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The Banach–Tarski Paradox: How Can One Ball Become Two?

The Banach–Tarski Paradox: How Can One Ball Become Two?

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The Banach–Tarski Paradox: How Can One Ball Become Two?

Take a solid ball. Cut it into five pieces. Using only rotations and sliding, with no stretching, bending or adding material, reassemble those pieces into two solid balls, each exactly the same size as the original.

That's not a riddle or a magic trick. It's a proven theorem, published in 1924 by Polish mathematicians Stefan Banach and Alfred Tarski. It sounds like it breaks the conservation of volume. Understanding why it doesn't reveals something deep about what "volume" even means.

The Pieces Don't Have a Volume

The paradox seems to say 1 = 2. The escape is that the pieces aren't like any shape you could cut with a knife. They're infinitely scattered clouds of points so strange that no consistent volume can be assigned to them at all.

Volume is preserved when you move and reassemble ordinary shapes. But these pieces aren't ordinary. They're non-measurable sets, and the usual rule "the volume of the whole equals the sum of the volumes of its parts" simply doesn't apply to them.

Warm-Up: Hilbert's Hotel in Geometry

You've seen a version of this before. In Hilbert's Hotel, a full hotel with infinitely many rooms can take a new guest by shifting everyone from room n to room n + 1.

Here's the geometric version. Mark points on a circle at angles of 1, 2, 3, 4, … radians from a starting point. Because π is irrational, none of these points ever lands on another; they never repeat. Now rotate this set of points backward by 1 radian. Every point shifts to the previous position, and the point at 1 radian moves onto the starting point, 0. So the rotated set is the original set plus one extra point. A rigid rotation turned the set "without 0" into the set "with 0."

A single rotation absorbed an extra point. That's the seed of the whole paradox, just with points instead of volume. Convert between radians and degrees with the radians to degrees calculator.

Ingredient 1: Rotations That Never Repeat

Choose two rotations of a sphere, A and B, around different axes and by carefully chosen irrational angles. Every combination, like A, B, AB, BA⁻¹, AABA⁻¹B, produces a different rotation. No sequence of moves ever accidentally undoes itself unless it literally cancels (like A followed by A⁻¹).

Mathematicians call this a free group on two generators. Now split all possible sequences by their first letter:

  • Words starting with A
  • Words starting with A⁻¹
  • Words starting with B
  • Words starting with B⁻¹
  • The empty word (doing nothing)

Here's the surprise: take all words starting with A⁻¹ and put A in front of each one. A·A⁻¹ cancels, so you get every word that doesn't start with A. That means the "A⁻¹ pile," shifted by A, plus the "A pile" rebuilds the entire group. The same works with B. From four piles you get two complete copies of the whole.

Ingredient 2: The Axiom of Choice

Next, apply those rotations to points on the sphere. Every point has an orbit: all the places it can reach under combinations of A and B. The sphere splits into uncountably many orbits.

To transfer the doubling trick from rotations to points, we need to pick one starting point from each orbit, all at once. No formula or rule can describe such a choice. It requires the axiom of choice, a principle Ernst Zermelo formalized in 1904:

Given any collection of non-empty sets, it's possible to choose one element from each.

For finite collections, this is obvious. For uncountably many sets with no rule, it's an assumption. The axiom of choice guarantees the chosen set exists, but it gives no way to actually construct it, which is exactly why the pieces are so wild.

From Sphere to Ball

Using the chosen points and the four piles, the surface of the sphere splits into pieces that reassemble into two complete sphere surfaces. Extending along lines from the center carries this to the solid ball, with a small extra adjustment for the center point. Refinements of the argument show that 5 pieces are enough. Raphael Robinson proved in 1947 that 5 is the minimum.

An Insider Reference: Why It Works in 3D but Not 2D

Oddly, the paradox can't happen in one or two dimensions. You can't cut a disk into finitely many pieces and rotate and slide them into two disks. Stefan Banach himself proved in 1923 that area can be extended consistently to all subsets of the plane, in a way that respects rigid motions.

The reason was clarified in 1929 by John von Neumann. He showed that everything depends on the group of motions. In 2D, rotations commute (rotating by 30° then 50° is the same as 50° then 30°), and that group can't contain a free group. In 3D, rotations don't commute, and free groups are possible. Von Neumann's study of this idea led to the concept of amenable groups.

Why You Can't Duplicate Gold

Real objects are made of a finite number of atoms, while the Banach–Tarski pieces require uncountably many points arranged in infinitely intricate patterns. You also can't physically carry out a choice the axiom of choice promises but never specifies. The theorem is a statement about mathematical space, not a recipe for matter.

Some mathematicians see the paradox as evidence that the axiom of choice is too strong. Most accept both: the axiom is enormously useful elsewhere, and Banach–Tarski simply shows that not every set of points has a meaningful volume.

Two Concepts Worth Knowing

Measure

A measure assigns sizes (length, area, volume) to sets in a consistent way. Banach–Tarski proves that in three dimensions, no measure can assign a volume to every set while respecting rotations and translations.

Axiom of Choice

The axiom of choice says you can always select one element from each set in any collection. It's independent of the other standard axioms of set theory, as Kurt Gödel (1938) and Paul Cohen (1963) showed.

Quick Answer: What Is the Banach–Tarski Paradox?

The Banach–Tarski paradox is a theorem stating that a solid ball can be split into five non-measurable pieces and reassembled, using only rotations and translations, into two balls identical to the original. It relies on the axiom of choice and doesn't violate physics, because the pieces have no well-defined volume.

Try Them Yourself

Hold a book and rotate it 90° around one edge, then 90° around another edge. Reset and do the rotations in the opposite order. The different results are the non-commuting rotations at the heart of Banach–Tarski.

Next → The Monty Hall Problem Explained