CHAOS THEORY, FRACTALS FRACTALS & ART

Frank Milordi(a.k.a. FAVIO)

The Start of Chaos Theory

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vs

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.506000

Difference = .000127

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Difference = .000127

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

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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

Lorenz’s Experiment

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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)

Chaos Definition

§ 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

§ The Butterfly Effect. A Butterfly flapping its wings in

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§ The Butterfly Effect. A Butterfly flapping its wings in Madagascar may cause a hurricane to hit the United States 3 weeks later!

Chaos Cartoon –The Project Selection Process

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Chaos That I Will Not Forget!!

§ As a young engineer I experienced chaos first-hand, in 1971 while testing the Navy’s F-14 fighter

§ Panel flutter at Mach 2.0 Plus at 40K ft

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§ AIM 54 store/wing flutter at Mach 1.05 at 5K ft

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

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§ 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

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

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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

Qualities of Fractals

§ Fractal Dimension

§ Complex Structure at All Scales

§ Infinite Branching

§ Self-Similarity

§ Chaotic Dynamics

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§ Chaotic Dynamics

* Note: all qualities do not apply to all Fractals

Fractal Dimensions

§ 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

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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

What Is A Fractal?

Approximating circle with polygons

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with polygons

The circumference of polygons inscribed in a unit circle

Approximating the length of the coastline of Britain

Estimation of the length of the coastline of Britain

What Is A Fractal? (contd)

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of the coastline of Britain

Big Things In A Small Package!

§ Newton’s Law § F = MA

§ Einstein’s Theory of Relativity§ E = MC2

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§ The Mendelbrot Set§ Zn+1 = Z2n + C (Z is a Complex Number)

The Mendelbrot Set

§ For Zn+1 = Z2n + C

(where Z is a complex number)

§ If Zn+1 is within the circle of Radius 2, it is in the set

§ If Zn+1 is outside the circle of Radius 2, it is not in the set

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§ The Mendelbrot Set is the most complex in mathematics!

Escaping & Test Orbit (Not Within Set)

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The escaping orbit of .37 + .41 Test orbit for .37 + .41

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

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

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§ 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

Statistically Self-Similar Fractal

(A) Brownian motion

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

Julia Sets via The Mendelbrot Set

30Julia Family

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

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§ “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.”

Creating Fractals

§ 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

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§ 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 may be the first person to view a specific image

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

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§ 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

3D View of Escape-Time Coloring

39Escape-time coloring of the Mandelbrot Set

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

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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

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

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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

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

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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

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

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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

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

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§ Music Generator

§ Turbulent Flow Modeling

§ Biological Modeling

§ Modeling of Space

§ Analysis of Organizational Systems

§ Modeling of Antenna

Fractal Programs

§ Fractal Programs for your consideration that are available via the WWW

§ Fractint (Free); lots of math, DOS-based, 8-bit color

§ ChaosPro (Free); great 24 bit color

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§ ChaosPro (Free); great 24 bit color

§ Ultra Fractal 3 ($59); integrates other Fractal programs, i.e., Fractint

QUESTIONS ?QUESTIONS ?QUESTIONS ?QUESTIONS ?

Legal Stuff!

Note:

*This summary is based on information from Fractal Creations; and Fractals, Chaos, Power Laws

*I want to thank Northrop Grumman for allowing

Note:

*This summary is based on information from Fractal Creations; and Fractals, Chaos, Power Laws

*I want to thank Northrop Grumman for allowing

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*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

*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