Composing two of these symmetries gives another symmetry. Goldstone bosons for the exact spontaneously broken chiral symmetries. An important subclass of continuous symmetries in physics are spacetime symmetries. Below this energy the symmetry is spontaneously broken to the standard model symmetries. Duality preserves the symmetries of a polyhedron. This aids in analytically studying motions in a spacetime with symmetries. Supersymmetry extends the Standard Model by adding another class of symmetries to the Lagrangian. Two other spacetime symmetries have not been accounted for. Felix Klein suggested to define geometries through their symmetries. The transformations describing physical symmetries typically form a mathematical group. These symmetries are called the minor symmetries of the stiffness tensor c. The set of rotational symmetries of a polygon forms a finite cyclic group. Components which are obtainable by the symmetries of the Riemann tensor are not displayed. These are called the major symmetries of the stiffness tensor. There are 219 possible crystal symmetries, called crystallographic space groups.