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Fractals and Chaos Theory презентация
Содержание
- 1. Fractals and Chaos Theory
- 2. Chaos Theory about disorder NOT denying
- 3. What is the chaos theory? Learning about
- 4. Wrong interpretations Society drew attention to the
- 5. Chaos Theory about disorder Truth that small
- 6. Usage of Chaos Theory Useful to have
- 7. Usage of Chaos Theory Simulation of biological
- 8. Brownian motion and it’s adaptation Brownian
- 9. Motion of billiard ball The slightest mistake
- 10. Motion of billiard ball Every single loop
- 11. Fusion of determined fractals Fractals are predictable.
- 12. Tree simulation using Brownian motion and fractal
- 13. Tree simulation using Brownian motion and fractal
- 14. Tree simulation using Brownian motion and fractal
- 15. Tree simulation using Brownian motion and fractal
- 16. Tree simulation using Brownian motion and fractal
- 17. Fractals and world around Branching, leaves on
- 18. What are fractals in reality? Fractal –
- 19. How chaos is chaotic? Fractals – part
- 20. Geometry of 21st century Pioneer, father of
- 21. Practical usage of fractals Computer systems (Fractal
- 22. Fractal dimension: hidden dimensions Mandelbrot called
- 23. Deterministic fractals First opened fractals. Self-similarity because
- 24. Sierpinskij lattice Triangles made of interconnection of
- 25. Sierpinskij sponge Plane fractal cell without square,
- 26. Sierpinskij fractal Don’t mix up this fractal
- 27. Koch Curve One of the most typical
- 28. Mandelbrot fractal Variant of Koch Curve Initiator
- 29. Snow Crystal and Star This objects are classical fractals. Initiator and generator is one figure
- 30. Minkovskij sausage Inventor is German
- 31. Labyrinth Sometimes called H-tree. Initiator and
- 32. Darer pentagon Pentagon as initiator Isosceles triangle
- 33. Dragon curve Invented by Italian mathematic Giuseppe
- 34. Hilbert curve Looks like labyrinth, but letter
- 35. Box Very simple fractal Made by adding
- 36. Sophisticated fractals Most fractals which you can
- 37. Sophisticated fractals Are generated by non linear
- 38. Mandelbrot multitude Most widespread sophisticated fractal
- 39. Mandelbrot multitude Z=Z*tg(Z+C). Because of Tangent function
- 40. Zhulia multitude Has the same formula as
Chaos Theory about disorder NOT denying of determinism NOT denying of ordered systems NOT announcement about useless of complicated systems Chaos is main point of order
Слайд 2Chaos Theory about disorder
NOT denying of determinism
NOT denying of ordered
systems
NOT announcement about useless of complicated systems
Chaos is main point of order
NOT announcement about useless of complicated systems
Chaos is main point of order
Слайд 3What is the chaos theory?
Learning about complicated nonlinear dynamic systems
Nonlinear –
recursion and algorithms
Dynamic – variable and noncyclic
Dynamic – variable and noncyclic
Слайд 4Wrong 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.
Because of it appeared a lot of wrong interpretations of chaos theory
Because of it appeared a lot of wrong interpretations of chaos theory
Слайд 5Chaos Theory about disorder
Truth that small changes could give huge consequences.
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.
Слайд 6Usage of Chaos Theory
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
Instead of X-Y graph -> phase-spatial diagrams
Instead of exact position of point -> general condition of system
Слайд 7Usage of Chaos Theory
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
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 couldn’t have realistic landscapes (clouds, rocks, shadows) without technology of fractal graphics
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
Fractal archivation claims in future coefficient of compression 600:1
Movie industry couldn’t have realistic landscapes (clouds, rocks, shadows) without technology of fractal graphics
Слайд 8Brownian motion and it’s 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 landscape creating
Output: frequency diagram
Could be transformed in music
Could be used for landscape creating
Слайд 9Motion of billiard ball
The slightest mistake in angle of first kick
will follow to huge disposition after few collisions.
Impossible to predict after 6-7 hits
Only way is to show angle and length to each hit
Impossible to predict after 6-7 hits
Only way is to show angle and length to each hit
Слайд 10Motion of billiard ball
Every single loop or dispersion area presents ball
behavior
Area of picture, where are results of one experiment is called attraction area.
This self-similarity will last forever, if enlarge picture for long, we’ll still have same forms. => this will be FRACTAL
Area of picture, where are results of one experiment is called attraction area.
This self-similarity will last forever, if enlarge picture for long, we’ll still have same forms. => this will be FRACTAL
Слайд 11Fusion of determined fractals
Fractals are predictable.
Fractals are made with aim to
predict systems in nature (for example migration of birds)
Слайд 12Tree 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.
Firstly, we need to generate Pythagor Tree.
Firstly, we need to generate Pythagor Tree.
Слайд 13Tree simulation using Brownian motion and fractal called Pythagor Tree
On this
stage Brownian motion is not used.
Now, every section is the centre of symmetry
Instead of lines are rectangles.
But it still looks like artificial
Now, every section is the centre of symmetry
Instead of lines are rectangles.
But it still looks like artificial
Слайд 14Tree simulation using Brownian motion and fractal called Pythagor Tree
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
Слайд 15Tree simulation using Brownian motion and fractal called Pythagor Tree
Rounded-up to
7 rank
Now it looks like logarithmic spiral.
Now it looks like logarithmic spiral.
Слайд 16Tree 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
Now numbers are rounded-up to 24 rank
Now numbers are rounded-up to 24 rank
Слайд 17Fractals and world around
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.
Discovery of fractals was discovery of new art aesthetics, science and math, and also revolution in humans world perception.
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.
Слайд 18What are fractals in reality?
Fractal – geometric figure definite part of
which is repeating changing its size => principle of self-similarity.
There are a lot of types of fractals
Not just complicated figures generated by computers.
Almost everything which seems to be casual could be fractal, even cloud or little molecule of oxygen.
There are a lot of types of fractals
Not just complicated figures generated by computers.
Almost everything which seems to be casual could be fractal, even cloud or little molecule of oxygen.
Слайд 19How chaos is chaotic?
Fractals – part of chaos theory.
Chaotic behaviour, so
they seem disorderly and casual.
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.
All world around is fractal-like
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.
All world around is fractal-like
Слайд 20Geometry of 21st century
Pioneer, father of fractals was Franco-American professor Benoit
B. Mandelbrot.
1960 “Fractal geometry of nature”
Purpose was to analyze not smooth and broken 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.
1960 “Fractal geometry of nature”
Purpose was to analyze not smooth and broken 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.
Слайд 21Practical usage of fractals
Computer systems (Fractal archivation, picture compressing 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)
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)
Слайд 22Fractal dimension: hidden dimensions
Mandelbrot called not intact dimensions – fractal
dimensions (for example 2.76)
Euclid geometry claims that space is straight and flat.
Object which has 3 dimensions correctly is impossible
Examples: Great Britain coastline, human body
Euclid geometry claims that space is straight and flat.
Object which has 3 dimensions correctly is impossible
Examples: Great Britain coastline, human body
Слайд 23Deterministic fractals
First opened fractals.
Self-similarity because of method of generation
Classic fractals, geometric
fractals, linear fractals
Creation starts from initiator – basic picture
Process of iteration – adding basic picture to every result
Creation starts from initiator – basic picture
Process of iteration – adding basic picture to every result
Слайд 24Sierpinskij lattice
Triangles made of interconnection of middle points of large triangle
cut from main triangle, generating triangle with large amount of holes.
Initiator – large triangle.
Generator – process of cutting triangles similar to given triangle.
Fractal dimension is 1.584962501
Initiator – large triangle.
Generator – process of cutting triangles similar to given triangle.
Fractal dimension is 1.584962501
Слайд 25Sierpinskij sponge
Plane fractal cell without square, but with unlimited ties
Would be
used as building constructions
Слайд 26Sierpinskij fractal
Don’t mix up this fractal with Sierpinskij lattice.
Initiator and generator
are the same.
Fractal dimension is 2.0
Fractal dimension is 2.0
Слайд 27Koch Curve
One of the most typical fractals.
Invented by german mathematic Helge
fon Koch
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
Fractal dimension is 1.261859507
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
Fractal dimension is 1.261859507
Слайд 28Mandelbrot fractal
Variant of Koch Curve
Initiator and generator are different from Koch’s,
but idea is still the same.
Fractal takes half of plane.
Fractal dimension is 1.5
Fractal takes half of plane.
Fractal dimension is 1.5
Слайд 29Snow Crystal and Star
This objects are classical fractals.
Initiator and generator is
one figure
Слайд 30Minkovskij sausage
Inventor is German Minkovskij.
Initiator and generator are quite
sophisticated, are made of row of straight corners and segments with different length.
Initiator has 8 parts.
Fractal dimension is 1.5
Initiator has 8 parts.
Fractal dimension is 1.5
Слайд 31Labyrinth
Sometimes called H-tree.
Initiator and generator has shape of letter H
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 1.333 to 1.6667
Because of changing thickness, dimension on the tip is 2.0, but elements between tips it is changing from 1.333 to 1.6667
Слайд 32Darer pentagon
Pentagon as initiator
Isosceles triangle as generator
Hexagon is a variant of
this fractal (David Star)
Fractal dimension is 1.86171
Fractal dimension is 1.86171
Слайд 33Dragon curve
Invented by Italian mathematic Giuseppe Piano.
Looks like Minkovskij sausage, because
has the same generator and easier initiator.
Mandelbrot called it River of Double Dragon.
Fractal dimension is 1.5236
Mandelbrot called it River of Double Dragon.
Fractal dimension is 1.5236
Слайд 34Hilbert curve
Looks like labyrinth, but letter “U” is used and width
is not changing.
Fractal dimension is 2.0
Endless iteration could take all plane.
Fractal dimension is 2.0
Endless iteration could take all plane.
Слайд 35Box
Very simple fractal
Made by adding squares to the top of other
squares.
Initiator and generator and squares.
Fractal dimension is 1.892789261
Initiator and generator and squares.
Fractal dimension is 1.892789261
Слайд 36Sophisticated fractals
Most fractals which you can meet in a real life
are not deterministic.
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.
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.
Слайд 37Sophisticated fractals
Are generated by non linear algebraic equations.
Zn+1=ZnІ + C
Solution involves
complex and supposed numbers
Self-similarity on different scale levels
Stable results – black, for different speed different color
Self-similarity on different scale levels
Stable results – black, for different speed different color
Слайд 38Mandelbrot multitude
Most widespread sophisticated fractal
Zn+1=Zna+C
Z and C – complex numbers
a
– any positive number.
Слайд 39Mandelbrot multitude
Z=Z*tg(Z+C).
Because of Tangent function it looks like Apple.
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.
Слайд 40Zhulia multitude
Has the same formula as Mandelbrot multitude.
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
