Von Neumann-Steinhaus decomposition of parallelepipeds

TL;DR

This paper discusses von Neumann's solution to Steinhaus's problem of partitioning intervals and extends the concept to parallelepipeds in higher-dimensional spaces, providing a generalized decomposition method.

Contribution

It introduces a generalization of Steinhaus's partition problem to parallelepipeds in  space, expanding the scope of the original problem.

Findings

Von Neumann's solution to Steinhaus's problem is detailed.

The generalization to parallelepipeds is established.

The paper provides a framework for decomposing higher-dimensional shapes.

Abstract

A problem of Steinhaus was to partition a finite interval $I$ of the real line into countably infinitely many pairwise disjoint sets that are congruent in the sense that each set is a translate of a fixed set $A$. This paper describes von Neumann's solution of the Steinhaus problem and generalizes the result to parallelepipeds in $\mathbf{R}^n$.