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.

Oscar Pistorius does not have an unfair advantage.

The fact is that amputee athletes have been using carbon fiber Cheetah feet for the last ten years. They are not new technology.

If they conferred a stable advantage over two-legged runners, Paralympians would have been beating Olympic times by now.

And they’re not.

After a decade, they are still running behind Olympic times.

I actually got curious about this a while back and ran some curve fits on Olympic and Paralympic men’s 200m times. They indicated that not only was there no advantage conferred to users of carbon fiber feet, but that there appeared to be a stable, persistent 1.5 second disadvantage accruing to their use.

In order to determine whether a coin is loaded, you don’t have to subject it to x-rays and MRIs. You just flip it a bunch of times. If it falls to either side of 50/50, it’s loaded.

Well, this particular coin has been getting flipped for a decade. There is no advantage to carbon fiber feet. There is in fact a stable disadvantage.

Oscar Pistorius is running with ankle weights, and he doesn’t even have ankles. He is laboring under a persistent second-and-a-half disadvantage, and yet still managed to run Olympic qualifying times two out of three times. In order to do this, he must be a staggering athlete, and if he had been born with ankles and feet, Usain Bolt would be selling encyclopedias someplace.

Still working an the article about this, but here’s a draft … shows the graphs and data, at least.

(This is where the mathematics side of my personality comes in. I don’t consider it significantly different from the musical side, so all three of my readers will have to tolerate what appears to be an upwelling of jockiness in the middle of my complaints about notating something in six flats when the relative major is a six-sharp key.)

Math and music

I find it a little irksome how musicians can often disparage math and the sciences when I’ve never known a single mathematician or scientist who didn’t play an instrument, often quite well. And yet, as scientists and math geeks, we’re supposed to be the narrow-minded ones. *sigh*

It’s not disparaging to compare music to math, nor does it cheapen it to do so. Math is some of the most delicious and colorful stuff in the world. Of course music is mathematical. Linguistic as well.

If I knew half as many musicians who can balance a checkbook as I know mathematicians who can play music, it would be a hell of a lot more musicians than I know now who can handle numbers, let’s just put it that way.

Math is beautiful, as much so as music. It’s bothersome to hear so many musicians who fail to see that, especially when compared to the number of mathematicians who don’t.