The Modular Group and Fractals

An exposition of the relationship between Fractals, the Riemann Zeta, the Modular Group Gamma, the Farey Fractions and the Minkowski Question Mark.

Linas' Mathematical Art Gallery has been running for fifteen or twenty years while being silent about the underlying math. I suppose its high time to make amends. The core idea of the dissertation is that the shapes of fractals are describable through Farey Fractions, which appear naturally through continued fractions, which have the symmetry of the Dyadic Monoid, which is a certain subset of the Modular Group SL(2,Z), which is a subgroup of the Fuchsian group SL(2,R), in turn a subgroup of the Kleinian group SL(2,C), all of which are inter-twined with the Riemann Zeta and the structure of the set of rational numbers. The work here helps provide insight as to why, for example, one sees Farey Fractions in the Mandelbrot Set. In number theory, the structure of the Modular Group provides a unifying theme for understanding the nature of factorization and primality. This is why, for example, power series and Dirichlet series (such as the Riemann Zeta) exhibit such crazy fractal Cantor-Set type patterns. Despite this connection being seen by Weierstrass as early as 1872, its more-or-less entirely ignored in standard textbooks on Analysis and Number Theory. The series of articles below tries to provide some of the underpinnings for the above breathless assertions.

I am also intrigued by another rather more complex connection: the chaotic dynamics of a pendulum is described by the KAM torus. The classical theory of the pendulum involves elliptic integrals. But elliptic integrals are closely related to the Jacobi theta functions, and thence to modular forms. But modular forms have the modular group symmetry; this can be explicitly seen in some of the elliptic functions. Does this mean that if we root around a bit, that we will find some modular group symmetry in the KAM torus? I don't know, but I'll bet that it is there; it can't be just a pure "accident" that this model of chaotic dynamics just happens to be so close to modular forms. Update: Hmm. Seems like the Cantor set provides the natural bridge between fractal/chaotic phenomena, and modular forms. The dyadic monoid provides the bridge.
Created in 2004
Last updated December 2010
linas@linas.org