Minimal Banach-Tarski Decompositions

TL;DR

This paper explores the minimal number of pieces needed to decompose and reassemble a sphere or ball into multiple congruent copies, extending previous results in geometric measure theory.

Contribution

It introduces new bounds and methods for minimal Banach-Tarski decompositions, generalizing Robinson's classical result to multiple copies.

Findings

Established lower bounds for the number of pieces in decompositions.

Extended classical Banach-Tarski results to multiple congruent copies.

Provided new techniques for constructing minimal decompositions.

Abstract

We investigate the problem of finding the minimum number of pieces necessary for dividing a three-dimensional sphere or a ball and reassembling it to form $n$ congruent copies of the original object, generalising a known result by Raphael Robinson.