Linas' Art Gallery

I will put Chaos into fourteen lines
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.
--- Edna St. Vincent Millay

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 'fractal art'. 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.

Hardcore Mathematics: 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? Details here!

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 circle map.

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 Fraction Mapping.

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) Equation.

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 ideas.

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 periodicity ...............

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 Arc Angles.

M-Set Derivations

The interior can be explored by taking derivatives. These derivatives have interesting analytic behaviors.

Undecidably So

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 come close.

Spectral Analysis

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 section.

Bud Sizes

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.

Sinai's Billiards

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 lattices.


A Bibliography, that is, further reading material and mathematical references for the maths explored in this gallery.

StudyWeb Award

Yes, we have source code! (Warning -- its very ugly -- its way hacked.)


Copyright (c) 1988, 1989, 1991, 1993-1998, 2001-2003, 2016 Linas Vepstas <>

Creative Commons License
Linas Vepstas' Art Gallery by Linas Vepstas is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

To contact Linas, see his Home Page.