Column: A pea, the Sun, and a million monkeys

When I was 15, I spent six weeks in western Massachusetts attending the Hampshire College Summer Studies in Mathematics program, a wild ride through the basics of group theory, fractal geometry , and all kinds of other topics my high school classes never would have touched on. It was there that I first heard the name of the prolific and illustrious mathematician John Horton Conway, who sadly died last month of COVID-19 complications . And I learned what remains the single weirdest math result I know: the Banach–Tarski paradox.

Banach–Tarski states that a three-dimensional solid sphere can be decomposed into a finite number of pieces that can be rearranged—without stretching or distorting any of them—and reassembled into two spheres, each the same size as the original. From there, it follows that any 3D shape can be cut up and reassembled into any other 3D shape, irrespective of volume. A sphere the size of a pea can be transformed into a sphere the size of the Sun.

It’s usually called the Banach–Tarski paradox, as opposed to the Banach–Tarski theorem, but not because there’s anything logically unsound about it. The proof is valid in every step, and it’s not even that hard to understand—YouTuber Vsauce has the most jargon-free exposition I’ve seen. Rather, the result is a “paradox” because of how it seems to go against our most deeply held intuitions of how 3D space works. When I first learned about it, it blew my 15-year-old mind. Had mathematicians found a loophole in the law of conservation of matter? Surely, all we had to do was get hold of a pea-size ball of gold and we’d all be rich, right?

Miller’s Diary

Physics Today editor Johanna Miller reflects on the latest Search & Discovery section of the magazine, the editorial process, and life in general.

These days, now that I’ve spent a couple of decades thinking about physics all the time, it’s hard for me to see what all the fuss was about. Not only is it obviously impossible to generate gold or anything else out of nothing just by cutting and rearranging pieces, but it’s easy to see why not.

The “pieces” in a Banach–Tarski decomposition aren’t continuous, solid chunks you can cut with a penknife; rather, they’re infinitely complex collections of scattered points. (If you imagine decomposing a segment of the number line so that all the rational numbers are in one piece and all the irrationals are in another, you won’t be far wrong.) To apply the decomposition to a real sphere made of matter, you’d have to separate every atom, every nucleon, and every bit of the electron cloud into an infinite array of points, with some points going into one piece and some into another.

Furthermore, the pieces don’t have well-defined volumes. Nor do they have masses, or energies, or anything else you’d expect to be conserved during rearrangement. The only way to sensibly measure their size is by the number of points they contain—their cardinality—which is infinite. And doing math with infinity can lead to some counterintuitive results.

Despite their vastly different sizes, a pea-size sphere and a Sun-size sphere have the same cardinality—also infinite—because all the points in one can be put in a one-to-one correspondence with all the points of the other. Just center both spheres at the origin in Cartesian space and multiply the coordinates of every point in the pea by the ratio of the two radii. Under that mapping, every point in the pea has a unique counterpart in the Sun, and vice versa. No point in the Sun is left out, and no point in the pea is used more than once.

Simple multiplication isn’t the only way to map the points in a pea onto the points in the Sun. There are (you guessed it) infinitely many ways to do it. That some of those ways involve the rigid translation of a finite number of infinitely complex pieces is still a surprising result, and all credit to Stefan Banach and Alfred Tarski for figuring it out in 1924. But that result shouldn’t be at odds with our physical intuition, because the inherent limits of the physical world—namely, the finite size of atoms—mean that it’s just not that applicable to any real-world peas or Suns.

The atomic structure of matter isn’t the only physical reality to throw a wet blanket on seemingly outlandish mathematical truths. In a textbook exercise titled “The meaning of ‘never,’” Charles Kittel and Herbert Kroemer made short work of the idea that a million monkeys banging randomly at a million typewriters could ever produce any, let alone all, of the works of William Shakespeare. A single monkey typing one key per second would produce on the order of 10¹⁸ different (overlapping) book-length strings in the 13.8-billion-year age of the universe. The probability that any one of them would be Hamlet is around 10^−164327.

At that rate, it doesn’t matter if you have an Avogadro’s number of monkeys rather than a million, if they could type a million keys per second rather than one, or if they could keep on typing for a million times longer than the universe has already existed. Any one of those corrections is just a rounding error on that gargantuan negative exponent. None of them bring the monkeys any closer to typing Hamlet. It’ll never happen.

On the other hand, if you could somehow employ an infinite number of monkeys, as the thought experiment is sometimes formulated, then some of them (infinitely many, in fact) would produce a flawless Hamlet on their first try. The universe might be infinite in size, so perhaps it could accommodate infinitely many monkeys. But even if it could, all but a finite number of them would have to sit outside the observable part of the universe—so far away, that is, that it would take longer than 13.8 billion years to hear of their accomplishment.