fractals and chaos theoryies
TRANSCRIPT
-
8/4/2019 Fractals and Chaos Theoryies
1/40
Fractals and Chaos
Theory
Ruslan Kazantsev
Rovaniemi Polytechnic, Finland
-
8/4/2019 Fractals and Chaos Theoryies
2/40
Chaos Theory about disorder
NOT denying of determinism
NOT denying of ordered systems
NOT announcement about useless ofcomplicated systems
Chaos is main point of order
-
8/4/2019 Fractals and Chaos Theoryies
3/40
What is the chaos theory?
Learning about complicated nonlineardynamic systems
Nonlinear recursion and algorithms
Dynamic variable and noncyclic
-
8/4/2019 Fractals and Chaos Theoryies
4/40
Wrong interpretations
Society drew attention to the chaostheory because of such movies asJurassic Park. And because of such
things people are increasing the fearof chaos theory.
Because of it appeared a lot of wrong
interpretations of chaos theory
-
8/4/2019 Fractals and Chaos Theoryies
5/40
Chaos Theory about disorder
Truth that small changes could givehuge consequences.
Concept: impossible to find exactprediction of condition, but it givesgeneral condition of system
Task is in modeling the system basedon behavior of similar systems.
-
8/4/2019 Fractals and Chaos Theoryies
6/40
Usage of Chaos Theory
Useful to have a look to thingshappening in the world differentfrom traditional view
Instead of X-Y graph -> phase-spatialdiagrams
Instead of exact position of point ->
general condition of system
-
8/4/2019 Fractals and Chaos Theoryies
7/40
Usage of Chaos Theory
Simulation of biological systems (most chaoticsystems in the world)
Systems of dynamic equations were used forsimulating everything from population growth
and epidemics to arrhythmic heart beating Every system could be simulated: stock
exchange, even drops falling from the pipe
Fractal archivation claims in future coefficient
of compression 600:1 Movie industry couldnt have realistic
landscapes (clouds, rocks, shadows) withouttechnology of fractal graphics
-
8/4/2019 Fractals and Chaos Theoryies
8/40
Brownian motion and its
adaptation 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 landscapecreating
-
8/4/2019 Fractals and Chaos Theoryies
9/40
Motion of billiard ball
The slightestmistake in angleof first kick willfollow to huge
disposition afterfew collisions.
Impossible to
predict after 6-7hits
Only way is toshow angle and
length to each hit
-
8/4/2019 Fractals and Chaos Theoryies
10/40
Motion of billiard ball
Every single loop ordispersion areapresents ball behavior
Area of picture, where
are results of oneexperiment is calledattraction area.
This self-similarity will
last forever, if enlargepicture for long, wellstill have same forms.=> this will be
FRACTAL
-
8/4/2019 Fractals and Chaos Theoryies
11/40
Fusion of determined fractals
Fractals are predictable.
Fractals are made with aim to predictsystems in nature (for example migrationof birds)
-
8/4/2019 Fractals and Chaos Theoryies
12/40
Tree simulation using Brownian motionand fractal called Pythagor Tree
Order of leaves andbranches is
complicated andrandom, BUT can beemulated by shortprogram of 12 rows.
Firstly, we need togenerate PythagorTree.
-
8/4/2019 Fractals and Chaos Theoryies
13/40
Tree simulation using Brownian motionand fractal called Pythagor Tree
On this stageBrownian motion isnot used.
Now, every sectionis the centre ofsymmetry
Instead of lines are
rectangles. But it still looks like
artificial
-
8/4/2019 Fractals and Chaos Theoryies
14/40
Tree simulation using Brownian motionand fractal called Pythagor Tree
Now Brownianmotion is usedto makerandomization
Numbers arerounded-up to 2
rank instead of39
-
8/4/2019 Fractals and Chaos Theoryies
15/40
Tree simulation using Brownian motionand fractal called Pythagor Tree
Rounded-up to 7
rank
Now it looks like
logarithmic spiral.
-
8/4/2019 Fractals and Chaos Theoryies
16/40
Tree simulation using Brownian motionand fractal called Pythagor Tree
To avoid spiral weuse Brownian
motion twice to theleft and only once tothe right
Now numbers are
rounded-up to 24rank
-
8/4/2019 Fractals and Chaos Theoryies
17/40
Fractals and world around
Branching, leaves on trees, veins in hand,curving river, stock exchange all thesethings are fractals.
Programmers and IT specialists go crazywith fractals. Because, in spite of itsbeauty and complexity, they can begenerated with easy formulas.
Discovery of fractals was discovery of newart aesthetics, science and math, and alsorevolution in humans world perception.
-
8/4/2019 Fractals and Chaos Theoryies
18/40
What are fractals in reality?
Fractal geometric figure definitepart of which is repeating changingits size => principle of self-similarity.
There are a lot of types of fractalsNot just complicated figures
generated by computers.
Almost everything which seems to becasual could be fractal, even cloud orlittle molecule of oxygen.
-
8/4/2019 Fractals and Chaos Theoryies
19/40
How chaos is chaotic?
Fractals part of chaos theory.
Chaotic behaviour, so they seemdisorderly and casual.
A lot of aspects of self-similarity insidefractal.
Aim of studying fractals and chaos topredict regularity in systems, whichmight be absolutely chaotic.
All world around is fractal-like
-
8/4/2019 Fractals and Chaos Theoryies
20/40
-
8/4/2019 Fractals and Chaos Theoryies
21/40
Practical usage of fractals
Computer systems (Fractal archivation, picturecompressing without pixelization)
Liquid mechanics Modulating of turbulent stream
Modulating of tongues of flame
Porous material has fractal structure Telecommunications (antennas have fractal form)
Surface physics (for description of surface curvature)
Medicine
Biosensor interaction Heart beating
Biology (description of population model)
-
8/4/2019 Fractals and Chaos Theoryies
22/40
Fractal dimension: hidden dimensions
Mandelbrot called not intactdimensions fractal dimensions (forexample 2.76)
Euclid geometry claims that space isstraight and flat.
Object which has 3 dimensionscorrectly is impossible
Examples: Great Britain coastline,human body
-
8/4/2019 Fractals and Chaos Theoryies
23/40
Deterministic fractals
First opened fractals.
Self-similarity because of method ofgeneration
Classic fractals, geometric fractals, linearfractals
Creation starts from initiator basicpicture
Process of iteration adding basic pictureto every result
-
8/4/2019 Fractals and Chaos Theoryies
24/40
Sierpinskij lattice
Triangles made ofinterconnection of middlepoints of large trianglecut from main triangle,generating triangle with
large amount of holes.
Initiator large triangle.
Generator process ofcutting triangles similar
to given triangle.
Fractal dimension is1.584962501
-
8/4/2019 Fractals and Chaos Theoryies
25/40
Sierpinskij sponge
Plane fractal cellwithout square, butwith unlimited ties
Would be used as
buildingconstructions
-
8/4/2019 Fractals and Chaos Theoryies
26/40
Sierpinskij fractal
Dont mix up thisfractal withSierpinskij lattice.
Initiator andgenerator are thesame.
Fractal dimension is2.0
-
8/4/2019 Fractals and Chaos Theoryies
27/40
Koch Curve
One of the most typical fractals.
Invented by german mathematic Helge fonKoch
Initiator straight line. Generator equilateral triangle.
Mandelbrot was making experiments with
Koch Curve and had as a result KochIslands, Koch Crosses, Koch Crystals, andalso Koch Curve in 3D
Fractal dimension is 1.261859507
-
8/4/2019 Fractals and Chaos Theoryies
28/40
Mandelbrot fractal
Variant of Koch Curve
Initiator andgenerator are
different from Kochs,but idea is still thesame.
Fractal takes half ofplane.
Fractal dimension is1.5
-
8/4/2019 Fractals and Chaos Theoryies
29/40
Snow Crystal and Star
This objects are classical fractals. Initiator and generator is one figure
-
8/4/2019 Fractals and Chaos Theoryies
30/40
Minkovskij sausage
Inventor is GermanMinkovskij.
Initiator and generator arequite sophisticated, aremade of row of straight
corners and segments withdifferent length.
Initiator has 8 parts.
Fractal dimension is 1.5
-
8/4/2019 Fractals and Chaos Theoryies
31/40
Labyrinth
Sometimes called H-tree.
Initiator and generator hasshape of letter H
To see it easier the H form isnot painted in the picture.
Because of changing
thickness, dimension on thetip is 2.0, but elementsbetween tips it is changingfrom 1.333 to 1.6667
-
8/4/2019 Fractals and Chaos Theoryies
32/40
Darer pentagon
Pentagon asinitiator
Isosceles triangle
as generator Hexagon is a
variant of thisfractal (David Star)
Fractal dimensionis 1.86171
-
8/4/2019 Fractals and Chaos Theoryies
33/40
Dragon curve
Invented by Italianmathematic GiuseppePiano.
Looks like Minkovskijsausage, because hasthe same generatorand easier initiator.
Mandelbrot called itRiver of DoubleDragon.
Fractal dimension is
1.5236
-
8/4/2019 Fractals and Chaos Theoryies
34/40
Hilbert curve
Looks like labyrinth,but letter U is usedand width is not
changing. Fractal dimension is
2.0
Endless iterationcould take all plane.
-
8/4/2019 Fractals and Chaos Theoryies
35/40
Box
Very simple fractal
Made by addingsquares to the top
of other squares. Initiator and
generator andsquares.
Fractal dimensionis 1.892789261
-
8/4/2019 Fractals and Chaos Theoryies
36/40
Sophisticated fractals
Most fractals which you can meet ina real life are not deterministic.
Not linear and not compiled fromperiodic geometrical forms.
Practically even enlarged part ofsophisticated fractal is different from
initial fractal. They looks the samebut not almost identical.
-
8/4/2019 Fractals and Chaos Theoryies
37/40
Sophisticated fractals
Are generated by nonlinear algebraicequations.
Zn+1=Zn + C
Solution involvescomplex and supposednumbers
Self-similarity ondifferent scale levels
Stable results black,for different speeddifferent color
-
8/4/2019 Fractals and Chaos Theoryies
38/40
Mandelbrot multitude
Most widespreadsophisticated fractal
Zn+1=Zna+C
Z and C complexnumbers
a any positive
number.
-
8/4/2019 Fractals and Chaos Theoryies
39/40
Mandelbrot multitude
Z=Z*tg(Z+C).
Because of Tangentfunction it looks like
Apple. If we switch Cosine it
will look like AirBubbles.
So there are differentproperties forMandelbrot multitude.
-
8/4/2019 Fractals and Chaos Theoryies
40/40
Zhulia multitude Has the same
formula asMandelbrotmultitude.
If building fractal
with different initialpoints, we will havedifferent pictures.
Every dot inMandelbrot
multitudecorresponds toZhulia multitude