Math 191b Section 2
Geometrical Paradoxes
Winter 2014-15
TR 1:00 PM // 153 Sloan
Course Description | Policies | Textbooks | Lecture Notes | Handouts | Homework | Sections

Instructor:  Alexander Kechris, 386 Sloan, 626-395-4368, kechris @ caltech dot edu




Course Description

This course will provide an introduction to the striking paradoxes that challenge our geometrical intuition. One of the most famous ones is the Banach-Tarski 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 Banach-Tarski Paradox (1924), the Laczkovich Paradox (1990, solving the Tarski Circle-Squaring Problem),  the Dougherty-Foreman Paradox (1992, solving the Marczewski Problem) and the continuous movement version of the Banach-Tarski 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).


  • There will be no formal textbook but the following book is an excellent reference: S. Wagon, The Banach-Tarski Paradox, Cambridge University Press, 1993.

Lecture Notes

Date Description


Date Description

  | California Institute of Technology | Questions?  kaubry @