Banach Tarski theorem

Back in the 1924, two mathematicians, Stefan Banach and Alfred Tarski stated that it is actually possible to decompose a ball into six pieces which can be reassembled by rigid motions to form two balls of the same size as the original.

Later on, number of pieces was reduced to five by Julia Robinson, which is also the minimal number of pieces required for this theorem.

You may thing we’re talking nonsense and this can’t be true, but according to Banach Tarski theorem it is completely possible.

What if you broke a glass ball? It is clear that you couldn’t just rearrange broken pieces and get two new balls identical to the original one.

Banach Tarski theorem works because it is not a physical sphere we’re talking about, but rather a mathematical sphere- an infinite collection of points.

Now let’s take a look at a set of natural numbers from one to infinity.

$N = {1, 2, 3, 4…}$
This set is obviously infinitely large. Now let’s take a look at a set which contains only even numbers from the set of natural numbers.

$E = {2, 4, 6, 8…}$
Also, the set of odd numbers:

$O = {1, 3, 5, 7…}$

At the first look you might say that there are half as many numbers in sets that contain only even or odd numbers as there are in the original set of natural numbers since we only took every other number, but every even number is made by multiplying every number of N by two, and every odd number is made by multiplying every number of N by two and subtracting 1. Therefore, sets of even and odd numbers are both infinitely large.

Let’s thing about what we just did. We created two infinitely large sets from one. This is why it is possible to break one sphere into two spheres identical to the original.

A generalization of this theorem is that any two bodies in R3 that do not extend to infinity and each containing a ball of arbitrary size can be dissected into each other.