Chaos Theory

CHAOS THEORY, FRACTALS & ART Frank Milordi (a.k.a. FAVIO)

The Start of Chaos Theory

.506127 vs .506000 Difference = .000127

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Edward Lorenz § 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

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Lorenz’s Experiment

Lorenz’s experiment: the difference between the start of these curves is only .000127 (Ian Stewart, Does God Play Dice? The Mathematics of Chaos, pg 141)

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Chaos Definition ยง A nonlinear system where feedback quickly magnifies small changes 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 ยง The Butterfly Effect. A Butterfly flapping its wings in Madagascar may cause a hurricane to hit the United States 3 weeks later!

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Chaos Cartoon – The Project Selection Process

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Chaos That I Will Not Forget!! § As a young engineer I experienced chaos firsthand, in 1971 while testing the Navy’s F-14 fighter § Panel flutter at Mach 2.0 Plus at 40K ft § AIM 54 store/wing flutter at Mach 1.05 at 5K ft

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Benoit Mendelbrot § 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

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Fractal Definition § 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

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Qualities of Fractals ยง Fractal Dimension ยง Complex Structure at All Scales ยง Infinite Branching ยง Self-Similarity ยง Chaotic Dynamics

* Note: all qualities do not apply to all Fractals

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Fractal Dimensions ยง Fractal Dimension (D) quantifies the scaling relation among patterns observed at different magnifications ยง For Euclidean shapes for a line (D = 1) and encapsulated filed area such as a square (D = 2) ยง For Fractal patterns, D lies between 1 and 2. D = 2 for the Mendelbrot set ยง As the complexity or irregularity of the repeating Fractal structure increases; D approaches 2 ยง D is related to how fast the estimated measurement of the object increases as the measurement device becomes smaller

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What Is A Fractal?

Approximating circle with polygons

The circumference of polygons inscribed in a unit circle 19

What Is A Fractal? (contd) Approximating the length of the coastline of Britain

Estimation of the length of the coastline of Britain

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Big Things In A Small Package! § Newton’s Law § F = MA § Einstein’s Theory of Relativity § E = MC2 § The Mendelbrot Set § Zn+1 = Z2n + C (Z is a Complex Number)

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The Mendelbrot Set ยง For Zn+1 = Z2n + C (where Z is a complex number) ยง If Zn+1 is within the circle of Radius 2, it is in the set ยง If Zn+1 is outside the circle of Radius 2, it is not in the set ยง The Mendelbrot Set is the most complex in mathematics!

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Escaping & Test Orbit (Not Within Set)

The escaping orbit of .37 + .41 23

Test orbit for .37 + .41

Non-Escaping & Test Orbit Define The Mendelbrot Set (Within Set)

The non-escaping orbit of .37 + .2i 24

Test orbit for .37 + .2i

Self-Similarity § 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 SelfSimilarity

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Statistically Self-Similar Fractal (A) Brownian motion

(B) The segment AB of the Brownian motion in part A sampled 100 times more frequently and magnified 10 times

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Julia Sets via The Mendelbrot Set

Julia Family 30

Chaos, Fractals and Computers § Chaos and Fractal formula solutions are bases on iteration § Computer are ideally suited for iteration § Computer Graphics are ideally suited to 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.”

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Creating Fractals ยง Explore various Fractal types and regions within each Fractal type ยง Look for regions that are very non-linear or chaotic because they are (extremely sensitive to minute parameter change) ยง These chaotic regions produce vastly different Fractal images for parameter change as small as .0001 ยง These chaotic regions can yield numerous interesting and unique images ยง Fractal space is infinite. Therefore you may be the first person to view a specific image

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Coloring a Fractal § A Black color (within the Mendelbrot Set) and White (outside the Mendelbrot Set) define the Mendelbrot Set boundary § Spectacular color Fractal images are a result of points near the set boundary, but not within it § 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 § Increase the repetition rate 38

3D View of Escape-Time Coloring

Escape-time coloring of the Mandelbrot Set 39

Fractals Relation to Art § Fractal are visually pleasing to those who enjoy color, texture and patterns § M.C. Escher’s “Circular Limit” images are examples of Exact Self-Similarity § Jackson Pollock’s “Drip Paintings” are examples of Statistical Self-Similarity § Pollock did not like the phrase, “Drip Painting.” He preferred something like - “Controlled Trajectories” § Numerous Van Gogh images have a Fractal basis § Quite a few artist use Fractals as a Form of Art, including FAVIO, and as a basis for Abstract Art § Basic Fractals § Annihilated Fractals 41

Fractals Relation to Art § Fractal are visually pleasing to those who enjoy color, texture and patterns § M.C. Escher’s “Circular Limit” images are examples of Exact Self-Similarity § Jackson Pollock’s “Drip Paintings” are examples of Statistical Self-Similarity § Pollock did not like the phrase, “Drip Painting.” He preferred something like - “Controlled Trajectories” § Numerous Van Gogh images have a Fractal basis § Quite a few artist use Fractals as a Form of Art, including FAVIO, and as a basis for Abstract Art § Basic Fractals § Annihilated Fractals 44

Fractals Relation to Art § Fractal are visually pleasing to those who enjoy color, texture and patterns § M.C. Escher’s “Circular Limit” images are examples of Exact Self-Similarity § Jackson Pollock’s “Drip Paintings” are examples of Statistical Self-Similarity § Pollock did not like the phrase, “Drip Painting.” He preferred something like - “Controlled Trajectories” § Numerous Van Gogh images have a Fractal basis § Quite a few artist use Fractals as a Form of Art, including FAVIO, and as a basis for Abstract Art § Basic Fractals § Annihilated Fractals 47

Fractals Relation to Art § Fractal are visually pleasing to those who enjoy color, texture and patterns § M.C. Escher’s “Circular Limit” images are examples of Exact Self-Similarity § Jackson Pollock’s “Drip Paintings” are examples of Statistical Self-Similarity § Pollock did not like the phrase, “Drip Painting.” He preferred something like - “Controlled Trajectories” § Numerous Van Gogh images have a Fractal basis § Quite a few artist use Fractals as a Form of Art, including FAVIO, and as a basis for Abstract Art § Basic Fractals § Annihilated Fractals 53

Da Vinci’s Vitruvian Man

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Potential Names § Jeannie Out of the Bottle § Spagettification § Black Hole § Waterfall § Ram’s Head § XXX

Application of Fractals § Compression § Weather Modeling § Stock Market Prediction § Art § Music Generator § Turbulent Flow Modeling § Biological Modeling § Modeling of Space § Analysis of Organizational Systems § Modeling of Antenna

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Fractal Programs ยง Fractal Programs for your consideration that are available via the WWW ยง Fractint (Free); lots of math, DOS-based, 8-bit color ยง ChaosPro (Free); great 24 bit color ยง Ultra Fractal 3 ($59); integrates other Fractal programs, i.e., Fractint

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Note: *This summary is based on information from Fractal Creations; and Fractals, Chaos, Power Laws *I want to thank Northrop Grumman for allowing me to develop this presentation in support of my participation in a local college advisory board resulting in lectures to several student groups. In addition this presentation was used to foster a bond with other companies that participated in national Engineer’s Week

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