circularequivalence

Circularequivalence is a concept used in mathematics and combinatorics to describe when two objects related to a circle are considered the same up to rotation. In formal terms, it is an equivalence relation defined by the action of the circle’s rotation group on a given set of objects. If the group acting is continuous (the circle group SO(2)), two objects are circularequivalent when one can be rotated into the other. In many discrete contexts, the relevant group is the cyclic group Cn, corresponding to discrete rotations by multiples of 360/n degrees.

A common setting is arrangements around a circle or cyclic sequences. For example, two strings of length n over an alphabet are circularequivalent if one string can be obtained from the other by a cyclic shift (rotating the positions around the circle). In geometry or graph labeling, two colorings or labelings of the vertices of an n-gon are circularequivalent if a rotation of the polygon maps one labeling to the other.

Key properties: circularequivalence is an equivalence relation, partitioning the set of objects into orbits under the rotation group. The quotient space formed by collapsing each orbit to a single representative captures objects up to circular symmetry. Counting distinct objects up to circular equivalence is a classical problem, often addressed with Burnside’s lemma or Polya enumeration.

Related concepts include cyclic words, necklaces, and symmetry in combinatorics, as well as group actions and orbit counting. Circularequivalence provides a framework for recognizing when two circularly arranged objects should be treated as identical due to rotational symmetry.