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1 Fractals and Chaos Theory Ruslan Kazantsev Rovaniemi Polytechnic, Finland

2 Chaos Theory about disorder NOT denying of determinism NOT denying of determinism NOT denying of ordered systems NOT denying of ordered systems NOT announcement about useless of complicated systems NOT announcement about useless of complicated systems Chaos is main point of order Chaos is main point of order

3 What is the chaos theory? Learning about complicated nonlinear dynamic systems Learning about complicated nonlinear dynamic systems Nonlinear – recursion and algorithms Nonlinear – recursion and algorithms Dynamic – variable and noncyclic Dynamic – variable and noncyclic

4 Wrong interpretations Society drew attention to the chaos theory because of such movies as Jurassic Park. And because of such things people are increasing the fear of chaos theory. Society drew attention to the chaos theory because of such movies as Jurassic Park. And because of such things people are increasing the fear of chaos theory. Because of it appeared a lot of wrong interpretations of chaos theory Because of it appeared a lot of wrong interpretations of chaos theory

5 Chaos Theory about disorder Truth that small changes could give huge consequences. Truth that small changes could give huge consequences. Concept: impossible to find exact prediction of condition, but it gives general condition of system Concept: impossible to find exact prediction of condition, but it gives general condition of system Task is in modeling the system based on behavior of similar systems. Task is in modeling the system based on behavior of similar systems.

6 Usage of Chaos Theory Useful to have a look to things happening in the world different from traditional view Useful to have a look to things happening in the world different from traditional view –Instead of X-Y graph -> phase-spatial diagrams –Instead of exact position of point -> general condition of system

7 Usage of Chaos Theory Simulation of biological systems (most chaotic systems in the world) Simulation of biological systems (most chaotic systems in the world) Systems of dynamic equations were used for simulating everything from population growth and epidemics to arrhythmic heart beating Systems of dynamic equations were used for simulating everything from population growth and epidemics to arrhythmic heart beating Every system could be simulated: stock exchange, even drops falling from the pipe Every system could be simulated: stock exchange, even drops falling from the pipe Fractal archivation claims in future coefficient of compression 600:1 Fractal archivation claims in future coefficient of compression 600:1 Movie industry couldnt have realistic landscapes (clouds, rocks, shadows) without technology of fractal graphics Movie industry couldnt have realistic landscapes (clouds, rocks, shadows) without technology of fractal graphics

8 Brownian motion and its adaptation Brownian motion – for example accidental and chaotic motion of dust particles, weighted in water. Brownian motion – for example accidental and chaotic motion of dust particles, weighted in water. Output: frequency diagram Output: frequency diagram Could be transformed in music Could be transformed in music Could be used for landscape creating Could be used for landscape creating

9 Motion of billiard ball The slightest mistake in angle of first kick will follow to huge disposition after few collisions. The slightest mistake in angle of first kick will follow to huge disposition after few collisions. Impossible to predict after 6-7 hits Impossible to predict after 6-7 hits Only way is to show angle and length to each hit Only way is to show angle and length to each hit

10 Motion of billiard ball Every single loop or dispersion area presents ball behavior Every single loop or dispersion area presents ball behavior Area of picture, where are results of one experiment is called attraction area. Area of picture, where are results of one experiment is called attraction area. This self-similarity will last forever, if enlarge picture for long, well still have same forms. => this will be FRACTAL This self-similarity will last forever, if enlarge picture for long, well still have same forms. => this will be FRACTAL

11 Fusion of determined fractals Fractals are predictable. Fractals are predictable. Fractals are made with aim to predict systems in nature (for example migration of birds) Fractals are made with aim to predict systems in nature (for example migration of birds)

12 Tree simulation using Brownian motion and fractal called Pythagor Tree Order of leaves and branches is complicated and random, BUT can be emulated by short program of 12 rows. Order of leaves and branches is complicated and random, BUT can be emulated by short program of 12 rows. Firstly, we need to generate Pythagor Tree. Firstly, we need to generate Pythagor Tree.

13 Tree simulation using Brownian motion and fractal called Pythagor Tree On this stage Brownian motion is not used. On this stage Brownian motion is not used. Now, every section is the centre of symmetry Now, every section is the centre of symmetry Instead of lines are rectangles. Instead of lines are rectangles. But it still looks like artificial But it still looks like artificial

14 Tree simulation using Brownian motion and fractal called Pythagor Tree Now Brownian motion is used to make randomization Now Brownian motion is used to make randomization Numbers are rounded-up to 2 rank instead of 39 Numbers are rounded-up to 2 rank instead of 39

15 Tree simulation using Brownian motion and fractal called Pythagor Tree Rounded-up to 7 rank Rounded-up to 7 rank Now it looks like logarithmic spiral. Now it looks like logarithmic spiral.

16 Tree simulation using Brownian motion and fractal called Pythagor Tree To avoid spiral we use Brownian motion twice to the left and only once to the right To avoid spiral we use Brownian motion twice to the left and only once to the right Now numbers are rounded-up to 24 rank Now numbers are rounded-up to 24 rank

17 Fractals and world around Branching, leaves on trees, veins in hand, curving river, stock exchange – all these things are fractals. Branching, leaves on trees, veins in hand, curving river, stock exchange – all these things are fractals. Programmers and IT specialists go crazy with fractals. Because, in spite of its beauty and complexity, they can be generated with easy formulas. Programmers and IT specialists go crazy with fractals. Because, in spite of its beauty and complexity, they can be generated with easy formulas. Discovery of fractals was discovery of new art aesthetics, science and math, and also revolution in humans world perception. Discovery of fractals was discovery of new art aesthetics, science and math, and also revolution in humans world perception.

18 What are fractals in reality? Fractal – geometric figure definite part of which is repeating changing its size => principle of self-similarity. Fractal – geometric figure definite part of which is repeating changing its size => principle of self-similarity. There are a lot of types of fractals There are a lot of types of fractals Not just complicated figures generated by computers. Not just complicated figures generated by computers. Almost everything which seems to be casual could be fractal, even cloud or little molecule of oxygen. Almost everything which seems to be casual could be fractal, even cloud or little molecule of oxygen.

19 How chaos is chaotic? Fractals – part of chaos theory. Fractals – part of chaos theory. Chaotic behaviour, so they seem disorderly and casual. Chaotic behaviour, so they seem disorderly and casual. A lot of aspects of self-similarity inside fractal. A lot of aspects of self-similarity inside fractal. Aim of studying fractals and chaos – to predict regularity in systems, which might be absolutely chaotic. Aim of studying fractals and chaos – to predict regularity in systems, which might be absolutely chaotic. All world around is fractal-like All world around is fractal-like

20 Geometry of 21 st century Pioneer, father of fractals was Franco-American professor Benoit B. Mandelbrot. Pioneer, father of fractals was Franco-American professor Benoit B. Mandelbrot Fractal geometry of nature 1960 Fractal geometry of nature Purpose was to analyze not smooth and broken forms. Purpose was to analyze not smooth and broken forms. Mandelbrot used word fractal, that meant factionalism of these forms Mandelbrot used word fractal, that meant factionalism of these forms Now Mandelbrot, Clifford A. Pickover, James Gleick, H.O. Peitgen are trying to enlarge area of fractal geometry, so it can be used practical all over the world, from prediction of costs on stock exchange to new discoveries in theoretical physics. Now Mandelbrot, Clifford A. Pickover, James Gleick, H.O. Peitgen are trying to enlarge area of fractal geometry, so it can be used practical all over the world, from prediction of costs on stock exchange to new discoveries in theoretical physics.

21 Practical usage of fractals Computer systems (Fractal archivation, picture compressing without pixelization) Computer systems (Fractal archivation, picture compressing without pixelization) Liquid mechanics Liquid mechanics –Modulating of turbulent stream –Modulating of tongues of flame –Porous material has fractal structure Telecommunications (antennas have fractal form) Telecommunications (antennas have fractal form) Surface physics (for description of surface curvature) Surface physics (for description of surface curvature) Medicine Medicine –Biosensor interaction –Heart beating Biology (description of population model) Biology (description of population model)

22 Fractal dimension: hidden dimensions Mandelbrot called not intact dimensions – fractal dimensions (for example 2.76) Mandelbrot called not intact dimensions – fractal dimensions (for example 2.76) Euclid geometry claims that space is straight and flat. Euclid geometry claims that space is straight and flat. Object which has 3 dimensions correctly is impossible Object which has 3 dimensions correctly is impossible Examples: Great Britain coastline, human body Examples: Great Britain coastline, human body

23 Deterministic fractals First opened fractals. First opened fractals. Self-similarity because of method of generation Self-similarity because of method of generation Classic fractals, geometric fractals, linear fractals Classic fractals, geometric fractals, linear fractals Creation starts from initiator – basic picture Creation starts from initiator – basic picture Process of iteration – adding basic picture to every result Process of iteration – adding basic picture to every result

24 Sierpinskij lattice Triangles made of interconnection of middle points of large triangle cut from main triangle, generating triangle with large amount of holes. Triangles made of interconnection of middle points of large triangle cut from main triangle, generating triangle with large amount of holes. Initiator – large triangle. Initiator – large triangle. Generator – process of cutting triangles similar to given triangle. Generator – process of cutting triangles similar to given triangle. Fractal dimension is Fractal dimension is

25 Sierpinskij sponge Plane fractal cell without square, but with unlimited ties Plane fractal cell without square, but with unlimited ties Would be used as building constructions Would be used as building constructions

26 Sierpinskij fractal Dont mix up this fractal with Sierpinskij lattice. Dont mix up this fractal with Sierpinskij lattice. Initiator and generator are the same. Initiator and generator are the same. Fractal dimension is 2.0 Fractal dimension is 2.0

27 Koch Curve One of the most typical fractals. One of the most typical fractals. Invented by german mathematic Helge fon Koch Invented by german mathematic Helge fon Koch Initiator – straight line. Generator – equilateral triangle. Initiator – straight line. Generator – equilateral triangle. Mandelbrot was making experiments with Koch Curve and had as a result Koch Islands, Koch Crosses, Koch Crystals, and also Koch Curve in 3D Mandelbrot was making experiments with Koch Curve and had as a result Koch Islands, Koch Crosses, Koch Crystals, and also Koch Curve in 3D Fractal dimension is Fractal dimension is

28 Mandelbrot fractal Variant of Koch Curve Variant of Koch Curve Initiator and generator are different from Kochs, but idea is still the same. Initiator and generator are different from Kochs, but idea is still the same. Fractal takes half of plane. Fractal takes half of plane. Fractal dimension is 1.5 Fractal dimension is 1.5

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

30 Minkovskij sausage Inventor is German Minkovskij. Inventor is German Minkovskij. Initiator and generator are quite sophisticated, are made of row of straight corners and segments with different length. Initiator and generator are quite sophisticated, are made of row of straight corners and segments with different length. Initiator has 8 parts. Initiator has 8 parts. Fractal dimension is 1.5 Fractal dimension is 1.5

31 Labyrinth Sometimes called H-tree. Sometimes called H-tree. Initiator and generator has shape of letter H Initiator and generator has shape of letter H To see it easier the H form is not painted in the picture. To see it easier the H form is not painted in the picture. Because of changing thickness, dimension on the tip is 2.0, but elements between tips it is changing from to Because of changing thickness, dimension on the tip is 2.0, but elements between tips it is changing from to

32 Darer pentagon Pentagon as initiator Pentagon as initiator Isosceles triangle as generator Isosceles triangle as generator Hexagon is a variant of this fractal (David Star) Hexagon is a variant of this fractal (David Star) Fractal dimension is Fractal dimension is

33 Dragon curve Invented by Italian mathematic Giuseppe Piano. Invented by Italian mathematic Giuseppe Piano. Looks like Minkovskij sausage, because has the same generator and easier initiator. Looks like Minkovskij sausage, because has the same generator and easier initiator. Mandelbrot called it River of Double Dragon. Mandelbrot called it River of Double Dragon. Fractal dimension is Fractal dimension is

34 Hilbert curve Looks like labyrinth, but letter U is used and width is not changing. Looks like labyrinth, but letter U is used and width is not changing. Fractal dimension is 2.0 Fractal dimension is 2.0 Endless iteration could take all plane. Endless iteration could take all plane.

35 Box Very simple fractal Very simple fractal Made by adding squares to the top of other squares. Made by adding squares to the top of other squares. Initiator and generator and squares. Initiator and generator and squares. Fractal dimension is Fractal dimension is

36 Sophisticated fractals Most fractals which you can meet in a real life are not deterministic. Most fractals which you can meet in a real life are not deterministic. Not linear and not compiled from periodic geometrical forms. Not linear and not compiled from periodic geometrical forms. Practically even enlarged part of sophisticated fractal is different from initial fractal. They looks the same but not almost identical. Practically even enlarged part of sophisticated fractal is different from initial fractal. They looks the same but not almost identical.

37 Sophisticated fractals Are generated by non linear algebraic equations. Are generated by non linear algebraic equations. Zn+1=ZnІ + C Zn+1=ZnІ + C Solution involves complex and supposed numbers Solution involves complex and supposed numbers Self-similarity on different scale levels Self-similarity on different scale levels Stable results – black, for different speed different color Stable results – black, for different speed different color

38 Mandelbrot multitude Most widespread sophisticated fractal Most widespread sophisticated fractal Zn+1=Zna+C Zn+1=Zna+C Z and C – complex numbers Z and C – complex numbers a – any positive number. a – any positive number.

39 Mandelbrot multitude Z=Z*tg(Z+C). Z=Z*tg(Z+C). Because of Tangent function it looks like Apple. Because of Tangent function it looks like Apple. If we switch Cosine it will look like Air Bubbles. If we switch Cosine it will look like Air Bubbles. So there are different properties for Mandelbrot multitude. So there are different properties for Mandelbrot multitude.

40 Zhulia multitude Has the same formula as Mandelbrot multitude. Has the same formula as Mandelbrot multitude. If building fractal with different initial points, we will have different pictures. If building fractal with different initial points, we will have different pictures. Every dot in Mandelbrot multitude corresponds to Zhulia multitude Every dot in Mandelbrot multitude corresponds to Zhulia multitude

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