Breaking Symmetry. One Point Theorem

Breaking symmetry is crucial in many areas of physics, mathematics, biology and engineering. We address symmetry of regular convex polygons, non-convex regular polygons (stars) and symmetric Jordan curves/domains. We demonstrate that removing of a single point from the boundary of the regular convex and non-convex polygons and symmetrical Jordan curves reduces the symmetry group of the polygon to the trivial C1 group, when the point does not belong to the axis of symmetry of the polygon. The same is true for solid and open 2D regular convex polygons and symmetric Jordan curves. The only exception is a circle. Removing of a single point from the boundary of a circle gives rise to the curve characterized by C2 group. Symmetry of circles is reduced to the trivial C1 group by removing a triad of non-symmetrical points. The same is true for a solid circle. The “effort” necessary for breaking symmetry of a circle is maximal. 3D generalization of the theorem is trivial. Thus, classification of symmetrical curves following the minimal number of points necessary for breaking their symmetry becomes possible. The demonstrated theorem shows that the symmetry group action on curves and domains becomes trivial when an asymmetric perturbation is introduced, when the curve is not a circle. Information interpretation of the demonstrated theorem, related to the Landauer principle is introduced.