On Euler’s Rotation Theorem

Abstract

Summary It is well known that a rigid motion of the Euclidean plane can be written as the composition of at most three reflections. It is perhaps not so widely known that a rigid motion of n-dimensional Euclidean space can be written as the composition of at most n + 1 reflections. The purpose of the present article is, firstly, to present a natural proof of this result in dimension 3 by explicitly constructing a suitable sequence of reflections, and, secondly, to show how a careful analysis of this construction provides a quick and pleasant geometric path to Euler’s rotation theorem, and to the complete classification of rigid motions of space, whether orientation preserving or not. We believe that our presentation will highlight the elementary nature of the results and hope that readers, perhaps especially those more familiar with the usual linear algebra approach, will appreciate the simplicity and geometric flavor of the arguments.