Dissecting brick into bars

TL;DR

This paper characterizes when an N-dimensional parallelepiped can be dissected into simpler 'bars' based on the linear span of its side lengths, extending Dehn's theorem on rectangles and squares.

Contribution

It generalizes Dehn's theorem to higher dimensions and introduces conditions for dissecting parallelepipeds into bars based on side length spans.

Findings

Dissection into bars iff side lengths span a low-dimensional space over Q

Extension of Dehn's theorem to N-dimensional parallelepipeds

Additional results on parallelepiped dissections

Abstract

An $N$-dimensional parallelepiped will be called a bar if and only if there are no more than $k$ different numbers among the lengths of its sides (the definition of bar depends on $k$). We prove that a parallelepiped can be dissected into finite number of bars iff the lengths of sides of the parallelepiped span a linear space of dimension no more than $k$ over $\QQ$. This extends and generalizes a well-known theorem of Max Dehn about partition of rectangles into squares. Several other results about dissections of parallelepipeds are obtained.