involutio

Involutio is a Latin noun meaning the act or process of turning inward, folding in, or rolling up. It derives from in- meaning “in” and volut- from volvere, “to roll or turn.” In modern English, the form involution is the common term, with involutio appearing mainly in historical or philological contexts as the Latin origin of that concept.

In mathematical usage, the corresponding idea is called an involution. An involution is a function f defined on a set such that applying it twice returns the original element: f(f(x)) = x for all x in the domain. Classic examples include reflections across a line in the plane, and the complex conjugation map on complex numbers (conjugate of the conjugate is the original number). Involutions are studied for their structural and symmetry properties.

In biology and medicine, involution refers to the process of returning to a former or normal size after a period of growth or enlargement. Common examples include postpartum uterine involution, the uterus shrinking back to its nonpregnant size after childbirth, and thymic involution, the age-related shrinking of the thymus. Involution can also describe the regression or simplification of tissues or organs as part of development, aging, or recovery.

In philosophy and other domains, involution may denote inward development or a turning inward of processes, contrasting with evolution or outward change. While involutio itself is mainly encountered in historical texts, the broader concept persists across disciplines under the term involution.