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.
has been running for over twenty 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,
which have the symmetry of the Dyadic Monoid, which is the
symmetry monoid of the Cantor set. 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.
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
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.
- Chapter 1: Distributions of
Rationals on the Unit Interval (or, How to (mis)-Count
(PDF) (19 pages)
is a high-giggle-factor review of
some so-called "facts" about fractions that you learned
as a child, and which every math teacher ever since has
repeated, but which are simply not true. I used to think
Number Theory was boring until I saw this. Reducing
a fraction to a relatively prime numerator and denominator
isn't as boring as its made out to be.
- Chapter 2: Continued Fractions
provides a very curious function that is
discontinuous on the rationals and whose discontinuities
seem to be perfectly randomly distributed. I find this
to be a rather dramatic result, possibly because I've never
heard of such a thing before. I've stared at a lot of fractals
and space-filling curves, but nothing like this.
- Chapter 2.1: Entropy of Continued
Fractions (Gauss-Kuzmin Entropy) (6 pages)
computes the entropy of the Gauss-Kuzmin distribution,
and finds it to be approximately 3.432527514776...
By contrast, the defacto entropy of small rationals
is considerably smaller, rising slowly and barely
getting as large as H=3 when computed for all rationals
with denominators smaller than 100,000.
- Chapter 3: The
Minkowski Question Mark and the Modular Group SL(2,Z)
shows that the distribution of Farey Fractions transforms
under a certain subset of the Modular Group, the dyadic monoid.
This monoid is defined, it's action on the infinite binary
tree (the dyadic tree) is explored. The relationship between
these ideas and the Cantor set is reviewed.
The Minkowski Question Mark Function is then constructed,
and it is shown how the self-similarities of this fractal
curve are given by the dyadic monoid. Also reviews the
hyperbolic rotations of binary trees; defines and reviews
the dyadic lattice.
This paper provides the core background material for the
structure of the dyadic monoid that is used in the other
papers of this series.
- On the Minkowski
measure (27 pages) (See also
points out that the derivative of the Minkowski Question
Mark function is given by the distribution of the Farey
fractions on the real number line. This is anchored by
a foundationally sound derivation of the Minkowski
measure, set in the terms of measure theory. This allows
an exact result to be presented for the measure,
as the infinite product of a set of piece-wise
differentiable functions, each piece being in the form
of a Mobius transform. Additional theoretical machinery is
developed to express transfer functions as push-forwards
on Banach spaces; this is used to demonstrate that the
Minkowski measure is an invariant measure, a Haar measure,
induced by a certain twisted Bernoulli operator. The theoretical
machinery allows the discussion of some of the eigenvectors of
the transfer operator. It is pointed out that the Minkowski
measure is also an eigenvector of the Gauss-Kuzmin-Wirsing
- Modular fractal measures
A working diary: a numerical exploration of the
Fourier transform of the Minkowski measure. Assorted
odds and ends, half-baked, poorly-expressed ideas.
- Chapter 6: Symmetries of
identifies a certain period-doubling monoid subset
of the Modular Group PSL(2,Z), the dyadic monoid,
as the basic symmetry
of a large class of fractals. Although this relationship
is obvious and not very deep, it never
ceases to amaze me that books on modular forms never mention
fractals, and that books on fractals never mention modular
forms. Even Mandelbrot, who put fractals into both the
popular and scientific limelight, doesn't breath even a
word of this in his most recent (2004) book.
This paper develops the Takagi or Blancmange Curve as
a fractal curve that transforms under the three-dimensional
matrix representation of the dyadic monoid. It then shows how to
build higher-dimensional representations out of the Bernoulli
polynomials. The Koch snowflake, the Peano space-filling curve
and the Levy C-curve all appear as special cases of the de Rham
- Chapter 6.5: A Gallery of de Rham
A brief definition of de Rham curves, followed by a gallery of
almost 50 images exploring the four-dimensional space of such
- Chapter 7: The Bernoulli Map
(PDF, 50 pages) applies Transfer Operator techniques to the
Bernoulli Map. Much of the material presented is "well-known"
in that Bernoulli processes are commonly studied in
many areas, from probability theory, (the simplest
Markov chains), the subshifts of finite type, the
one-dimensional Potts model, the dyadic Wavelet transforms,
the Cantor set, etc: all these are connected via a dyadic,
representation, the set of strings in two letters.
Perhaps the only new result here is the discussion of
the continuous spectrum of the Bernoulli operator. It is given
by the Hurwitz zeta function, which can be written as a linear
combination of the Takagi curves.
- Chapter 8: The Gauss-Kuzmin-Wirsing
Operator (PDF, 47 pages)
is the Transfer Operator of the Gauss Map. An incomplete
review of some of the facts concerning this operator is
presented; some results are new.
The new results here are the presentation of a "topologically
equivalent" map, which is exactly solvable. However, the
topological equivalence does not preserve the spectrum of
the GKW, so this cannot be considered to be a solution of the
GKW. This failure is attributed to the fact that the
conjugating function is the Minkowski Question Mark function.
The Jacobian of the transform is the infamous, prototypical
"multi-fractal measure" discussed previously.
- Chapter 9: Spectrum of the
Beta Transform (62 pages) and On
the Beta Transform - Diary and Notes (123 pages).
The beta transformation is the iterated map βx mod 1.
This text explores the transfer operator of the beta
transform and discovers that it has eigenvalues resting
on the unit circle of radius 1/β in the complex plane.
For certain values of β, these are finite in number; these
correspond to a generalzation of the Golden ratio, and are
the zeros of a generalized series of "golden polynomials".
These can be counted by Moreau's necklace-counting function,
but they do not appear to be related to other structures
that the necklace-counting function counts(!)
These special values of β are in one-to-one correspondence
with the famous "Islands of Stability" (the period-coubling
regions) in the logistic map. In this sense, they "explain"
why the period-doubling regions are where they are, and how
to control them.
The paper on the spectrum is condensed and readable;
the diary contains additional material that is scattered,
incomplete and poorly organized. (Some of the material is
boring, it's just dead-ends.) The diary does contain some
good stuff, though. This includes:
- Connections to Bergman space and Bergman polynomials, via
the Hessenberg matrix operator form are sketched, as well
as a move towards Jacobi polynomials.
- The complicated structure of the iterated logistic map,
tent map, etc. is entirely due to the chaotic dynamics of
the carry bit in multiplication. If the carry bit is
suppressed, then one obtains only reshufflings, which have
a completely uniform distribution lacking in structure.
- Chapter 10: Linas' Art Gallery
Unlike everything else on this page, this points not at some
PDF's filled with ... stuff, but rather at an Art Gallery
filled with pretty pictures. The PDF's use fancy words like
"subshifts of finite type" but this animated gif actually
shows what some random typically atypical shift really looks
like: in this case, the shift applied to the
greateest prime factor
exponential generating function. (Right before the end of the
strip, observe how two zeros, left of center, merge and then
split! Cool, huh? What's going on here is not quite as simple
as you might first think.)
- The Newton Series Representation
for the Riemann Zeta derived from the Gauss-Kuzmin-Wirsing
Operator (2004/2005) (PDF, 18 pages).
It is well known that the
Riemann zeta function is the Mellin Transform of the Gauss Map.
As shown above, the GKW operator is the Transfer operator of
the Gauss Map. Putting these together leads to a curious
representation of the Riemann zeta function as a Newton Series
(a finite difference series) of the Riemann zeta. One of the
peculiar and interesting results is that the coefficients
of this expansion are exponentially small.
This paper is cited by:
- On Differences of
Zeta Values is a cleaned-up, expanded, published variant
of the above, co-authored with
The second half of this paper gives several statements that are
equivalent to the Riemann hypothesis. (Unfortunately, neither
the abstract nor the introduction make this clear, which is an
- Notes Relating to Newton
Series for the Riemann Zeta Function (PDF, 34 pages)
(2006) Extended working notes and observations made during
the development of the above-mentioned joint paper with
An evaluation of the asymptotic form for the Newton series
of the Riemann zeta function and the Dirichlet L-functions
are given. The asymptotic form is obtained by performing a
saddle-point analysis of the Norlund-Rice integrals that
correspond to the series. Similar series for other
number-theoretic functions, such as the Mobius, Liouville
and Euler Totient are explored. This extends results previously
given by Lagarias, Coffey, Baez-Duarte and Maslanka.
- Yet Another Riemann Hypothesis
(PDF, 10 pages)
considers the action of the permutation group on the
continued fraction expansion of the real numbers.
This action generalizes a certain integral representation
the Riemann zeta function, and leads to a set of functions that
resemble the zeta in that they appear to have their zeros
in the critical strip and probably on the critical line.
The exploration is purely numerical.
- An efficient algorithm for computing
the polylogarithm and the Hurwitz zeta functions
(PDF, 33 pages) (See also:
This paper develops an extension of the techniques given by
efficient algorithm for computing the Riemann zeta function",
to the polylogarithm and the Hurwitz zeta function.
The algorithm provides a rapid means of evaluating
Lis(z) for general values of complex
s and the region of complex z values given by
|z2/(z-1)|<3.3. This region includes the the
Hurwitz zeta ζ(s,q) for general complex s
and real 1/4≤ q ≤3/4. By using the duplication
formula, the range of
convergence for the Hurwitz zeta can be extended to the whole
real interval 0<q<1, although the algorithm does run
logarithmically slower as it approaches the endpoints. In
particular, this algorithm allows the exploration of the
Hurwitz zeta in the critical strip, where fast algorithms are
Includes a discussion of the monodromy group of the
- On Plouffe's Ramanujan
Ramanujan Journal: Volume 27, Issue 3 (2012), Page
gave a series of identities discovered numerically
and Part II)
for the Riemann zeta at odd integer values, inspired by an
identity for Apery's constant zeta(3) in the Ramanujan
notebooks. This text presents an analytic derivation
of these identities (both those from 1998, and the new ones from
April 2006), showing their full generality. Turns out that
these identities are "well known", and there have been over
half-a-dozen independent, published re-discoveries of these
identities in the century since Ramanujan's time. Add my name
to the list!
of Famous Number Theoretic Functions. This one is
another tease; and worse than the first.
I just don't get the underpinnings yet, so there is
no explanation to go with these pretty pictures.
There is something one can conclude, though:
Graphing the Maclaurin series for random arithmetic
function on the unit disk will reveal hyperbolic Riemann
surfaces with visually evident Fuchsian group symmetries.
Any random function will do. I suspect this is at the heart
of the Riemann zeta and the Dirichlet L-functions:
the actual series don't matter. Almost any randomly
generated series will be hyperbolic, and exhibit
a Fuchsian-group symmetry. I used to believe I knew what
an analytic function was; I now realize how nearly total
my ignorance is of such matters. Equally shameful is a
prevailing academic attitude attitude that a few semesters
of undergraduate analysis is all one will ever need to know
about analytic functions; clearly, there is more to it than
- Divisor Operator
(PDF, 12 pages)
The “divisor operator” is defined as an infinite-dimensional
matrix operator, encoding Dirichlet convolution in a linear
algebra setting. The finite-dimensional variant is known as
the Redheffer matrix. As a matrix operator, it naturally acts
on the Banach space l1 of summable sequences. On
this space, it is not a bounded operator. It's point spectrum
consists of all completely multiplicative arithmetic series
(that are l1-summable).
- Euler Re-summation of Multiplicative Series
(PDF, 6 pages)
...doesn't work. Which is a surprising result. The Euler
transformation of alternating series is known to improve
numeric convergence. Sometimes. Applied to a zeta-like
series constructed from a completely multiplicative arithmetic
function, it fails. The intended question to be posed is: what
classes of completely multiplicative arithmetic functions
result in sums obeying the Riemann hypothesis? A numerical
survey addressing this question seems straightforward, if only
the summation can be re-written to converge quickly in the
critical strip. Euler re-summation is a basic, simple trick
for achieving this. It works like a charm, for the Riemann zeta,
and utterly fails otherwise.
- Measure of the Very Fat Cantor Set
This brief note defines the idea of a "very fat" Cantor set,
and briefly examines the measure associated with such a very
fat Cantor set. The canonical Cantor set is "thin" in
that it has a measure of zero. There are a variety of methods by which
on can construct "fat" Cantor sets (also known as Smith-Volterra-Cantor
sets) which have a measure greater than zero. One of the commonest
constructions, based on the dyadic numbers, has a continuously-varying
parameter that is associated with the measure. The Smith-Volterra-Cantor
set attains a measure of one for a finite value of the parameter; this
paper then explores what happens when the parameter is pushed beyond
this value. These are the "very fat" Cantor sets referred to in the
title. The results consist almost entirely of a set of graphs showing
- The Mandelbrot Set and
(PDF) (27 pages)
examines the limit cycles of iterated points in the interior of
the Mandelbrot set, and discovers that these limit cycles seem
to be some sort of modular form. The interior is compared
visually to the Dedekind Eta (a modular form of weight 12) and
the Weierstrass elliptic invariant g2 (a modular form of
weight 2), as well as to sums built from the number-theoretic
divisor function. The visual resemblance is remarkable, but
an explicit expression for the modular form is not obtained.
The statement here is that this provides an even more direct
link between modular forms and fractals, exhibiting explicitly
a modular group symmetry of the Mandelbrot Set. This is
an expansion and revision of the
old draft (2000)
in the art gallery.
- The Simple Harmonic Oscillator
A glance at the non-square-integrable eigenfunctions of the
quantum simple harmonic oscillator. Unlike the square-integrable
eigenfunctions, these form a continuous spectrum. The
eigenfunctions themselves are given by the confluent
hypergeometric series (Kummer’s function). Some pretty pictures
are graphed (November 2006).
Theory develops some basic relationships between continued
fractions, fractals and Farey Numbers.
- Algorithms in Analytic
Number Theory is the set of routines I wrote to explore
some of the math above. Written for the Gnu MP
arbitrary-precision math library, these implement the
Hurwitz zeta, Riemann zeta, polylogarithm, Minkowski question
mark, confluent hypergeometric function, as well as
assorted trig functions (sine, cosine, exp, log), the
gamma function, binomial coefficients, a complex number type,
and assorted high-precision constants. All written in C for
the GMP math library.
to Abramowitz & Stegun
includes a set of annotations to the classic Handbook of
Mathematical Functions, including new integrals over
Bessel functions, and some sums over the Riemann Zeta function.
Created in 2004
Last updated December 2017