Symmetry Breaking: One-Point Theorem

Symmetry breaking is crucial in many areas of physics, mathematics, biology, and engineering. We investigate the symmetry of regular convex polygons, non-convex regular polygons (stars), and symmetric Jordan curves/domains. We demonstrate that removing a single point from the boundary of 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 a single point from the boundary of a circle creates a curve characterized by the C2 group. The 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 to break the symmetry of a circle is maximal. A 3D generalization of the theorem is exemplified. Thus, the classification of symmetrical curves following the minimal number of points necessary to break 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. An informational interpretation of the demonstrated theorem, which is related to the Landauer principle, is provided.