The Banach–Tarski Paradox
Cut a ball, end up with two of the same size
Using the Axiom of Choice you can decompose a solid ball in three-dimensional space into finitely many pieces and reassemble them — without stretching or distorting — into two solid balls each identical to the original. It is rigorously proven and impossible to do with anything physical. The 'pieces' are not measurable sets; that is where the strangeness lives.
What you are seeing
The infinite tree is the Cayley graph of the free group F₂ generated by two rotations a and b. Every reduced word in a, a⁻¹, b, b⁻¹ is a different node; the root is the identity rotation. Each non-root node has three children (it can't go back via the inverse of the step that produced it).
Colour each node by the first letter of its word and you split F₂ into four disjoint pieces W(a), W(a⁻¹), W(b), W(b⁻¹). The paradox is purely combinatorial: multiplying W(a⁻¹) on the left by a (its inverse) covers every word that doesn't start with a — i.e. the three other pieces plus the identity. One piece, shifted, is three pieces plus a point. Lift this from the group to the sphere via Hausdorff (1914), then to the ball — and one ball becomes two.
