Can we really create something from nothing? It’s the strange—and terribly counterintuitive—promise of a mathematical result known as the Banach–Tarski paradox. According to it, one could cut a sphere into a finite number of pieces and then reassemble those pieces to produce… two spheres of the same size and the same volume as the original. In other words, 1 would become 2. Absurd? Yes. Legitimate? Also.
An April Fools’ story that became serious
The tale of this paradox even sparked a minor panic. In 1989, a mathematician published in Scientific American a column explaining how, thanks to this result, one could manufacture gold from a simple ball. The International Council for Gold raised official concerns, fearing the collapse of the global monetary system. The joke: the “inventor” behind the discovery was… Arlo Lipof, an anagram of “April Fool.”
But behind the prank lies a genuine theorem, proven in 1924 by two Polish mathematicians, Stefan Banach and Alfred Tarski. And it says something fascinating: mathematics does not always operate as our physical intuition would expect.
When infinity clouds our logic
To understand, we must dive into the world of infinite sets. Take the integers: 1, 2, 3, 4… There are infinitely many. Now take the perfect squares: 1, 4, 9, 16… There are also infinitely many. But at first glance, it seems obvious that there are “fewer” squares than integers.
Galileo had already noted this paradox: in reality, one can map each integer to its square. This perfect correspondence shows that these two sets have the same mathematical “size,” which we call a countable infinity.
There exists, however, another type of infinity: uncountable infinity, such as that of real numbers (all possible decimals between 0 and 1, for example). There, it’s impossible to “count” them, because you can always find a new one between any two numbers, no matter how close they are. This infinity is even larger.
And it is precisely this type of infinity that plays a role in the Banach–Tarski paradox.
An orange that doubles itself
Imagine a perfect orange, mathematically represented as a sphere made up of an infinite number of points. Banach and Tarski show that by using certain carefully chosen rotations, one can partition the points of this sphere into a small number of very particular groups.
These groups do not resemble ordinary pieces at all: they are not orange wedges you could hold in your hand, but infinite sets of points, impossible to visualize. And yet, once cut in this way, these sets can be reassembled to form… two oranges identical to the original.
That is where the illusion arises: it seems as if matter has been created from nothing, but in reality, everything rests on how mathematics handles infinity.
Why it doesn’t work in real life
Rest assured: no one will ever double their fortune by cutting a gold coin. The Banach–Tarski paradox does not apply to the physical world. First, because it assumes matter is made of infinitely small, infinitely divisible points—something that is not the case. Our atoms cannot be cut without limit.
Secondly, because this demonstration rests on a mathematical tool called the axiom of choice. This axiom is widely accepted because it allows proving many useful results, but it is also… unprovable. In short: we choose to believe it, somewhat like a postulate. Without it, the paradox does not hold.
A mirror of mathematical oddities
So this paradox does not undermine the laws of physics, but it reveals how mathematics can sometimes unsettle us. In their abstract world, they follow an impeccable logic, but it does not always correspond to our experience of the world.
That is also what makes them so fascinating: they compel us to question our intuitions, to accept that 1 can, under certain conditions, yield 2.
So, no, no one will create gold or oranges by multiplying spheres. But the Banach–Tarski paradox reminds us of an essential truth: infinity does not obey our rules. And when mathematics takes hold of it, reality can quickly feel very narrow.
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