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A Disorienting Look at Euler's Theorem on the Axis of a Rotation

1. INTRODUCTION. 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 1775–1776, Leonhard Euler [8] 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 .