SLIDE 1
Fractals Self Similarity and Fractal Geometry presented by Pauline - - PowerPoint PPT Presentation
Fractals Self Similarity and Fractal Geometry presented by Pauline - - PowerPoint PPT Presentation
Fractals Self Similarity and Fractal Geometry presented by Pauline Jepp 601.73 Biological Computing Overview History Initial Monsters Details Fractals in Nature Brownian Motion L-systems Fractals defined by linear algebra operators
SLIDE 2
SLIDE 3
History
Euclid's 5 postulates:
- 1. To draw a straight line from any point to any other.
- 2. To produce a finite straight line continuously in a straight line.
- 3. To describe a circle with any centre and distance.
- 4. That all right angles are equal to each other.
- 5. That, if a straight line falling on two straight lines make the interior angles on the same
side less than two right angles, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
SLIDE 4
History
Euclid ~ "formless" patterns Mandlebrot's Fractals
"Pathological" "gallery of monsters" In 1875:
Continuous non-differentiable functions, ie no tangent
La Femme Au Miroir 1920 Leger, Fernand
SLIDE 5
Initial Monsters
1878 Cantor's set 1890 Peano's space filling curves
SLIDE 6
Initial Monsters
1906 Koch curve 1960 Sierpinski's triangle
SLIDE 7
Details
Fractals :
are self similar fractal dimension
A square may be broken into N^2 self-similar pieces, each with magnification factor N
SLIDE 8
Details
Effective dimension
Mandlebrot: " ... a notion that should not be defined precisely. It is an intuitive and potent throwback to the Pythagoreans' archaic Greek geometry"
How long is the coast of Britain?
Steinhaus 1954, Richardson 1961
SLIDE 9
Brownian Motion
Robert Brown 1827 Jean Perrin 1906
Diffusion-limited aggregation
SLIDE 10
L-Systems and Fractal Growth
Packing efficiency Axiom & production rules
Axiom: B Rules: B->F[-B]+B F->FF B F[-B]+B FF[-F[-B]+B]+F[-B]+B
SLIDE 11
L-Systems and Fractal Growth
Turtle graphics
Seymour Papert
SLIDE 12
L-Systems and Fractal Growth
SLIDE 13
L-Systems and Fractal Growth
SLIDE 14
L-Systems and Fractal Growth
SLIDE 15
Affine Transformation Fractals
"It has a miniature version of itself embedded inside it, but the smaller version is slightly rotated." Transofrmations: translation, scale, reflection rotation
SLIDE 16
Affine Transformation Fractals
Michael Barnsley Multiple Reduction Copy Machine Algorithm (MRCM)
SLIDE 17
Affine Transformation Fractals
Multiple Reduction Copy Machine Algorithm (MRCM)
SLIDE 18
Affine Transformation Fractals
Iterated Functional Systems (IFS)
SLIDE 19
Affine Transformation Fractals
Iterated Functional Systems (IFS)
SLIDE 20
Affine Transformation Fractals
Iterated Functional Systems (IFS)
SLIDE 21
Affine Transformation Fractals
Iterated Functional Systems (IFS)
SLIDE 22
The Mandelbrot & Julia sets
Iterative Dynamical Systems
SLIDE 23
The Mandelbrot & Julia sets
For each number, c, in a subset of the complex plane Set x0 = 0 For t = 1 to tmax Compute xt = x2
t + c
If t< tmax, then colour point c white If t = tmax, then colour point c black
SLIDE 24
The Mandelbrot & Julia sets
SLIDE 25
The Mandelbrot & Julia sets
M-Set and computability
cardoid: x = 1/4(2 cos t - cos 2t ) y = 1/4(2 sint t - sin 2 t )
SLIDE 26
The Mandelbrot & Julia sets
The M-Set as the Master Julia set.
Set c to some constant complex value For each number, x0 in a subset of the complex plane For t = 1 to tmax Compute xt = xt
2 + c
If |xt| > 2 then break out of loop If t < tmax then colour point c white It t = tmax thencolour point c black
SLIDE 27
The Mandelbrot & Julia sets
SLIDE 28