# The Pea and the Sun: A Mathematical Paradox

Every year in November we organize at Charles University the so-called Day of Open Doors. The interested public is encouraged to come and listen to lectures and watch physical experiments, shows and exhibitions, etc. Departments send their representatives to advertise their subjects. In our department (mathematical analysis), people take turns in trying to prepare an interesting lecture on analysis. The choice of topic largely depends on the personal taste of the lecturer, but there is one issue that is almost always present: the Banach-Tarski paradox. The message to a student goes somehow as follows: if you decide to study analysis, we will teach you in the second year measure course how to partition a 100 Crown banknote to pieces and then to reassemble them to form a 1000 Crown banknote (special scissors are recommended).

The Banach-Tarski paradox is something really special. Some call it the most surprising result in theoretical mathematics. Many respectable citizens feel irate and call for a law against it. Even among professional mathematicians the claim ignited a lot of controversy immediately after the result was published. The remarkable Banach-Tarski theorem was obtained by two great stars of 20th century Polish mathematics, Stefan Banach and Alfred Tarski, in 1924. The two had very little in common otherwise and this was their only meeting point, their only joint work (imagine what they could have discovered had they worked together longer!). Banach's main subject was analysis; he is considered to be the founder of functional analysis and the list of objects bearing his name is rather impressive. Tarski was an equally spectacular logician. When the general public gradually learned about the theorem, the controversy mentioned above spread and people (including mathematicians) formed two antagonistic camps, one accepting the beautiful discovery and tolerating its counterintuitive nature and the other rejecting it right from the very start as outrageous nonsense.

A good part of the controversy and the misunderstandings was caused by the fact that the result is hardly accessible for a layman. It is a deep measure-theoretic theorem whose understanding requires a thorough study of several abstract disciplines and perfect knowledge of the Axiom of Choice and non-measurable sets. Therefore, naturally, despite the avalanche of highly technical papers in academic journals, little has been written for the general public. Most experts would consider it a very difficult task, many impossible. Well, not any more. The book by Leonard M. Wapner shows that the task is possible and it achieves the goal in a most satisfactory way. One does not need to have a degree in mathematics in order to follow the lively and readable, highly intriguing story of the paradox. Yet the exposition is serious, correct and comprehensive, and it presents a detailed proof of the result. The presentation is light-hearted, highly entertaining and illustrated with many examples, puzzles, etc. This book is (already) a classic in an area that needed one.

**Submitted by Anonymous |

**21 / Oct / 2011