Michiel Smid, School of Computer Science, Carleton

In 1924, Banach and Tarski proved that a 3D ball of volume 1 can be decomposed into a finite number of pieces, such that these pieces can be reassembled (using rotations and translations) into two balls of volume 1.

Earlier, Vitali proved the following weaker result: A 2D disk of area 1 can be decomposed into a countable number of pieces, such that these pieces can be reassembled (again using rotations and translations) into two disks of area 1.

I will present the proof of Vitali's construction, which uses only elementary results taught in COMP 1805.