A rotation in two dimensions (or other even dimensions) does not in general leave any direction fixed, and even in three dimensions it is not immediately obvious that the composition of rotations about distinct axes is equivalent to a rotation about a single axis. However, in [1775], Leonhard Euler published a remarkable result stating that in three dimensions every rotation of a sphere about its center has an axis, and providing a geometric construction for finding it.
In modern terms, we formulate Euler’s result in terms of rotation matrices as follows.
If \( \mathbf{R} \) is a \( 3{\times}3 \) orthogonal matrix (\( \mathbf{R}^T\mathbf{R} = \mathbf{R}\,\mathbf{R}^T = \mathbf{I} \)) and \( \mathbf{R} \) is proper (\( \det\mathbf{R} = +1 \)), then there is a nonzero vector \( \mathbf{v} \) satisfying \( \mathbf{Rv} = \mathbf{v} \).
This important fact has a myriad of applications in pure and applied mathematics, and as a result there are many known proofs. It is so well known that the general concept of a rotation is often confused with rotation about an axis.
In the next section, we offer a slightly different formulation, assuming only orthogonality, but not necessarily orientation preservation. We give an elementary and constructive proof that appears to be new that there is either a fixed vector or else a “reversed” vector, i.e., one satisfying \( \mathbf{Rv} = -\mathbf{v} \). In the spirit of the recent tercentenary of Euler’s birth, following our proof it seems appropriate to survey other proofs of this famous theorem. We begin with Euler’s own proof and provide an English translation from the original Latin. Euler’s construction relies on implicit assumptions of orientation preservation and genericity, and leaves confirmation of his characterization of the fixed axis to the reader. Our current tastes prefer such matters to be spelled out, and we do so in Section 4. There, we again classify general distance preserving transformations, this time using Euler’s spherical geometry in modern dress instead of linear algebra. We note that some constructions present in Euler’s original paper correspond to those appearing in our proof with matrices. In the final section, we survey several other proofs.
