fractals and chaos theory

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Fractals and Fractals and Chaos Theory Chaos Theory Ruslan Kazantsev Rovaniemi Polytechnic, Finland

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Chaos originating from Order and Order originating from Chaos

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Page 1: Fractals and Chaos Theory

Fractals and Chaos Fractals and Chaos TheoryTheory

Ruslan KazantsevRovaniemi Polytechnic, Finland

Page 2: Fractals and Chaos Theory

Chaos Theory about disorder Chaos Theory about disorder NOT denying of determinismNOT denying of determinism NOT denying of ordered systemsNOT denying of ordered systems NOT announcement about useless of NOT announcement about useless of

complicated systemscomplicated systems

Chaos is main point of orderChaos is main point of order

Page 3: Fractals and Chaos Theory

What is the chaos theory?What is the chaos theory? Learning about complicated Learning about complicated

nonlinear dynamic systemsnonlinear dynamic systems

Nonlinear – Nonlinear – recursionrecursion and algorithms and algorithms Dynamic – Dynamic – variablevariable and and noncyclicnoncyclic

Page 4: Fractals and Chaos Theory

Wrong interpretationsWrong interpretations Society drew attention to the chaos Society drew attention to the chaos

theory because of such movies as theory because of such movies as Jurassic Park. And because of such Jurassic Park. And because of such things people are increasing the fear things people are increasing the fear of chaos theory.of chaos theory.

Because of it appeared a lot of wrong Because of it appeared a lot of wrong interpretations of chaos theoryinterpretations of chaos theory

Page 5: Fractals and Chaos Theory

Chaos Theory about disorderChaos Theory about disorder Truth that small changes could give Truth that small changes could give

huge consequences.huge consequences. Concept: impossible to find exact Concept: impossible to find exact

prediction of condition, but it gives prediction of condition, but it gives general general conditioncondition of system of system

Task is in modeling the system based Task is in modeling the system based on behavior of similar systems.on behavior of similar systems.

Page 6: Fractals and Chaos Theory

Usage of Chaos TheoryUsage of Chaos Theory Useful to have a look to things Useful to have a look to things

happening in the world different happening in the world different from traditional viewfrom traditional view

– Instead of X-Y Instead of X-Y graphgraph -> phase- -> phase-spatialspatial diagrams diagrams

– Instead of exact position of point -> Instead of exact position of point -> general general conditioncondition of system of system

Page 7: Fractals and Chaos Theory

Usage of Chaos TheoryUsage of Chaos Theory SSimulationimulation of of biologicalbiological systems (most chaotic systems (most chaotic

systems in the world)systems in the world) Systems of dynamic equations were used for Systems of dynamic equations were used for

simulating everything from population growth simulating everything from population growth and and epidemicepidemics to s to arrhythmicarrhythmic heart beating heart beating

Every system could be simulated: stock Every system could be simulated: stock exchange, even drops falling from the pipeexchange, even drops falling from the pipe

Fractal archivation claims in future coefficient Fractal archivation claims in future coefficient of of compressioncompression 600:1 600:1

Movie industry couldn’t have realistic Movie industry couldn’t have realistic landscapes (clouds, rocks, shadows) without landscapes (clouds, rocks, shadows) without technology of fractal graphicstechnology of fractal graphics

Page 8: Fractals and Chaos Theory

Brownian motion and it’s Brownian motion and it’s adaptationadaptation

Brownian motion – for example Brownian motion – for example accidentalaccidental and chaotic and chaotic motion of dust particles, weighted in water.motion of dust particles, weighted in water.

Output: frequency diagramOutput: frequency diagram

Could be transformed in musicCould be transformed in music Could be used for landscape creatingCould be used for landscape creating

Page 9: Fractals and Chaos Theory

Motion of Motion of billiardbilliard ball ball The slightest The slightest

mistake in angle of mistake in angle of first kick will follow first kick will follow to huge disposition to huge disposition after few after few collisioncollisions.s.

Impossible to Impossible to predict after 6-7 predict after 6-7 hitshits

Only way is to Only way is to show angle and show angle and length to each hitlength to each hit

Page 10: Fractals and Chaos Theory

Motion of Motion of billiardbilliard ball ball Every single loop or Every single loop or

dispersion area dispersion area presents ball behaviorpresents ball behavior

Area of picture, where Area of picture, where are results of one are results of one experiment is called experiment is called attraction area.attraction area.

This self-similarity will This self-similarity will last forever, if enlarge last forever, if enlarge picture for long, we’ll picture for long, we’ll still have same forms. still have same forms. => this will be => this will be FRACTALFRACTAL

Page 11: Fractals and Chaos Theory

Fusion of determined fractalsFusion of determined fractals Fractals are Fractals are predictablepredictable.. Fractals are made with aim to predict Fractals are made with aim to predict

systems in nature (for example migration systems in nature (for example migration of birds)of birds)

Page 12: Fractals and Chaos Theory

Tree simulation using Brownian motion Tree simulation using Brownian motion and fractal called and fractal called PythagPythagor Treeor Tree

Order of leaves and Order of leaves and branches is complicated branches is complicated and random, BUT can be and random, BUT can be emulated by short emulated by short program of 12 rows.program of 12 rows.

Firstly, we need to Firstly, we need to generate generate PythagPythagor Tree.or Tree.

Page 13: Fractals and Chaos Theory

Tree simulation using Brownian motion Tree simulation using Brownian motion and fractal called and fractal called PythagPythagor Treeor Tree

On this stage On this stage Brownian motion is Brownian motion is not used.not used.

Now, every section Now, every section is the centre of is the centre of symmetrysymmetry

Instead of lines are Instead of lines are rectanglerectangles.s.

But it still looks like But it still looks like artificialartificial

Page 14: Fractals and Chaos Theory

Tree simulation using Brownian motion Tree simulation using Brownian motion and fractal called and fractal called PythagPythagor Treeor Tree

Now Brownian Now Brownian motion is used motion is used to make to make randomizationrandomization

Numbers are Numbers are rounded-up to 2 rounded-up to 2 rank instead of rank instead of 3939

Page 15: Fractals and Chaos Theory

Tree simulation using Brownian motion Tree simulation using Brownian motion and fractal called and fractal called PythagPythagor Treeor Tree

Rounded-up to 7 Rounded-up to 7 rankrank

Now it looks like Now it looks like logarithmic spiral.logarithmic spiral.

Page 16: Fractals and Chaos Theory

Tree simulation using Brownian motion Tree simulation using Brownian motion and fractal called and fractal called PythagPythagor Treeor Tree

To avoid spiral we To avoid spiral we use Brownian use Brownian motion twice to the motion twice to the left and only once to left and only once to the rightthe right

Now numbers are Now numbers are rounded-up to 24 rounded-up to 24 rankrank

Page 17: Fractals and Chaos Theory

Fractals and world aroundFractals and world around Branching, leaves on trees, veins in hand, Branching, leaves on trees, veins in hand,

curving river, stock exchange – all these curving river, stock exchange – all these things are fractals.things are fractals.

Programmers and IT specialists go crazy Programmers and IT specialists go crazy with fractals. Because, in spite of its with fractals. Because, in spite of its beauty and complexity, they can be beauty and complexity, they can be generated with easy formulas.generated with easy formulas.

Discovery of fractals was discovery of new Discovery of fractals was discovery of new art aesthetics, science and math, and also art aesthetics, science and math, and also revolution in humans world revolution in humans world perceptionperception..

Page 18: Fractals and Chaos Theory

What are fractals in reality?What are fractals in reality? Fractal – geometric figure definite Fractal – geometric figure definite

part of which is repeating changing part of which is repeating changing its size => principle of self-similarity.its size => principle of self-similarity.

There are a lot of types of fractalsThere are a lot of types of fractals Not just complicated figures Not just complicated figures

generated by computers.generated by computers. Almost everything which seems to be Almost everything which seems to be

casualcasual could be fractal, even cloud or could be fractal, even cloud or little molecule of little molecule of oxygenoxygen..

Page 19: Fractals and Chaos Theory

How chaos is chaotic?How chaos is chaotic? Fractals – part of chaos theory.Fractals – part of chaos theory. Chaotic bChaotic behaviourehaviour, so they seem , so they seem disorderlydisorderly

and and casualcasual.. A lot of aspects of A lot of aspects of self-similarityself-similarity inside inside

fractal.fractal. Aim of studying fractals and chaos – to Aim of studying fractals and chaos – to

predict regularity in systems, which might be predict regularity in systems, which might be absolutely chaotic.absolutely chaotic.

All world around is fractal-likeAll world around is fractal-like

Page 20: Fractals and Chaos Theory

Geometry of 21Geometry of 21stst century century Pioneer, father of fractals was Franco-American Pioneer, father of fractals was Franco-American

professor Benoit B. Mandelbrot.professor Benoit B. Mandelbrot. 1960 “Fractal geometry of nature”1960 “Fractal geometry of nature” Purpose was to analyze not smooth and Purpose was to analyze not smooth and brokenbroken

forms.forms. Mandelbrot used word “fractal”, that meant Mandelbrot used word “fractal”, that meant

factionalismfactionalism of these forms of these forms Now Mandelbrot, Now Mandelbrot, Clifford AClifford A. . Pickover, James Pickover, James

Gleick, HGleick, H..OO. . Peitgen are trying to Peitgen are trying to enlargeenlarge area area of fractal geometry, so it can be used practical of fractal geometry, so it can be used practical all over the world, from prediction of costs on all over the world, from prediction of costs on stock exchange to new discoveries in stock exchange to new discoveries in theoretical physics.theoretical physics.

Page 21: Fractals and Chaos Theory

Practical usage of fractalsPractical usage of fractals Computer systems Computer systems (Fractal archivation, picture (Fractal archivation, picture

compressing without pixelization)compressing without pixelization) Liquid mechanicsLiquid mechanics

– Modulating of turbulent streamModulating of turbulent stream– Modulating of tongues of flameModulating of tongues of flame– Porous material has fractal structurePorous material has fractal structure

Telecommunications Telecommunications (antennas have fractal form)(antennas have fractal form) Surface physics Surface physics (for description of surface curvature)(for description of surface curvature) MedicineMedicine

– Biosensor interactionBiosensor interaction– Heart beatingHeart beating

Biology Biology (description of population model)(description of population model)

Page 22: Fractals and Chaos Theory

Fractal dimension: hidden dimensions Fractal dimension: hidden dimensions

Mandelbrot called not intact dimensions Mandelbrot called not intact dimensions – fractal dimensions (for example 2.76)– fractal dimensions (for example 2.76)

Euclid geometry claims that space is Euclid geometry claims that space is straight and flat.straight and flat.

Object which has 3 dimensions Object which has 3 dimensions correctly is impossiblecorrectly is impossible

Examples: Great Britain coastline, Examples: Great Britain coastline, human bodyhuman body

Page 23: Fractals and Chaos Theory

DeterministicDeterministic fractals fractals First opened fractals.First opened fractals. SSelf-similarityelf-similarity because of method of because of method of

generationgeneration Classic fractals, geometric fractals, linear Classic fractals, geometric fractals, linear

fractalsfractals Creation starts from initiator – basic Creation starts from initiator – basic

picturepicture Process of iteration – adding basic picture Process of iteration – adding basic picture

to every resultto every result

Page 24: Fractals and Chaos Theory

Sierpinskij Sierpinskij latticelattice Triangles made of Triangles made of

interconnection of middle interconnection of middle points of large triangle points of large triangle cut from main triangle, cut from main triangle, generating triangle with generating triangle with large amount of holes.large amount of holes.

Initiator – large triangle.Initiator – large triangle. Generator – process of Generator – process of

cutting triangles similar cutting triangles similar to given triangle.to given triangle.

Fractal dimension is Fractal dimension is 1.584962501 1.584962501

Page 25: Fractals and Chaos Theory

Sierpinskij Sierpinskij spongesponge Plane fractal cell Plane fractal cell

without square, but without square, but with unlimited tieswith unlimited ties

Would be used as Would be used as building building constructionsconstructions

Page 26: Fractals and Chaos Theory

Sierpinskij fractalSierpinskij fractal Don’t mix up this Don’t mix up this

fractal with fractal with Sierpinskij Sierpinskij latticelattice..

Initiator and Initiator and generator are the generator are the same.same.

Fractal dimension is Fractal dimension is 2.02.0

Page 27: Fractals and Chaos Theory

Koch CurveKoch Curve

One of the most typical fractals.One of the most typical fractals. Invented by german mathematic Helge fon Invented by german mathematic Helge fon

Koch Koch Initiator – straight line. Generator – Initiator – straight line. Generator –

equilateral triangle.equilateral triangle. Mandelbrot was making experiments with Mandelbrot was making experiments with

Koch Curve and had as a result Koch Islands, Koch Curve and had as a result Koch Islands, Koch Crosses, Koch Crystals, and also Koch Koch Crosses, Koch Crystals, and also Koch Curve in 3DCurve in 3D

Fractal dimension is Fractal dimension is 1.2618595071.261859507

Page 28: Fractals and Chaos Theory

Mandelbrot fractalMandelbrot fractal

Variant of Koch CurveVariant of Koch Curve Initiator and generator Initiator and generator

are different from are different from Koch’s, but idea is still Koch’s, but idea is still the same.the same.

Fractal takes half of Fractal takes half of plane.plane.

Fractal dimension is Fractal dimension is 1.51.5

Page 29: Fractals and Chaos Theory

Snow Crystal and StarSnow Crystal and Star This objects are classical fractals.This objects are classical fractals. Initiator and generator is one figureInitiator and generator is one figure

Page 30: Fractals and Chaos Theory

Minkovskij Minkovskij sausagesausage Inventor is German Inventor is German

Minkovskij.Minkovskij.

Initiator and generator are Initiator and generator are quite sophisticated, are quite sophisticated, are made of row of straight made of row of straight corners and segments with corners and segments with different length.different length.

Initiator has 8 parts.Initiator has 8 parts. Fractal dimension is 1.5Fractal dimension is 1.5

Page 31: Fractals and Chaos Theory

Labyrinth Labyrinth Sometimes called H-tree.Sometimes called H-tree. Initiator and generator has Initiator and generator has

shape of letter Hshape of letter H To see it easier the H form is To see it easier the H form is

not painted in the picture.not painted in the picture. Because of changing Because of changing

thicknessthickness, dimension on the , dimension on the tip is 2.0, but elements tip is 2.0, but elements between tips it is changing between tips it is changing from 1.333 to 1.6667from 1.333 to 1.6667

Page 32: Fractals and Chaos Theory

Darer Darer pentagonpentagon Pentagon as Pentagon as

initiatorinitiator Isosceles triangle Isosceles triangle

as generatoras generator Hexagon is a Hexagon is a

variant of this variant of this fractal (David Star)fractal (David Star)

Fractal dimension Fractal dimension is 1.86171is 1.86171

Page 33: Fractals and Chaos Theory

Dragon curveDragon curve Invented by Italian Invented by Italian

mathematic Giuseppe mathematic Giuseppe Piano.Piano.

Looks like Minkovskij Looks like Minkovskij sausagesausage, because has , because has the same generator the same generator and easier initiator.and easier initiator.

Mandelbrot called it Mandelbrot called it River of Double River of Double Dragon.Dragon.

Fractal dimension is Fractal dimension is 1.5236 1.5236

Page 34: Fractals and Chaos Theory

Hilbert curveHilbert curve Looks like labyrinth, Looks like labyrinth,

but letter “U” is used but letter “U” is used and and widthwidth is not is not changing.changing.

Fractal dimension is Fractal dimension is 2.02.0

Endless iteration Endless iteration could take all plane.could take all plane.

Page 35: Fractals and Chaos Theory

BoxBox Very Very simplesimple fractal fractal Made by adding Made by adding

squares to the top squares to the top of other squares.of other squares.

Initiator and Initiator and generator and generator and squares.squares.

Fractal dimension Fractal dimension is 1.892789261is 1.892789261

Page 36: Fractals and Chaos Theory

SSophisticatedophisticated fractals fractals Most fractals which you can meet in Most fractals which you can meet in

a real life are not deterministic.a real life are not deterministic. Not linear and not compiled from Not linear and not compiled from

periodic geometrical forms.periodic geometrical forms. Practically even enlarged part of Practically even enlarged part of

ssophisticatedophisticated fractal is different from fractal is different from initial fractal. They looks the same initial fractal. They looks the same but not almost identical.but not almost identical.

Page 37: Fractals and Chaos Theory

SSophisticatedophisticated fractals fractals Are generated by non Are generated by non

linear algebraic linear algebraic equations.equations.

ZnZn+1=+1=ZnZnІ + І + CC Solution involves Solution involves

complex and complex and supposed numberssupposed numbers

SSelf-similarityelf-similarity on on different scale levelsdifferent scale levels

Stable results – black, Stable results – black, for different speed for different speed different colordifferent color

Page 38: Fractals and Chaos Theory

Mandelbrot Mandelbrot multitudemultitude Most widespread Most widespread

sophisticated fractal sophisticated fractal ZnZn+1=+1=ZnaZna++CC Z and C – complex Z and C – complex

numbersnumbers a – any positive a – any positive

number.number.

Page 39: Fractals and Chaos Theory

Mandelbrot Mandelbrot multitudemultitude Z=Z*tg(Z+C).Z=Z*tg(Z+C). Because of Tangent Because of Tangent

function it looks like function it looks like Apple.Apple.

If we switch Cosine it If we switch Cosine it will look like Air will look like Air Bubbles.Bubbles.

So there are different So there are different properties for properties for Mandelbrot multitude.Mandelbrot multitude.

Page 40: Fractals and Chaos Theory

ZhuZhulia multitudelia multitude Has the same Has the same

formula as formula as Mandelbrot Mandelbrot multitude.multitude.

If building fractal If building fractal with different initial with different initial points, we will have points, we will have different pictures.different pictures.

Every dot in Every dot in Mandelbrot Mandelbrot multitude multitude corresponds to corresponds to Zhulia multitudeZhulia multitude