# Involution

See involution (philosophy) for the philosophy meaning.

In mathematics, an involution, or an involutary function, is a function that is its own inverse, so that

f(f(x)) = x for all x in the domain of f.

## General properties

The identity map is a trivial example of an involution. Common examples in mathematics of more interesting involutions include multiplication by −1 in arithmetic, the taking of reciprocals, complementation in set theory and complex conjugation.

Other examples include include circle inversion, the ROT13 transformation, and the Beaufort polyalphabetic cipher.

An involution is a kind of bijection.

## Involutions in Euclidean geometry

A simple example of an involution of the three-dimensional Euclidean space is reflection against a plane. Doing a reflection twice, brings us back where we started.

This transformation is a particular case of an affine involution.

## Involutions in differential geometry

In differential geometry, an involutive distribution is a certain type of subbundle of a vector bundle. According to Frobenius theorem, involutive distributions are completely integrable. They often used to generate a foliation.

## Involutions in ring theory

In ring theory, the word involution is customarily taken to mean an antihomomorphism that is its own inverse function. Examples include complex conjugation and the transpose of a matrix.