## The chaotic series of fractal articles

## Preface:

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:

- 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.
- The science of fractals, and the mathematics behind, is a part of the new science of Chaos.
- 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:

- Lack of time and energy.
- Lack of knowledge of how to create web sites. (Different persons has been a good help for my main site).
Now, however the time has come. I decided to publish the articles as pdf-documents. I think most of you have

Adobe Readeras 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:

- It was possible to perform the layout using my favorite application,
ClarisWorks 5.0for Windows, and thereafter print them out as pdf-files.- It made it possible to write exponents and indexes in a correct manner instead of using the signs (^) and (_). I think that would be much more difficult using html-editors.
- I think it would be much easier for the dear reader to print them out on papers. Moreover I've designed the documents to be nice readable (I hope) when printed out in duplex (on both sides).
- An important thing when you read the articles on the screen. In some images there are text etc to illustrate certain things. To see these illustrations clearly, please put the size of the document to about 125%.

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.That's for you who are lucky to be owners of this application.The parameter files to these images can be downloaded from here.

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 isCubic Parameterspace3. It also containsQuartic Parameterspace3,andPentic 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.Some lessons regarding cubics are also slightly rewritten.All his modules, as well as mine, can be downloaded from here.

I myselfwho 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 ofcomplex numbersand the concept ofiteration.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:http://www.ddewey.net/mandelbrot

http://hypertextbook.com/chaos/11.shtml## 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 TUTORIALandPictures from Cubic Parameterspace.Can also be reached from mymain pageAnd here myself on

deviantARTRegards,

Ingvar KullbergUnique visitors since 2006.05.20