Linas' Art Gallery
I will put Chaos into fourteen lines
--- Edna St. Vincent Millay
And keep him there; and let him thence escape
If he be lucky; let him twist, and ape
Flood, fire, and demon --- his adroit designs
Will strain to nothing in the strict confines
Of this sweet order, where, in pious rape,
I hold his essence and amorphous shape,
Till he with Order mingles and combines.
Past are the hours, the years of our duress,
His arrogance, our awful servitude:
I have him. He is nothing more nor less
Than something simple not yet understood;
I shall not even force him to confess;
Or answer. I will only make him good.
Artistic Apologies, Dogmatic Disclaimers:
This material was originally presented (in the 1994 web page, and
in the earlier 1985-1993 pre-web posters and booklets)
with graphic design sensibilities that were more
sparse, neutral, mystical and suggestive, which helped highlight
the intricately twisted patterns.
I now think (as of 2000) that more literate readers would rather
enjoy a bit of
the actual mathematical background that goes into the making of
And so, I've been adding doses of math, here and there, to the
presentation, sometimes running deep.
As there are now hundreds of web sites with fractal artworks
far prettier than my own, I am hoping that this tying together
proves refreshing and adds poise and grace.
My apologies to the visual artists (and fractal ring operators)
who would prefer the poetry
of unvoiced mystery and the directness of visual impact.
Art must suffer for the sake of the artist.
My apologies to the mathematicians who think this is all
a hokey waste of time, and look down on anything that is not
all theorems and proofs, sober sweetness and light.
Math does suffer in the hands of this mathematician.
These images are grasping at a general concept, which I can now
put into words: iteration is not required to generate fractals,
the symmetry group of fractals is the Modular Group SL(2,Z),
portions of the Modular Group have a set-theoretic representation
in the rationals, and thus, the real numbers are deeply
and inherently fractal. This fractal symmetry of the reals
is a bit obscure and not instantly visible, and so it takes
a bit of work, such as iteration, continued fractions, or
various infinite series to make this symmetry group manifest.
The following set of papers, part of a series (of six papers,
still being written), develop these concepts:
Source code: Almost all of the source code used to
create these images is far to hacky and hacked and undocumented,
to be published publicly. However, Some of these do make use of the
Anant -- Algorithmic 'n
Analytic Number Theory multi-precision math library.
Polylogarithm, The Movie! (Newly Revised!)
Come watch the orbiting, leap-frogging zeroes of the
Riemann zeta function play out in the full glory of
the complex z-plane of the polylogarithm! Will any of them
ever miss? The Riemann Hypothesis says no! Huh?
Arnold Tongues, Revisted (New!)
Other parts of this web site previously explored Arnold tongues.
This page revisits some of this old ground, but provides a different
perspective and emphasis, and explores maps that are simpler than the
Greatest Prime Factor (New!)
Graphs of the analytic generating functions of the greatest prime factor
function. Very remarkably, there seems to be very little or nothing
known about these functions: they do not appear to have been studied
before! (Here is a PDF version of this page.)
Arithm-exp-ic Series (New!)
Graphs of the exponential generating functions of various popular
arithmetic functions from number theory. These are just ... awesome!
I promise, you've never seen anything like this before!
The Farey Room
The Farey Room contains pictures generated by means transformations of
the Farey Number Mapping, and through transformations of the Continued
The Gap Theory Page reviews
some of the mathematical underpinnings.
The Gap Room explores images
based on gaps.
The F2 Room holds some additional,
miscellaneous continued-fraction-based images.
The Circle Map Room
The Circle Map Room contains pictures of various portions
of the Hausdorff Measure of the iterated Forced Oscillator (Circle Map)
The Hausdorff Room
This room contains pictures of the Hausdorff measure of the iterated
Logistic Equation, and the Iterated Tent Equation.
The Damped Circle Room
This room contains pictures of the Poincare Recurrence Time
of the damped circle-map.
The Interior Mandelbrot Room
Decorating the interior of a well know exterior -- an (increasingly fat) sketchbook of
The Logistic Variations
The Logistic Equation x[n+1] = lambda * x[n] * (1-x[n]) + omega
is in some ways a simplified version of the Circle Map
x[n+1] = lambda * sin (2*pi*x[n]) + x[n] + omega. In this room,
we explore further variations.
The Drip Drop Room
More games with continued fraction mappings result in
constructed figures that bear a vague and superficial
resemblance to Poincare maps. This twisted mess is what
happens when you just say no to iteration.
The Number Theory Room
Images of famous number-theoretic functions. The goal is to provide
a visual overview of the SL(2,Z) modular group symmetry of these
functions. In development.
The Hacked Mandelbrot Room
Ahh ... the Post-Industrial Era. Err.. Futurist ... Ahh Dada ?
Well ... The usual Mandelbrot set, z(n+1) = z(n)*z(n) + c,
where c = r * exp (i*theta). Well, we've plotted theta along x-axis, and
r along y axis. A bit of a different view ... Hmmm?
The Circle Period
The artist's early period ........ wherein he experimented early with
The Movie Room
Circle Map Movies
The Escape Theory Room
A smooth, fractionally-valued, mathematically exact iteration count
can be defined for any iterated equation. This
Renormalized Iteration Count
removes the dependency of results on the number of
iterations and on the escape-radius. This theory is useful for
producing smooth, non-banded images.
Note that its closely related to the
Douady-Hubbard potential whose field lines or
external rays underpin
the theoretical analysis of the M-set.
Here's a brief Atlas of Rays,
and an accompanying table of
The interior can be explored by taking derivatives. These derivatives have
interesting analytic behaviors.
Penrose conjectures that the inside of the Mandelbrot set is
computationally undecidable, and Smale tries to prove the same.
We none-the-less provide an algorithm that seems to at least
Interior points converge to limit cycles. A limit cycle that repeats
after N iterations can be averaged together to get an average value.
This page explores a technique for reliably computing smooth 'average
values' out of very unsmooth sequences. It appears that the interior
of the Mandelbrot Set is described by the Dedekind eta, providing yet
another tie between modular forms and fractals. A revised, updated
treatment of this topic can be found in the math
An ongoing attempt to measure the size of, and characterize the shape of
the Mandelbrot set. We present a simple estimate of the bud sizes that is
good to about 5%; unfortunately, the buds aren't quite circular.
Includes supporting visual documentation.
Ray-traced images of light passing through a simple cubic lattice
of reflective 'atoms' shows clearly the strongly ergodic nature
of Sinai's Billiards. An atlas of different sizes of balls and
Bibliography, that is, further reading material and mathematical references
for the maths explored in this gallery.
Yes, we have
source code! (Warning -- its very ugly -- its way hacked.)
Copyright (c) 1988, 1989, 1991, 1993-1998, 2001-2003, 2016 Linas Vepstas
Linas Vepstas' Art Gallery
by Linas Vepstas is licensed under a
Attribution-ShareAlike 4.0 International License.
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