that that was also prime, and thus that 1,061 and 1,063 were a prime pair — a set of numbers two apart from one another, both of which are prime. They pop up from time to time.

Anyhow, there’s a neat hypothesis (which means “it smells true but we can’t be sure without proof”) about how the distribution of prime numbers in general — not just prime pairs — may be governed by a function called the Riemann zeta function. You can read about this here; it’s a good basic primer for the whole topic. So anyway, this function may govern how prime numbers are distributed.

So anyway, a few years back I visited my old math professor, who has a habit of waiting for about three seconds for me to come into his house before he ambushes me with some really cool, totally unsolvable problem in mathematics that proceeds to eat my brain for about two weeks. This is a big part of my definition of what constitutes a “fun guy.”

So he tells me about this goofy thing in math that I’ve been idly ruminating on since then. Take a sphere, just a ball shape. Now, take a knife. How many cuts do you make in it — straight, full cuts, like you’re cutting a piece totally off — before there are no pieces left that are as wide as the sphere at any point? Say the sphere is two feet in diameter. You’d have to pass the knife through it at minimum three times before none of the resulting pieces were two feet across in any direction.

So okay, now what about n-dimensional spheres? Like, a circle is a 2-dimensional sphere — you would need to make a minimum of two cuts before no resulting pieces were the same width as the original sphere along any direction.

Now, what about a 4-dimensional sphere? A 7-dimensional sphere? A 691-dimensional sphere? An N-dimensional sphere where N is any integer? (Let’s leave aside the idea of fractional dimensions, although it’s a neat thought.) From the above examples, it’s looking like you might need to cut the N-dimensional sphere N times, right?

Well, it turns out there are some dimensions where this isn’t the case, where for some reason, something goes pear-shaped and the “cut the sphere N times” thing fails. It’s really strange; it’ll work fine up to — for example — dimensional a zillion and one, and then for that dimension, it doesn’t work. (BTW, this is a big part of why “well, it’s true in most cases and anyway, it looks true” and “it is true” are not the same thing for a mathematician. They aren’t the same thing for anyone really, but most people blow it off to the species’ great detriment.)

And sometimes, these dimensions come in pairs that are two off from one another, like this cutting business will fail in dimensions 691 and 693 for some impenetrable reason.

So anyway, it struck me that if this is one more weird example of some sort of subdivision going unpredictably pear-shaped in integer measures of magnitude (whole numbers, dimensions), and even sometimes doing so in two-off pairs … maybe there is a function someplace that governs the dimensional failure of this theorem as well.

And my math prof also told me that there are a ton of other theorems (not just about cleaving numbers or cutting spheres) that also fail in certain dimensions. What about those? Are there functions that fulfill the role of the Riemann zeta function for those failures as well?

Let’s keep in mind that we don’t even know for sure that the Riemann hypothesis is even true. Shitloads of mathematicians hope so, though. It’d be right purty if it were.

If this is the case, provided there are functions that govern dimensional (or other!) theorem failures, is there a branch of mathematics that can govern which functions are assigned to the dimensional failures of which theorems?

Can we do a Noether’s Theorem sort of thing like they do in physics, where a broken symmetry demands a conservation law — if there is a function someplace that works in a Riemann-function-esque way, might there be a (yet unknown) theorem someplace whose dimensional failures are governed by it?

So anyhow. Shit like this is what my head generates when I’m bored. This is why conversations with me often drift off into unintelligibility and awkward looks. (Well, it’s one reason.) I should ask my math prof about this. In terms of the other weirdos that study math, these ruminations are really not amazing; someone else must have thought of this before. There’s probably a whole branch of mathematics that examines dimensional theorem failures.