selfsymmetries

Self-symmetries are the symmetries of a single object that map the object onto itself. In mathematical terms, these are automorphisms: structure-preserving transformations of the object to itself. The collection of all such automorphisms forms the automorphism group, or symmetry group, of the object, and it encodes the object’s intrinsic invariances.

In geometry, self-symmetries are isometries that fix the object setwise. For example, a square has eight self-symmetries: four rotations and four reflections, forming the dihedral group D4. A circle has infinitely many self-symmetries, consisting of all rotations about its center and reflections in any diameter. A line segment has several basic self-symmetries, typically including the identity, a 180-degree rotation about its midpoint, and reflections across the line containing the segment and across the perpendicular bisector through its center.

Beyond geometry, many mathematical structures admit self-symmetries in the form of automorphisms. A finite group, ring, field, or graph has an automorphism group comprised of structure-preserving bijections from the object to itself. These groups act on the object and help classify invariants, orbits, and redundancies, which can simplify analysis and computation.

Applications of self-symmetries appear in crystallography, chemistry, physics, and computer science. They underpin predictions of material properties, guide model reduction, enable graph isomorphism testing, and provide a framework for understanding invariants under symmetry. See also automorphism, symmetry group, isomorphism, and self-similarity.