It’s Monday again, and it’s also the first day of August in 2022. This makes it interesting, in a sense, but I always need to remind myself that, no, Monday being the first of the month does not mean that we’re going to have a Friday the 13th. It’s when Sunday is the first day of the month that we have a Friday the 13th.
That’s not really important, of course—I have no superstitious beliefs about 13 or Friday the 13th. In fact, 13 is one of my favorite numbers, so I rather like a Friday the 13th. This is partly because some people think it’s an unlucky day, and partly because of the bad reputation 13 has with the public among the primes, especially when compared to 7, which is not even as interesting a prime, in my opinion, as 13. Though, when added together, 13 and 7 do make 20.
This is not a big deal, though. According to Goldbach’s Conjecture, every even number greater than 2 can be made from the sum of 2 prime numbers. As far as I know this still hasn’t been proven in a rigorous mathematical sense, but I also don’t think they’ve been able to find any exceptions, and since they have supercomputers and the like with which to work these problems, they’ve gone pretty darn high.
Similarly, they’ve solved π (pi) to about 62 trillion digits or whatnot. Think about that incredible number of digits. By comparison, a googol—which is a number larger by far* than the number of elementary particles in the accessible universe—is only 100 digits long. The Planck length itself is 1.6 x 10-35m. So, it has 35 digits of significance, really, taking the most generous possible meaning of “significance”. And that’s the fundamental, measurable minimum sensible distance quantum mechanics, in its current best form, says exists for reality**.
In other words, even if we had the greatest possible precision that is physically within the realm of reasonable speculation, we could not measure the radius and circumference of any instantiated circle precisely enough to come close to telling if it matched the current figured length of π.
Of course, no actual, physical circle is going to be a perfect, mathematical circle. See above regarding the Planck length; that alone will screw up how perfect a circle can be. Also, spacetime itself is not perfectly flat (although it can locally be extremely close to flat, and on the largest scales it appears to be flat). Even the presence of the person doing the measuring would probably be enough of a spacetime distortion to make a circle’s ratio of circumference to diameter mismatch against π.
Don’t even start trying to compare the ratio of circumference to radius in a massive body like the Earth or the Sun. Those ratios are measurably (in principle at least) below π because of spacetime distortion as described by General Relativity. And, of course, a black hole’s radius, as measured from within, would be functionally infinite. So, its local equivalent of π would go to zero.
But π is a mathematical constant, describing ratios of mathematical objects that are precisely defined in flat, Euclidian geometry, and as such, π is a real thing…indeed, a transcendental thing, you might say. It is known to have an infinite number of non-repeating digits. Which is not to say that there are no repeats at all, just that there is no repetitive pattern. Obviously, in base ten we have only ten digits with which to work, so there are quite a lot of reuses of each digit—an infinite number of them, in fact.
In fact, I suspect—though I don’t know—that if you πcked any finite number of contiguous digits of π, you would eventually be able to find a recurrence of them somewhere down the line, though it might be far beyond what’s been calculated to date. The reasoning, at least as I’m thinking about it, is similar to the reasoning that demonstrates that the “Level 1 Multiverse” is a real thing, if spacetime is infinite in spatial extent. The best data we have now seem to indicate that either space really is infinite or at least it’s waaaaaaaaaaaaaaaaaaaaaaaaaaaay bigger than the 93 billion light-years-across visible universe. This is part of that spacetime flatness I mentioned before.
To think about it from a more mundane point of view: the Earth, locally, looks flat (ish), but if you start precisely measuring the angles of bigger and bigger triangles, you’ll find that they’re adding up to more and more above 180 degrees, showing—even if nothing else did show it***—that the Earth is not flat, and in fact has “positive” curvature in Riemannian geometry. But if you kept on measuring the sums of the angles of bigger and bigger triangles and they all stayed at 180 degrees to the greatest precision you could possibly muster even at immense size, you’d come to the conclusion that, well, either the world on which you stand is flat, or if it’s a spheroid, it’s a really, really BIG spheroid.
The analogous measurements (in three spatial dimensions, obviously) have been done on the scale of the microwave background radiation, which is as far back (and thus as far away) as we can see with light (microwaves, specifically): about 300,000 years after the Big Bang. They are consistent with a flat spacetime. So, as I said, the universe is either spatially infinite, or WAAAAY bigger than what we can see.
This infinity doesn’t really do us any good, of course. We still couldn’t reach almost any of it, even if we were traveling at the speed of light, since the expansion of the universe appears to be accelerating, and thus distant regions are moving away from us faster than light. But, since quantum mechanics appears to dictate that, within any closed region of space, there is a maximum number of possible configurations (defined, at its upper limit, by the event horizon of a black hole with that apparent volume, the number of possible states (or its entropy) of which is related to the surface area of the horizon expressed in Planck lengths squared…so, it is big, but it is finite), then if space is infinite, there will be regions of space “out there” that are precisely the same as any finite region you might choose to compare them to, from the size of a human to the size of the accessible universe. Indeed, in a spatially infinite universe, there are an infinite number of them.
If it helps, you can think of decks of cards being shuffled. There are 52! (read as “fifty-two factorial”****) ways for a deck of cards to be ordered if they are shuffled randomly…that’s about 8.06 x 1067 ways. It’s a big number, and though it’s nowhere near the number of elementary particles in the visible universe, it’s so big that we can be mathematically all but certain that no two fairly shuffled decks of cards have ever in human history come out the same.
However, if we have an infinite number of decks being shuffled, not only will any given ordering be repeated, it will be repeated an infinite number of times, though there might be quite a large average distance between repeats. So it will be with iterations of any person or planet or galaxy or locally causally connected “universe”.
Don’t worry about it too much. Though in an infinite universe there are an infinite number of any given person (as well as every possible variation thereof), these doppelgängers will have no effect upon you, except perhaps to blow your mind, as your existence will blow theirs.
Well, I don’t want to keep going on forever (har!), so I’ll call it to a close now, by noting in passing that this month (August, see above) is named (as many know) for the first “official” Roman emperor, as July was named (also as many know) for the first “de facto” Roman emperor, and June was named (as very few know) for June Cleaver, the empress of late 1950’s to early 1960’s American television.
*How far? About 10,000,000,000,000,000,000 or 100,000,000,000,000,000,000 times as big.
**That doesn’t mean there are no finer distances, necessarily—though it might—but unless quantum gravity or whatever changes things significantly, it sure seems to be a limit…a physical one, not a mathematical one.
***Other things do. The Earth is round, and people have known this for thousands of years, contrary to popular belief. 2200 years ago, Eratosthenes figured out the Earth’s circumference by measuring shadow lengths at different latitudes and doing some basic trigonometry. He got the right answer, too.
****Which is 52 x 51 x 50 x 49 x … x 3 x 2 x 1. The 1 is a really superfluous, since anything times one is just the thing itself, but it’s there for completeness.