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 self-similarity.


Click on an image or link to see the mathematics behind each fractal type.
Divergent Fractals

Divergent Fractals 
These are the "classic" fractal functions, typified by the Mandelbrot set and the corresponding Julia sets. "Divergent" relates to the regions that are most interesting in creating striking images and art.

Convergent 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 Tessellation Fractals  
Hyperbolic geometry is a non-Euclidean geometry developed independently by Nikolai Lobachevski and Farkas Bolyai. The work of the artist M.C. Escher contains examples of Hyperbolic Tessellation Fractals.
JuliaBrot Fractals 3D Fractals and Higher Dimensions
A Mandelbrot set and its corresponding Julia sets comprise the 4-dimensional Juliabrot. Quaternions and Hypercomplex objects are also 4D fractals. 3D fractals are 3-dimensional cuts through these 4-D objects. 
Circle and Sphere Inversion Fractals 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 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 Height Field Fractals  
For a 2-dimensional fractal the 3rd dimension is a function of the number of iterations, the fractal magnitude, or the orbit trap value.
Strange Attractors 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. 
IFS Fractals 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 self-similarity of the fractal. IFS fractals can also be created using Möbius transformations.
L System Fractals L System Fractals
Developed by  Aristid Lindenmayer to model the morphology of organisms. It is an iterative turtle graphics system.