 Helmholtz presents the idea of “rigid motions” to account for these invariant properties. “Rigid motions” are those motions that preserve a set of properties of the objects. For instance, when a sphere rotates around its central vertical axis, that motion preserves the sphere’s symmetries about the x and y axes, and thus these symmetries are invariant under that particular transformation (rotation). A given object may have a group of properties that remain invariant under transformation and a group that change, that is, when objects move in a certain way they may stretch or lose symmetry about an axis. After the publication of Helmholtz’s work, Sophus Lie argued that rigid motions form groups, and described the set of rigid motions in the mathematical terms of group theory.