By Don Koks, 2025.
The Banach–Tarski theorem was published in 1924 by the mathematicians Stefan Banach and Alfred Tarski. It's a statement of pure mathematics that says "A sphere can be cut into as few as five pieces that can be rearranged to produce two spheres identical to the first". Can we use that idea to create two material spheres from one? Or to create gold? Or food?
We most certainly cannot. The theorem refers to what we might call a "mathematical" sphere, and not a real, material sphere. A mathematical sphere is the set of all points whose distance from some particular point (the sphere's "centre") is less than or equal to some given number (the sphere's "radius"). In contrast, a real, material sphere is a bunch of atoms. The difference between these two objects is fundamental: between any two points, there always exists another point; whereas between any two atoms, there does not always exist another atom. This means that a mathematical sphere is fundamentally different from a material sphere.
The Banach–Tarski theorem involves some advanced ideas of sets. Its essence is built on the idea that between any two points, there always exists another point. That concept gives rise to results in pure mathematics that do not apply to the world of atoms. For example, consider the function y = 2x that maps, say, the interval [0,1] to [0,2]. Every point in the x interval [0,1] maps to a unique point in the y interval [0,2], and every point in the y interval [0,2] maps to a unique point in the x interval [0,1]. Mathematicians use this idea to define what we'll call an infinite number of points, and so the intervals are said to have the same (infinite) number of points. (Different levels of such an infinite number exist.) And yet, the y interval is twice as long as the x interval. You can perform such magic with points, but not with the atoms on two axes drawn with ink. When you draw with ink, there really are twice as many molecules of ink on the [0,2] line segment as there are on the [0,1] line segment. Mathematical ideas about points simply don't apply to atoms.
The same discussion holds for spheres. Two mathematical spheres contain, in a sense, the same number of points as one mathematical sphere, and so it's not surprising that we can create two spheres from one with some mathematical juggling of the points. But two material spheres contain twice the number of atoms as one material sphere, and no amount of juggling of the atoms in a single material sphere is going to double their number to make two such spheres.
Here is another example of playing around with an infinite number of points, of which a more sophisticated version is used to prove the Banach–Tarski theorem. Consider the infinite set of points on the real-number line, that corresponds to the natural numbers and is denoted as {1, 2, 3, …}. If you shift each point in this set one unit to the left, then after an infinite time you'll arrive at the set {0, 1, 2, 3, …}. This shift has created an extra point. Although it's fine to talk about this in principle, you can't do it in practice because the task requires an infinite time. Can you do something similar to create a new atom on the left of a line of atoms? Only if you have an infinite number of them, and an infinite time for the task. But perhaps I can make this job easier for you than asking you to spend an infinite amount of time shifting each of an infinite number of atoms one unit to the left. I'll simply suggest that you sit and wish for a new atom to appear on the left of the line. I'll bet you a million dollars that if you wait for an infinite time, that atom will just appear. Of course, I can never lose that bet; but is it any more or less nonsensical than the infinitely more difficult physical job of shifting each of the infinity of atoms one unit to the left? I can even formalise this idea by calling my claim a "conjecture" and having it published in a maths journal. It can never be proved or disproved.
The simple fact that atoms are not points puts an end to the idea that the Banach–Tarski theorem might help us create matter. But perhaps we can use the theorem to go the other way, by suggesting that matter must be composed of atoms. After all, if matter were made of mathematical points, then we really could make two golf balls from one golf ball, and there is just something intrinsically odd about that. For example, it messes with our ideas of mass being a source of gravity.
The 2005 book "The Pea and the Sun" gives a proof of the Banach–Tarski theorem, and ends with a couple of chapters that discuss the strangeness of the result and whether it might apply to the real world. And yet it never recognises the core fact that material objects are not a continuum of matter that can be treated as a set of mathematical points. It mentions the flights of fancy of various physicists and mathematicians who conjecture that the theorem has the power to explain cosmology, chaos, particle physics, and goodness knows what else. But their conjectures are just words, with no grounding in physical reality.
L. Wapner, "The Pea and the Sun, A Mathematical Paradox", A.K. Peters Ltd, 2005.