fractals and chaos theory
DESCRIPTION
Fashion, apparel, textile, merchandising, garmentsTRANSCRIPT
Fractals and Chaos Fractals and Chaos TheoryTheory
Ruslan KazantsevRovaniemi Polytechnic, Finland
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
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
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
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.
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
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
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
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
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
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)
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.
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
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
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.
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
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..
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..
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 predict regularity in systems, which might be absolutely chaotic.be absolutely chaotic.
All world around is fractal-likeAll world around is fractal-like
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.
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)
Fractal dimension: hidden dimensions Fractal dimension: hidden dimensions
Mandelbrot called not intact Mandelbrot called not intact dimensions – fractal dimensions (for dimensions – fractal dimensions (for example 2.76)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
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
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
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
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
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
Mandelbrot fractalMandelbrot fractal
Variant of Koch CurveVariant of Koch Curve Initiator and Initiator and
generator are generator are different from Koch’s, different from Koch’s, but idea is still the but idea is still the same.same.
Fractal takes half of Fractal takes half of plane.plane.
Fractal dimension is Fractal dimension is 1.51.5
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
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
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
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
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
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.
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
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.
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
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.
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.
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