Various Fractal Types and Categories of 'Chaos'
COMPLEX PLANE FRACTALS
fractals that can be produced by iterating a fractal formula with respect to a sample point on the complex plane and coloring each sample point based on the size and characteristics of the resulting iteration. Related fractal types include Mandelbrot fractals, Julia fractals, Convergent fractals, Newton fractals, and Orbit Traps.
Julia Sets
Julia Sets are produced with the same formula as the Mandelbrot set, but the starting values are different. That is c is some constant, and z0 is the starting point on the plane. From this definition, you can see that there is an infinite number of Julia Sets; one for each value of c. In fact, there is a Julia Set that corresponds to each point on complex plane.
There is an interesting relationship between the Mandelbrot Set and the Julia Sets. In a way, you can think of the Mandelbrot Set as an index for the Julia Sets. For values of c that are inside the Mandelbrot Set, you will get connected Julia Sets. That is all the black regions are connected. Conversely, those values of c outside the Mandelbrot Set, you get unconnected sets.
Mandelbrot Sets
The Mandelbrot set is the subset of the complex plane consisting of those parameters for which the Julia set of is connected.
The Mandelbrot set can also be defined as the set of parameters for which the set has a finite upper bound.
Kleinian Group Fractals
Kleinian Group fractals are fractals based on 2 pairs of Mobius transformations and allow you to produce Quasifuchsian, Single Cusp, and Double Cusp, Two-Generator Group fractals described in the book Indra's Pearls - The Vision of Felix Klein by David Mumford, Caroline Series, and David Wright.
Newton Method Fractals
Isaac Newton discovered what we now call Newton's method around 1670. Although Newton's method is an old application of calculus, it was discovered relatively recently that extending it to the complex plane leads to a very interesting fractal pattern.
Quaternion 3D Fractals
.Quaternion Julia fractals are created by the same principle as the more traditional Julia set
except that
it uses 4 dimensional complex numbers instead of 2 dimensional complex numbers
Hyper-Complex and Three Dimensional Fractals
There are many methods for visualizing the maps of 2 dimensional fractals in 3 or more dimensions.
The results are often surprising, as in the case of the 'Mandelbulb' discovery.
Bifurcation Diagrams & Chaotic Attractors
Bifurcation diagrams were one of the first ways of exploring chaos visually. The bifurcation diagram nicely shows the forking of the possible periods of stable orbits from 1 to 2 to 4 to 8 etc. Each of these bifurcation points is a period-doubling bifurcation. The ratio of the lengths of successive intervals between values of r for which bifurcation occurs converges to the first Feigenbaum constant. These diagrams represent many processes involving period-doubling, such as populaiton growth, weather patterns and chaotic fluid dynamics.
Strange Attractors
Strange Attractors are defined by an equation or system of equations. The orbit points are generated by passing the current orbit point through the equations to obtain the next orbit point. This process is repeated thousands (or millions) of times to produce the fractal data. Of course, most equations will not produce a fractal and the challenge is to find equations that do. Quadratic Attractors and Cubic Attractors are examples of these fractal types.
IFS Fractals:
IFS stands for Iterated Function System. Fractals of this type are created by applying one of a number of functions, chosen randomly from the rules set up for the IFS, repeatedly to an intitial point, and graphing each new point. With IFS fractals, it can be seen that the starting point does not effect the shape of the fractal too much. This means that a particular fractal can be defined by the rules used to find the next point, and the probabilities that an individual function will be chosen. This fractal type consists in a series of affine transformations (scaling, rotation, shear and translation) that are iterater thought the chaos game..
Polynomial Root Fractal
The Strange fractal patterns emerge when you plot the complex roots of high order polynomials. This picture shows all the roots for all possible combinations of 18th order polynomials with coefficients of ±1. Notice that some parts bear a remarkable resemblance to the Heighway Dragon Curve (seen in blue at far right above this text) .
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Quadratic Rational Fractals
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Mandelbrot Variations
The Mandelbrot set is the subset of the complex plane consisting of those parameters for which the Julia set of is connected.
The many alternate Mandelbrot like fractals can be produced by altering the equation being mapped on the complex plane
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Not sure of the name of this fractal, but i've seen a few versions around lately
