DescriptionIn the 1920's, S. Banach and A. Tarski proved a result which can be described as follows:It is possible to cut up a pea into finitely many pieces that can be rearranged to form a ball the size of the sun! To repeat this in more mathematical terms, the statement is that given any two balls B, B' ⊆ R³ there exists a decomposition
B = X1 ∪...∪ Xn
gi(Xi) = Yi for all i=1,...,n. Before you try this at home, observe that each gi applied to a rectangle gives a rectangle of the same volume. Therefore the sets Xi and Yi have to somehow defy the concept of volume.Further analysis shows that the Banach-Tarski Paradox relies on a seemingly innocent statement called the Axiom of Choice, which has caused a lot of controversy in mathematics during the last century, partly due to statements of the above type. Indeed, in order to define the sets Xi, Yi one has to make an uncountable number of choices, a task one should not waste ones spare time on. However, from a logical point of view there is nothing wrong with the above statement, provided one accepts the Axiom of Choice. Besides the obvious links to set theory, logic and measure theory, the topic has close ties to geometry and group theory, which can be exploited in many different directions, such as finding minimal decompositions or deciding whether such a paradox exists in the plane (it does not). Prerequisites2H core modules, and a willingness to suppress intuition to a certain extend.ResourcesThe monograph of S. Wagon, The Banach-Tarski Paradox, Cambridge University Press, Cambridge 1993, gives a good introduction to the subject.The Wikipedia page of the Banach-Tarski Paradox contains also useful information, including a link to the original paper of Banach and Tarski. |
email: D Schuetz