This chapter is a survey of the involution of the real line induced by Dyer’s outer automorphism of the group PGL(2,Z). This ‘modular’ involution is discontinuous at the rationals but satisfies a surprising collection of functional equations. It preserves the set of real quadratic irrationals mapping them in a non-obvious way to each other. It commutes with the Galois action on real quadratic irrationals. It restricts to a non-trivial involution of the set of elements of norm +1 in real quadratic number fields.
More generally, it preserves set-wise the orbits of the modular group, thereby inducing an involution of the moduli space of real rank-two lattices. We give a description of this involution as the boundary action of a certain automorphism of the infinite trivalent tree. It is conjectured that algebraic numbers of degree at least three are mapped to transcendental numbers under this involution.