Linas' Fractals

Analytic Combinatorics and Dynamical Systems

This git repo contains a large collection of software and dozens of formal papers exploring a range of concepts in self-similarity. This is the code that was used to create the pictures found in Linas Art Gallery, as well as the text source for the PDF's in Linas Math page (subtitled The Modular Group and Fractals).

Most of the content is closely related to one-another, exploring the relationships between the following topics:

• Continued fractions
• Rational numbers
• The Modular Group SL(2,Z)
• The Cantor Set
• The Gauss-Kuzmin-Wirsing (GKW) operator
• The Minkowski Question Mark function
• The Minkowski measure (the "derivative" of the Question Mark)
• Symmetries of period-doubling maps
• de Rham curves
• The Bernoulli Map
• The Beta Transform bx mod 1
• Subshifts
• Analytic Number Theory
• Newton Series
• Riemann Hypothesis
• Hurwitz zeta function
• Ramanujan identities
• The Divisor Operator
• Mode locking in the Circle Map

All of the above are deeply inter-related with one-another, and, in a certain sense, are all "isomorphic", exhibiting different manifestations of "the same thing": the Cantor set. There's so much here, mostly because the Cantor set is a very rich system.

There's also some unrelated content in here, having to do with either linguistics, with quantum mechanics, or with general relativity.

Table of Contents

Some of the directories contain a lot of content, and are deeply nested; others contain a lot, and are shallow. Yet others are nearly empty. There's a fair amount of source code in here; however, it was never intended to be used/usable by others. It is sloppy, hacky and minimalist, just enough to do what I wanted, but not presentable to the public. This is my private workshop, and not my showroom. I am publishing this in response to requests that I make certain parts available. Well, here's all of it.

FYI, some of the code is very old (dating to 1987), and might not actually compile. Some of the libraries and dependencies have moved, breaking this old code. I will accept bug reports and pull requests to fix what's broken.

In alphabetical order:

• experiments -- This is a HUGE directory, containing various experiments in various subdirectories. See the README file there, for an overview.

• generate -- Code for filling in 2D pixmaps of a broad assortment of dynamical systems. This includes code for making short animated movies of the same. The "main" workhorse/driver is brat.C, which will fill in a pixmap with floating-point numbers, given a range of parameters and a function implementing the dynamical system. The shell script cvt will then assign colors to the floats and create an RGB image. The oldest code, for generating various explorations related to the Mandelbrot set, is in here.

• generate/samples -- A variety of images that were created with the above. Most of these are test runs, and are incomplete or damaged in some way. You will find final versions of these in the art gallery.

• image -- A single utility for converting flo files to mtv files. A flo file contains one floating point number per pixel. An mtv file contains one 24-bit RGB per pixel. Both formats include the pixmap width and height, and no other metadata.

• lang -- Some very simple natural-language and AI experiments from 1997. This is perhaps my earliest work on AI. The diary.txt file writes up my thoughts from that era.

• paper -- Another HUGE directory, containing LyX source for all of the papers that I've published, and a bunch of others that I have not published. Please keep in mind that using this stuff is plagiarism. I've been plagiarized before. If you think that you can just grab this stuff, and publish it under your own name, well, that would be wrong. You will get caught, sooner or later. The stuff here is a historical record, and is here for your entertainment. It's not here for theft.

• tools -- A collection of numeric libraries to support the various experiments. Includes implementations of various number-theoretic functions, such as computing divisors, finding primes, working with continued fractions. Also includes some basic image manipulation code. Much of this code was written in the 1990's and 2000's, long before python existed, long before math libraries were available, and long before you could just download code to do image processing. Forget finding number-theory functions. Some or much of what is here can now be found in standard libraries (but with a different API). The code in the experiments directory makes very heavy use of the code here.