During the early 1960’s, Edward Lorenz, a meteorologist, began to examine how natural systems such as the weather changed with time. He found that:
- These systems are not haphazard
- Lurking underneath is a remarkable subtle form of order
- A new scientific field called Chaos Theory to a certain degree explains nature’s dynamics
- Chaos Theory is based on non-linear mathematics
- A nonlinear system where feedback quickly magnifies small changes as that so that the effect is out of proportion to the cause. The system is so webbed with positive feedback that the slightest twitch anywhere may become amplified into an unexpected conclusion or transformation
- The Butterfly Effect. A Butterfly flapping its wings in Madagascar may cause a hurricane to hit the United States 3 weeks later.
In the 1970’s, a new form of Geometry was discovered to describe patterns associated with chaotic processes. This new Geometry called Fractals was discovered by Benoit Mandelbrot of IBM. Fractals:
- Look nothing like traditional smooth Euclidean shapes
- Consist of patterns that recur or repeat at finer and finer magnification
- Define shapes of immense complexity and chaos
- Are a visual representation of Chaos Theory
- If the estimated length of a curve becomes arbitrarily large as the measuring stick becomes smaller and smaller, then the curve is called a fractal curve
- Fractals describe the roughness of the world, its energy, its dynamical changes and transformations. Fractals are images of the way things fold and unfold, feeding back into each other and themselves. The study of fractals has confirmed many of the chaologists’ insights into chaos, and has uncovered some unexpected secrets of nature’s dynamical movements as well.
The Mandelbrot Formula. Big things in a small package.
Zn+1 = Z2n + K - Mandelbrot (Z is a Complex Number)
F = MA - Newton
Fractals display self-similarity – that is they have a similar appearance at any magnification. A small part of the structure looks very much like the whole.
Self-Similarity comes in two flavors: Exact and Statistical.
- Exact repetition of patterns at different magnifications
- Statistical patterns don’t repeat exactly, instead statistical quantities of patterns repeat
- Most of nature’s patterns obey Statistical Self-Similarity
Fractal Dimension (D) quantifies the scaling relation among patterns observed at different magnifications.
- For Euclidean shapes for a line (D = 1)) and encapsulated filed area such as a square (D = 2)
- For Fractal patterns, D lies between 1 and 2. D = 2 for the Mendelbrot set.
- As the complexity or irregularity of the repeating Fractal structure increases; D approaches 2
- D is related to how fast the estimated measurement of the object increases as the measurement device becomes smaller
Fractals Relation to Art.
- M.C. Escher’s “Circular Limit” images are examples of Exact Self-Similarity
- Jackson Pollock’s “Drip Paintings” are examples of Statistical Self-Similarity.
- Note: Pollock did not like the phrase, “Drip Painting.” He preferred something like -“Controlled Trajectories”
Chaos, Fractals and Computers
- Chaos and Fractal formula solutions agree bases on iteration
- Computer are ideally suited for iteration
- Computer Graphics are ideally suited too visually represent Chaotic Systems and Fractals
- “The computer is the silent hero of Chaos Theory and Fractals. Computers have acted as the most forceful forceps in extracting Fractals (and Chaotic Systems) from the dark recesses of abstract mathematics and delivering their geometric intricacies into the bright daylight.
- Explore various Fractal types and regions within each Fractal type
- Look for regions that are very non-linear or chaotic because they are (extremely sensitive to minute parameter change)
- These chaotic regions produce vastly different Fractal images for parameter change as small as .0001
- These chaotic regions can yield numerous interesting and unique images
- Fractal space is infinite. Therefore you are probably the first person to view these images
Coloring a Fractal
- A Black (within the Mandelbrot Set) and White (outside the Mandelbrot Set)
define the Mandelbrot set boundary.
- Spectacular color Fractal images are a result of points near the set, but not
- Color those points outside the set based on number of iterations required to
escape (E.G., 10 iteration – Red, 11 Iterations – Yellow, 12 Iterations – Blue, ---)
- Additional image variation will result as a functioning of the color assignment associated with the number of iterations required to calculate the Fractal
- Cycle the color gradient
Application of Fractals
- Compression, weather modeling, stock market prediction, art, music generator, turbulent flow modeling, biological modeling, modeling of space, analysis of organizational systems
Fractal Programs for your consideration that are available via the WWW.
- Fractint (Free)
- ChaosPro (Free)
- Ultra Fractal 3 ($59)
Note: This summary is based on information from Fractal Creations and Fractals, Chaos, Power Laws