Rotational Symmetry

Abstract

This chapter is about rotational symmetry. A corresponding n-ary operation is defined on the Gestalt domain. The resulting Gestalten will have periodicity n; i.e., when rotated by $2\pi /n$, they remain the same. They have a phase feature, which is used as their orientation. The optimal a posteriori value is assigned to this attribute by an iterative least squares optimization. Although there is no linear one-step solution, rotational Gestalten are of great algebraic elegance and beauty, which also evidently show in their appearance. Their value in applications is less dominant. Most examples are either flowers, mechanical parts, or religious symbols. The chapter investigates how rotational symmetry must be fused with similarity and proximity. A greedy search rationale is presented avoiding the otherwise combinatorically growing search efforts. Many if not most rotational patterns occurring in the visual world are also reflection symmetric. Algebraically, a different group applies the dihedral group.