This
course will provide an introduction to the striking paradoxes that
challenge our geometrical intuition. One of the most famous ones is the
BanachTarski Paradox: A pea can be decomposed into finitely many
pieces which can be rearranged in space to form a ball the size of the
sun. A recent popular account of these paradoxes can be found in the
book: The Pea and the Sun, A Mathematical Paradox, by Leonard M.
Wapner, A.K. Peters, 2005.
Topics to be discussed include: geometrical transformations, especially
rigid motions; free groups; amenable groups; equidecomposability and
invariant measures; Tarski's Theorem; the role of the Axiom of Choice;
old and new paradoxes, including the BanachTarski Paradox (1924), the
Laczkovich Paradox (1990, solving the Tarski CircleSquaring
Problem), the DoughertyForeman Paradox (1992, solving the
Marczewski Problem) and the continuous movement version of the
BanachTarski Paradox (2005, due to Trevor Wilson, a former Caltech
undergraduate, solving the de Groot Problem).
The treatment of the material will be as elementary as possible. The
course should be accessible to students who are familiar with basic
algebra (Math 5 or equivalent).
