Created with Snap
SWE 596
Dynamic Web
Final Project
Hasan Kerem Ocakoğlu
Fractals
10.02.2021

Order in Randomness

Chaotic systems show maximum sensitivity to their original state.
“This expression has been introduced to denote the property of a chaotic system that small differences in the initial conditions, however small, are persistently magnified because of the dynamics of the system.”[2]

Philip Holmes
From Philip Holmes perspective chaotic systems are: "The complicated aperiodic attracting orbits of certain, usually low-dimensional dynamical systems."


Sub-Systems of Chaos Theory

  1. Feedback loops
  2. Repetition
  3. Self-similarity
  4. Fractals
  5. Self organization
Chaos does not mean to be randomness.
Fractals are the most obvious clue that chaos has order within itself.

Fractals

Describing the chaotic system is not easy. It is as same as fractals.
Fractals has several descriptions. Such as;
  • Never ending patterns.

  • Infinitely repeating patterns.

  • Any geometric object with chaotic structure

  • Fractals are the geometry of chaos.

Benoit Mandelbroth
Benoit Mandelbroth is one of the key mathematicians about fractals and fractal geometry. In fact, he would like to reffered himself as "fractalist".

Once he said, “Fractals are shapes that perfectly self similar”
Another Definition
“A characteristic feature of chaotic dynamical systems is the property of pathological sensitivity to initial positions. This means that starting the same process from two different—but frequently indistinguishable—initial states generally leads to completely different long-term behaviour”[1]


You are seeing the Koch Snowflake at right side since beginning of the course.

It is also known as Koch curve. It is developed by Helge von Koch

Koch curve is one of the versions of fractals. There are several equilateral triangles into a Koch curve.

Simply, we are seeing same pattern over and over again.

Dimension
Koch snowflake has 1.262 dimensional

If we do this infinite number of times, we could have we will have infinitely many sides
Perimeter of Koch snowflake would be infinite but area of Koch snowflake is not infinite

The formula to find perimeters of a Koch Snowflake:
Nn=N(n1)x4=3x4n *n represents for repetition number.
Example perimeter numbers for different numbers of Koch Snowflake

Repetition Number Total Perimeters
2 48
3 192
4 768
5 3072
∞ ∞








We have 5 repetition and 3072 perimeters at the left side of Koch Snowflake.


Koch fractal has infinite perimeter but finite area. We have an infinite line closing on itself and enclosing a finite region. For example if we draw a circle with a finite area around the snowflake, it would fit the completely inside no matter how many sides fractal has.

History

Nathan Cohen used fractal while designing radio antennas at 90s. Regular antennas for 90s would cut for one type of signal. For example FM antennas could only pick FM channels on the other hand, TV antennas would pick only TV channels.
However fractal antennas are different. Because fractals repeats itself more and more and they could pick multiple type of signal as a result lots of channels.

Menger Sponge

Menger Sponge is 3D version of Koch snowflake. It is used for cellphone anttenas. It can receive all kind of signals.

Where do we see the Fractals?
We can see fractals
  • Antennas
  • Snowflakes
  • River systems
  • Lightning bolts
  • Seashells

References

  1. The Role Of Chaos And Fractals
  2. Werner Lauterborn, in Encyclopedia of Physical Science and Technology (Third Edition), 2003
  3. Order in Randomness: Fractals and Chaotic Systems
  4. Chaos Theory
  5. Fractals: The Geometry of Chaos - Christmas Lectures with Ian Stewart
  6. Fractals are typically not self-similar
  7. What Is A Fractal (and what are they good for)?
  8. Benoit Mandelbrot
  9. Helge von Koch
  10. Nathan Cohen BIO
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