Post on 22-Jan-2018
transcript
Playing With ChaosFractals and Strange Attractors
Recreational MathematicsPuzzles, games, etc.
Algorithmic/Generative ArtDon't get too serious about the word “art”
What is a fractal?
A shape
Made from other shapes
Each part a copy of the whole
Theoretically infinite
Sierpinski Gasket
Koch Curve
Koch Curve
Koch Snowflake
Length
4.0
5.3
7.1
+1.3
+1.8
1 iteration = length 4
20 iterations = length 946
Infinite perimeter. Finite area.
Infinite triangles. Zero area.
Fractal tree
Symmetry and Regularity
Non-symmetry
Irregularity
Pythagorean Fractal
A
B
C
How long is the coast of Britain?
1 iteration = length 4
20 iterations = length 946
How long is the coast of Britain?
It depends on the size of your measuring stick.
The Richardson Effect
Dimensions
Line:
Square:
1.0
2.0
Fractal Dimensions
Line:
South Coast of Africa:
West Coast of Britain:
Koch Curve:
Sierpinski Gasket:
Square:
1.0
1.02
1.25
1.2619
1.5849
2.0
Chaos Game
Chaos Game
Chaos Game
Chaos Game
Chaos Game
Chaos Game
Barnsley Fern
IFS Tree
Diffuse Limited Aggregation
The Mandelbrot Set
Benoit Mandelbrot
Imaginary Number
i = ⎷-2
Complex Numbers
Real Number + Imaginary Number
4 + 5i
x
-y
y
-x
4, 2
-2, -1
real
-imaginary
imaginary
-real
4, 2i
-2, -i
The Mandelbrot Set
z1 = z2 + c
z and c are complex numbers
z begins as 0+0i
c is a point on the complex plane
iterate for each point on the plane
● z goes above a certain value ● color it based on how fast it got there
● z stays in range ● color it black
Result of iteration:
-r r
i
-i
-r r
i
-i
c = -2, -i z = 0, 0i
Strange Attractors
Edward Lorenz
The Lorenz Attractor
Fractals in Nature
Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line... Nature exhibits not simply a higher degree but an altogether different level of complexity.
- Benoit Mandelbrot
Thank You
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