Which one result in mathematics has surprised you the most?

Abstraction: Community poll on most surprising mathematical results

Key points:

• Math Stack Exchange community question asking for the most surprising mathematical results.
• The Banach-Tarski Paradox is cited as the question's motivating example: a unit ball can be decomposed into finitely many disjoint parts, then reassembled via rotation and translation into two complete balls of the same radius.
• The paradox relies on the Axiom of Choice to construct non-measurable sets; the pieces have no well-defined volume.
• Voting system is used to surface the results most widely found surprising by the mathematical community.

Connections: Set Theory · Mathematics · Paradoxes

Source: http://math.stackexchange.com/questions/2949/which-one-result-in-maths-has-surprised-you-the-most