MVEX01-22-20 The Banach-Tarski paradox

​​The Banach-Tarski paradox asserts that it is possible to cut an apple into a finite number of pieces, and rearrange them to get two identical copies of the original apple. An alternative formulation says that a small pea can be chopped up and reassembled into the Sun.

This result is called a paradox because it seems to contradict basic intuition. Indeed, one should not be able to double the volume by applying rigid motions (rotations and translations), since these preserve the volume. The reason why this contradiction is only apparent and not real, is that the sets in which one divides these solids are not measurable, and have no well-defined volume. They existence depends on the axiom of choice, and one should think of them as a scattered family of points. Reassembling them reproduces a set that has a volume, which happens to be different from the volume at the start.

The result was proved by Banach and Tarski in 1924, building on earlier work of Hausdorff. Besides the axiom of choice, the theorem depends crucially on the fact that the free group on two generators acts via rotations on the 3-dimensional Euclidean space.

The project will begin by developing the basic theory of free groups, and understanding the Lebesgue measure (volume) on the Euclidean space. By introducing the idea of a paradoxical decomposition, we will be able to prove the Hausdorff paradox, then build up to prove the Banach-Tarski paradox.

The report will be written in English.


Projektkod: MVEX01-22-20
Gruppstorlek: 3-6 studenter
GU- och Chalmersstudenter. För GU-studenter räknas projektet som ett projekt i Matematik (MMG900/MMG910).
Projektspecifika förkunskapskrav: Algebraiska strukturer (MMG500/MVE150)
Se respektive kursplan för allmänna förkunskapskrav. Utöver de allmänna förkunskapskraven i MVEX01 ska Chalmersstudenter ha avklarat kurser i en- och flervariabelanalys, linjär algebra och matematisk statistik.
Handledare: Eusebio Gardella,
Examinator: Maria Roginskaya, Ulla Dinger
Institution: Matematiska vetenskaper

Sidansvarig Publicerad: må 25 okt 2021.