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. Well, all that, and more.

Linas' Mathematical Art Gallery has been running for over thirty years while being silent about the underlying math. At some point, this became untenable, and this page attempts 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. These have the same symmetry of the infinite binary tree. This is termed the "dyadic monoid" below; but it is known by other names. It is the self-similarity (symmetry) monoid of the Cantor set, aka Cantor space. Via a run-length encoding, aka the "continued fraction encoding", aka "the Minkowski question mark function", it is the similarity monoid of Baire space.

The dyadic monoid 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 (via the continued-fraction mapping).

This provides an interesting knot of ideas to study, in part because they relate together so many different things. Most obviously, it offers some hints as to why Farey Fractions are prominent in the Mandelbrot Set and other fractals.

In number theory, the structure of the Modular Group provides a unifying theme for understanding the nature of factorization and primality. This offers glimmers as to why, for example, power series and Dirichlet series (such as the Riemann Zeta), and even the prime numbers exhibit such crazy fractal Cantor set style patterns.

The Cantor space space and Baire space are both examples of "Polish spaces", along with the real numbers (including Euclidean space) and seperable Banach spaces (including Hilber spaces). Large parts of topology, including explorations of the Borel hierarchy and even subsets of descritive set theory, happen on Polish spaces, if not directly on Cantor or Baire space. Those analyses rarely (if ever?), explore the self-similarity underlying these spaces, and yet its an underlying property. So one can go explore. Yummy! Like a candy store!

Last but not least, the Bernoulli shift on Cantor space provides one of the simplest available measure-preserving dynamical systems that exhibits many of the characteristic features of more complex dynamical systems, including periodic orbits, chaotic orbits, ergodicity and an invariant measure. It is simple enough to be exactly solvable, yet rich enough to be foundational.

Any one of the above topics seems worthy of examination. An exquisite tangle of all of them, intertwined, is too much to be spurned.

This page provides several dozen tracts on these topics. They are presented in a sequence of increasing sophistication and complexity. Many, perhaps most of these require little or no formal education in mathematics. They can be read by anyone exicted by the topics being presented. There are almost no theorems or proofs at all; instead, there are a lot of images, figures and illustrations.

Despite this attempt at general readability, a lot of complicated material does show up in here. In the end, I wrote all this for myself, and not for you, dear reader. Alas. My favorites? Hard to say. The newer stuff in the middle. I saved the worst for last; the bottom of this list is populated by assorted dregs that didn't quite work out. I've attempted to mark the latest stuff with a bright red (New!) marker.

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.

Of course, everything is even crazier than it all seems at first. Elliptic integrals show up in a number of rather distinct, seemingly unrelated places. One is in a class of (exact) solutions for the three-wave equation. This is interesting, because the three-wave equation provides one of the simplest systems in which to study chaotic dynamics and ergodicity in wave mechanics. In short: three (or more) waves can mix and exchange energy between modes whenever their dispersion equation allows conservation of both energy and momentum. This is the primary avenue from turbulence in high-dimensional systems to thermodynamic equilibrium.

A second, utterly unrelated appearence of the elliptic functions are in providing exact solutions for geodesics in the Schwarzschild vacuum. This is not odd at all; this is merely just some geometric coincidence, a side effect of the particular form that the Hamilton-Jacobi equations just happen to have in this spacetime metric. This has nothing to do with anything else, whatsoever, at all. Nada, Zero. Zippo, Zilch. Not in the slightest, not in the least. Just some unrelated phenomenon. Right, guys? Right? Guys? Uh ... guys? Hey, where are you?

Created in 2004
Last updated December 2017
Last updated December 2023