The chaotic series of fractal articles


At the end of 2001 I begun to write this series of fractal articles.They were under ways posted to three different fractal mail groups. The images were attached on a normal way. They achieved a great response, and after a while, the moderator for one of the groups, Joseph Trotsky, I am very grateful to him, started to upload them on his site FAME on a very nice way. Today 22 of the more than 30 articles are there. The last article was released at the end of 2002. The name "Chaotic series" is due to the following circumstances:

  1. I had since a long time planed to write a series of lessons about the elementary math regarding Mandel and Julia type of fractals. However the reason for the commencement at just that time was a little bit of sensitive dependence on initial conditions. The starting point was that I discovered a bug in the internal outside color routine "Binary decomposition" in Ultra Fractal, which made a dear friend of my to write a better routine now included in my ik color routine-folder.
  2. The science of fractals, and the mathematics behind, is a part of the new science of Chaos.
  3. Some formal rules for the layout etc may be applied inconsequentially. However this is a chaotic series..

However for a long time I've planed to put the articles on a site of my own. This has taken a long time for the following reasons:

Now, however the time has come. I decided to publish the articles as pdf-documents. I think most of you have Adobe Reader as all modern computers, both Mac and PC, have it preinstalled (in other case you can download it from Adobe). In most cases the Adobe Reader will open within your browser when you click the links to the articles below. The reasons for my choice are:

Fractal applications: Some of the images in the beginning of the series, it's those with alternating colors, are drawn with MandelZot (for earlier versions of MacOS) by Dave Platt. The rest are drawn with Ultra Fractal. The parameter files to these images can be downloaded from here. That's for you who are lucky to be owners of this application.
New module from my friend for UF: (För those of you who have UF) This year (2006) my friend released his third formula folder, sp3. The formula folder contains the third version of Cubic Parameterspace that is Cubic Parameterspace3. It also contains Quartic Parameterspace3, and Pentic Parameterspace3. The number "3" for the two later indicates the parallel's of Cubic parameterspace3. In fact Pentic Parameterspace3 is the very first version for drawing pictures from all slices of the eight-dimensional pentic parameter space. Note: All the parameter files for cubics and quartics in this series are converted in order to be opened by these later modules. All his modules, as well as mine, can be downloaded from here. Some lessons regarding cubics are also slightly rewritten.
I myself who has written these articles am NOT a mathematician. The content is merely based on my own fractal exploration. Following these articles presuppose a very elementary knowledge of complex numbers and the concept of iteration. News! From September 2007 I've uploaded three very short basic articles about iteration and complex numbers:

Intro I) Iteration
Intro II) The imaginary Numbers and the Complex Number Plane
Intro III) Iteration in the Complex Number Plane

Two other sites dealing with the above very basics are:

Below the articles:

1) Level sets
2) Binary decomposition of level sets
3) Binary decomposition, acupuncture points and secondary decorations
4) The complete Lesbian Mystery
4b) The complete Lesbian Mystery, more obvious exposed
5) The Parameter Plane and Quadratic Connectedness Locus
6) Exercises with the Julia Switch feature
7) Relations between the Julia sets and the Mandelbrot set
8) The path to the mystery
9) Binary decomposition around the Mandelbrot set
10) The combinatorial rules of the filaments 1
11) The location of the acupuncture points
12) Julia-like barriers
13) Buds are minibrots (The combinatorial rules of the filaments 2)
14) Emptiness - Fullness (What constitutes the filaments)
15) Perturbed M sets and the Juliabrot
16) Introduction to Cubics
17) The six perpendiculars slices intersecting origo
18) The perpendiculars slices not intersecting origo
19a) Dual critical points manual
19b) Short manual to the module Cubic Parameterspace 3
20a) Cracks, where do they come from?
20b) The reason why the Trolls burst
21) A 4D Mystery
22) Rotations in 4D-space
23) Every Connectedness Locus is connected
24) Mysteries in the Cubic Mandelbrot set
25) Other Views of the ”Great Cubic Mandelbrot Mystery”
26) Another zenbuddhistic koan
27) Compasses
28) To derive the optimal iteration formulas for polynomials of degree ”d”
29) Four-dimensional quadratic parameter space (a little bit of fiction)
30) Short Introduction to Quartics
31) More about Quartics
32) Yet more about Quartics with a first attempt to Pentics

And here my CUBIC TUTORIAL and Pictures from Cubic Parameterspace. Can also be reached from my main page

And here myself on deviantART


Ingvar Kullberg

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