In n-dimensional Euclidean space no reflection with respect to a hyperplane can be realised by a rigid motion. But this is possible if we allow rigid motions in (n + 1)-dimensional space. These notes show a way to visualise a rigid motion of a cube in 4-dimensional space that flips the cube ‘as the page of a book’.

The two terms rigid motion and isometry are sometimes used as synonyms. Yet they do refer to different concepts. The first one has a purely kinematic connotation: the swing of a door or the movement of a piece of furniture pushed over the floor are described by rigid motions. On the other hand to ensure that two figures are isometric it is enough that there exists a correspondence between their points that maintains the relative distances.