Fractals And Chaos Theory

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1. Fractals and Chaos Fractals and Chaos
TheoryTheory
Ruslan Kazantsev
Rovaniemi Polytechnic, Finland

2. 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

3. What is the chaos theory?What is the chaos theory?
Learning about complicated nonlinear Learning about complicated nonlinear
dynamic systemsdynamic systems
Nonlinear – Nonlinear – recursionrecursion and algorithms and algorithms
Dynamic – Dynamic – variablevariable and and noncyclicnoncyclic

4. 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

5. 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.

6. 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

7. 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 of Fractal archivation claims in future coefficient 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

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

9. Motion of Motion of billiardbilliard ball ball
The slightest The slightest
mistake in angle mistake in angle
of first kick will of first kick will
follow to huge follow to huge
disposition after disposition after
few 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

10. 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

11. 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)

12. 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 branches is
complicated and complicated and
random, BUT can be 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 or
Tree.Tree.

13. 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

14. 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

15. 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.

16. 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

17. 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..

18. 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..

19. 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 predict regularity in systems, which
might be absolutely chaotic.might be absolutely chaotic.
All world around is fractal-likeAll world around is fractal-like

20. Geometry of 21Geometry of 21
stst
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.

21. 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)

22. 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

23. 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

24. 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 to cutting triangles similar to
given triangle.given triangle.
Fractal dimension is Fractal dimension is
1.584962501 1.584962501

25. 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

26. 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

27. 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 Koch Curve and had as a result Koch
Islands, Koch Crosses, Koch Crystals, and Islands, Koch Crosses, Koch Crystals, and
also Koch Curve in 3Dalso Koch Curve in 3D
Fractal dimension is Fractal dimension is 1.2618595071.261859507

28. 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

29. 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

30. 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

31. 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

32. 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

33. 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

34. 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.

35. 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

36. 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.

37. 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 supposed complex and supposed
numbersnumbers
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

38. 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.

39. 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.

40. 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