Banach–Tarski paradox

The Banach–Tarski paradox is a theorem in set-theoretic geometry that shows something very strange. It says that you can take a solid ball in three-dimensional space and cut it into a few pieces. Then, by just moving and turning those pieces, you can put them back together to make two balls that are exactly the same size as the original one. This seems impossible because moving and turning things usually doesn’t change how much space they take up.

One famous version of this idea is called the "pea and the Sun paradox". It says you could take a tiny object, like a pea, cut it into pieces, and rearrange those pieces to make something huge, like the Sun. This sounds like magic, but it’s based on a real mathematical idea.

This paradox works because the pieces aren’t normal solids. They are made of infinite points that don’t have a usual “volume”. Because of this, the normal rules about size and space don’t apply. The proof of this idea depends on something called the axiom of choice, which lets mathematicians create very unusual collections of points.

Even though it seems to break our everyday ideas about size, the Banach–Tarski paradox doesn’t actually break any rules in advanced mathematics. It just shows that some things in math can be very surprising!

Banach and Tarski publication

In 1924, mathematicians Stefan Banach and Alfred Tarski wrote a paper about a surprising idea. They showed that a solid ball can be split into pieces and then rearranged to make two identical balls. This is called the Banach–Tarski paradox.

They used earlier work by other mathematicians and studied how shapes can be broken apart and moved in space. They found that this surprising result only works in three dimensions or higher, not in one or two dimensions. This is because the rules for moving shapes are different in higher dimensions.

Formal treatment

The Banach–Tarski paradox shows that a ball in space can be cut into pieces and then put back together to make two balls the same size as the original.

Mathematicians study this by looking at how shapes can match each other after moving and turning them. They use special rules for these moves and talk about sets that can be split and rearranged. This helps make the surprising result easier to understand.

Connection with earlier work and the role of the axiom of choice

Banach and Tarski used ideas from earlier work by Giuseppe Vitali, Hausdorff, and Banach. They needed something called the axiom of choice, or AC, to prove their surprising results. This idea is also used to prove other true statements in geometry.

Later, A. P. Morse showed that one geometry result could be proved without AC. Paul Cohen proved that AC cannot be proven from the basic rules of set theory. Even a weaker version of AC, called the axiom of dependent choice or DC, is not enough to prove the Banach–Tarski paradox.

The paradox has inspired important research in mathematics, especially in the study of groups. In 1991, work by Matthew Foreman and Friedrich Wehrung helped Janusz Pawlikowski show that the paradox can also be proved using the Hahn–Banach theorem. This theorem uses a weaker form of AC known as the ultrafilter lemma.

A sketch of the proof

Here a proof is sketched that is similar but not identical to that given by Banach and Tarski. Essentially, the paradoxical decomposition of the ball is achieved in four steps:

Find a paradoxical decomposition of the free group in two generators.
Find a group of rotations in 3-d space isomorphic to the free group in two generators.
Use the paradoxical decomposition of that group and the axiom of choice to produce a paradoxical decomposition of the hollow unit sphere.
Extend this decomposition of the sphere to a decomposition of the solid unit ball.

These steps are discussed in more detail below.

Step 1

The free group with two generators a and b consists of all finite strings that can be formed from the four symbols a, a⁻¹, b and b⁻¹ such that no a appears directly next to an a⁻¹ and no b appears directly next to a b⁻¹. Two such strings can be joined and turned into a string of this type by replacing certain parts with nothing. For example: abab⁻¹a⁻¹_ concatenated with abab⁻¹a becomes abab⁻¹a⁻¹abab⁻¹a, which then becomes abab⁻¹bab⁻¹a, and finally abaab⁻¹a. The set of these strings with this joining operation forms a group with identity element the empty string e. This group may be called F₂.

The group F₂ can be "paradoxically decomposed" in a certain way. This is the key part of the proof.

Step 2

In order to find a free group of rotations of 3D space, i.e., one that behaves just like (or "is isomorphic to") the free group F₂, two orthogonal axes are taken (e.g., the x and z axes). Then, A is taken to be a rotation about the x axis, and B to be a rotation about the z axis.

The group of rotations made from A and B will be called H.

This step cannot be done in two dimensions because it needs rotations in three dimensions.

Step 3

The unit sphere S² is split into orbits by the action of our group H: two points belong to the same orbit if and only if there is a rotation in H which moves the first point into the second.

The axiom of choice can be used to pick exactly one point from every orbit.

The (majority of the) sphere has now been split into four sets, and when two of these are turned, the result is twice what was there before:

Step 4

Finally, connect every point on S² with a half-open segment to the origin; the paradoxical decomposition of S² then gives a paradoxical decomposition of the solid unit ball minus the point at the ball's center.

Some details, fleshed out

In Step 3, the sphere was split into orbits of our group H.

What remains to be shown is the Claim: S² − D is the same size as S².

For step 4, it has already been shown that the ball minus a point can be paradoxical. It remains to be shown that the ball minus a point is the same size as the ball.

The proof sketched above needs many pieces. But with more work, the number of pieces can be made smaller.

Obtaining infinitely many balls from one

Using the Banach–Tarski paradox, we can get as many copies of a ball as we want from just one ball. This works in spaces with three or more dimensions. The ball can be divided into pieces that can be rearranged to form the same ball again, or even many copies of it.

Because of special math properties, we can also split a sphere into countless pieces. Each piece can be turned and moved to match the whole sphere.

Von Neumann paradox in the Euclidean plane

Main article: Von Neumann paradox

In flat space, two shapes that can be split and rearranged using only straight moves and turns must have the same size. Because of this, it's impossible to take a square or circle and split it into pieces that can be rearranged into two copies of the original shape.

Von Neumann studied why this works in flat space but not in three dimensions. He found that the rules for moving shapes in flat space allow for a special kind of measurement that always gives the same result, making such tricky rearrangements impossible. However, if we use more complex ways to move shapes, it becomes possible to create similar surprising results in flat space. He showed this by splitting a square into pieces that could be rearranged into two copies of the original square using these special moves.

Researchers have continued to explore these ideas, finding new ways to split shapes in different spaces and with different rules for moving them. These studies help us understand more about geometry and the ways shapes can be transformed.