# How do you describe a fractal mathematically?

The math background

Quote: "Benoit Mandelbrot 1975 - The fractal geometry of nature" (according to MathePrisma):

Clouds are not spheres, mountains are not cones, coastlines are not circles. The bark is not smooth - and lightning does not make its way straight either ...
The existence of such forms challenges us to study what Euclid leaves aside as formless, leads us to the morphology of the amorphous. So far, however, mathematicians have avoided this challenge. By developing theories that are no longer related to visible things, they have moved away from nature. In response to this, we will develop a new geometry of nature and demonstrate its usefulness in various fields. This new geometry describes many of the irregular and fragmented shapes around us - with a family of figures we will call fractals.

Fractal Type: IFS Fern, found on James Henstridge's Java Fractals
Everything you need to know about theory and still have fun with MathePrisma at the Bergische Universität Wuppertal.
Fractals and the game of chaoshttp://www.MathePrisma.uni-wuppertal.de/Module/Fraktal/index.htm
An introduction to the variety and mathematical backgrounds of fractals that is easy to follow.
Self-similarity, feedback, iterated function systems (IFS), affine mapping, geometric
Iteration, order in chaos, ornaments and fractals, lots of Java applets to try out.

Fractals for viewing on the internet

Very nicely designed gallery with stunning pictures: ´You have to call that.
The HOP Galleryhttp://www.mpeters.de/mpeweb/hop/gallery/gallery.htm

There are Competitions (Contests) for fractal art.

One of the prizewinners in 2000 is the picture "Bistred Mandorla", Artist: Tina Oloyedehttp://www.contest2k.com/en/entry-050-6.htm LOOK AT!

If you want to search for fractals on the web yourself, you can start here:

Spanky's Fractal Database "What's new?"http://spanky.triumf.ca/www/whats-new.html

Interactive fractals on the Internet

On z., B. On the following pages you will find Java applets to play with fractals and create your own pictures

Mandelbrot Set, Julia Set, Fractal Planet Generator, Java Applet http://www.ehynan.com/java_applet/fractal_applet/FractApplet/
I created the following with the planet generator there:

Planets and clouds created with the Fractal Planet Generator

A compilation and overview of different types of fractals:
Java fractals by James Henstridgehttp://daa.com.au/~james/fractals/

Classic fractals that can be zoomed in any depth
Fractal maphttp://frac.samurajdata.se

Mandelbrot Volcano, multiple zoom, found on Fractal-Map

Mandelbrot Blues, multiple zoom, found on Fractal-Map

My software favorites

Here are some software links. I tried the following programs and with them the
most of the images created here on the site. There are a lot more offers if you are looking for it then
check out The Spanky Fractal Database New Fractal Programs or Specific Fractal Programs at Other Sites.
A search on cnet also brings plenty of results.
I've only considered programs that run on Windows NT4.0 or 2000.
The software is freeware or shareware. You can find the details on the respective download page.

Ultra Fractal, shareware softwarehttp://www.ultrafractal.com
Most extensive formula library, i.e. an extremely large number of different basic patterns. Can use layers, color gradients.
Anyone who knows Java can write their own IFS functions and classes and thus expand the formula library.

Two layers Julia, made with Ultra-Fractal

Fractal eXtremehttp://www.cygnus-software.com/index.html
Various basic objects. Can zoom movie and iteration movie. Interesting color selection dialogs (gradient)

Two of the Cubic Newton type created with Fractal eXtreme

Aros Fractals for Windows95 and Windows NThttp://www.arosmagic.com
Only 3 types of images. In addition to Mandelbrot and Newton, beautiful textures are also possible

Two of the type Plasma Cloud created with Aros Fractals

Diverse, brings some more basic types and design ideas, but no preview when selecting and changing parameters.
You just have to click a lot: Color dialog on - change something - press ok - see result.
With this program the picture is created at the top of the page (pair with circles).

Finally

I am fascinated by the possibilities these programs offer. In a short time I have created the best pictures.
Here are a few more. I've lost track of what is done with which program.

A great peculiarity of the fractal images is that you can zoom in - without end. The small picture is an enlargement of one of the small black dots in the first picture.

Have fun discovering yourself! Martin Wohlgemuth for Matroids Matheplanet 8.4.2001.
Update of the external links on July 20th, 2002