Fractal Mathematics
Benoit Mandelbrot is generally considered to be the father of fractals. He coined the term fractal to describe curves, surfaces and objects that have some very peculiar properties. You learned in school that simple curves, such as a line, have one dimension. Squares, rectangles, circles, polygons, etc. have two dimensions, while solid objects such as a cube, have three dimensions. The three dimensions define space. Time can be considered a fourth dimension. We normally think of dimensions as integers: 1, 2, 3, . . .
What is so peculiar about about fractals is that they have fractional dimensions! A fractal curve could have a dimensionality of 1.4332, for example, rather than 1. Fractals are not just a mathematical curiosity. Most natural objects are fractal by nature, and can be best described using fractal mathematics. Clouds, leaves, the blood vessel system, coastlines, particles of lint, etc. have fractal shapes.
Fractals are generated by an iterative process  doing the same thing again and again. Fractals also have the property that when you magnify them they still look much the same. This is called selfsimilarity.
Click on an image or link to see the mathematics behind each fractal type.  
Divergent Fractals


Convergent Fractals
These are fractals generated using convergent iterative methods such as Newton's Method. You may be surprised  some of the images may look like they came from a "divergent" function! 

Hyperbolic Tessellation Fractals
Hyperbolic geometry is a nonEuclidean geometry developed independently by Nikolai Lobachevski and Farkas Bolyai. The work of the artist M.C. Escher contains examples of Hyperbolic Tessellation Fractals. 

3D Fractals and Higher
Dimensions
A Mandelbrot set and its corresponding Julia sets comprise the 4dimensional Juliabrot. Quaternions and Hypercomplex objects are also 4D fractals. 3D fractals are 3dimensional cuts through these 4D objects. 

Circle and Sphere Inversion Fractals
Circle and sphere inversions are closely related to Möbius transformations. The best known fractal example is the Apollonian Gasket. 

Kleinian Group
Fractals
Mobius transformations, which are also known as fractional linear transformations, can be used to generate a variety of fractals. This has been popularized by the book "Indra's Pearls" by Mumford, Series and Wright.  
Height Field Fractals
For a 2dimensional fractal the 3rd dimension is a function of the number of iterations, the fractal magnitude, or the orbit trap value. 

Strange Attractors
Dynamical systems are models with rules that describe the way a quantity changes with time. These systems can behave a strange attractor. It can be graphed but its behavior is complex and unpredictable. Like leaves in the wind, it is impossible to predict where the leaves will end up. 

Iterated Function System (IFS) Fractals
IFS fractals were developed by Michael Barnsley. The original IFS formulas are called contractive affine transformations, which provide specifications for the selfsimilarity of the fractal. IFS fractals can also be created using Möbius transformations. 

L
System
Fractals
Developed by Aristid Lindenmayer to model the morphology of organisms. It is an iterative turtle graphics system. 
