commensurability

Commensurability is a concept describing when two quantities or structures can be measured by a common unit, or when their relationship is a rational number.

In elementary geometry, two line segments are commensurable if there exist positive integers m and n such that m times one segment equals n times the other; equivalently, their lengths have a rational ratio. If no such integers exist, the segments are incommensurable. The ancient Greek discovery that the diagonal and side of a square are incommensurable helped motivate the development of irrational numbers.

Beyond basic geometry, commensurability arises in more abstract settings. In group theory, two subgroups H and K of a group G are commensurable if their intersection has finite index in both H and K. In geometry and number theory, two lattices in Euclidean space are commensurable if their intersection is again a lattice of finite index in each.

In physics and materials science, commensurability often refers to the relationship between periodicities. A periodic modulation or pattern is commensurate with a crystal lattice if its period is a rational multiple of the lattice period; if the ratio is irrational, the arrangement is described as incommensurate. This distinction helps characterize phases and electronic or magnetic structures, such as charge or spin density waves.

Overall, commensurability denotes a compatible, discrete alignment between scales or structures, typically captured by the existence of a finite common measure or a finite-index intersection. It provides a unifying lens across geometry, algebra, and physical systems to describe when two objects can be reconciled by a shared discrete framework.