Transcript Sierpinski`s Triangle

Chaos Theory and Fractals
By Tim Raine and Kiara Vincent
Chaos Theory
• About finding order in disordered systems
• ‘(Math.) Stochastic behaviour occurring in a
deterministic system.’
• Initial Conditions
• Butterfly Effect
• E.g. x2+1, 2x2+1
Early Chaos
“Chaos often breeds life, when order breeds
habit” Henry Adams
Ilya Prigogine showed,
• Complex structures come from simpler
ones.
• Like order coming from chaos.
The Prize of King Oscar II of Norway
“To prove or disprove the solar system is stable.”
• Henri Poincaré offered his solution.
• A friend found an error in his calculations.
• The prize was taken away until he found a new
solution
• Poincaré found that there was no solution.
• Sir Isaac Newton's laws could not provide a
solution
• It was a system where there was no order
Chaos in the Real World
• There are plenty of chaotic systems in the
real world:
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Weather
Flight of a meteorite
Beating heart
Electron flow in transistors
Dripping tap
Double pendulum
Edward Lorenz
• Meteorologist at MIT
• Weather patterns on a computer.
• Stumbled upon the butterfly effect
• How small scale changes affect large scale things.
• Classic example of chaos, as small changes lead to
large changes.
• A butterfly flapping its wings in Hong Kong could
change tornado patterns in Texas.
• Discovered the Lorenz Attractor
• An area that pulls points towards itself.
Sierpinski’s Triangle
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Simple fractal
Formed by cutting out equilateral triangles
Has 1.58496 dimensions
Sierpinski & Pascal
1
1
1
1
10
1
1
1
1
1
1
1
16
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120
136
153
171
364
455
560
680
816
969
286
91
105
3876
210
330
495
715
1001
1365
1820
2380
3060
165
220
78
14
15
17
66
13
120
55
12
126
2002
3003
4368
1287
3003
5005
8008
1716
6435
6435
9
120
495
11440 12870 11440
10
715
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1001
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1
78
13
364
1365
4368
6188 12376 19448 24310 24310 19448 12376
1
66
286
3003
8008
1
55
220
2002
5005
1
45
165
1287
3003
1
36
330
792
3432
8
84
210
924
1
28
462
1716
7
56
252
792
1
21
126
462
1
6
35
70
84
45
11
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15
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1
1
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1
1
1
1
1
455
1820
6188
1
91
105
560
2380
8568 18564 31824 43758 48620 43758 31824 18564 8568
14
120
680
3060
11628 27132 50388 75582 92378 92378 75582 50388 27132 11628
1
15
1
16
136
816
3876
1
17
153
969
1
18
171
1
19
1
The Menger Sponge
• Fractal made using cubes
– divide a cube into 27
smaller cubes (3x3x3)
then remove the middle
one and the one in the
centre of each face
• Has 2.72683 dimensions
Jurassic Park Fractal
• Made by folding a strip of paper in half,
always the same way, then opening up
• With each iteration, the area gets less, yet
the length of the line is the same
• By the 20th iteration, a 1km long piece of
paper would cover less area than a pin point
Benoit Mandelbrot
• Polish born French mathematician
• Believed fractals were found everywhere in nature
• Showed fractals cannot have whole-number
dimensions
•Fractals must have fractional dimensions
Mandelbrot Set
• Simplest non-linear Function
• f(x)=x^(2+c)
Cantor Set
• Produced with a line and removing the
middle third
• If we add up the amount removed to
infinity, we get 1 (using geometric series),
this tells us the whole of the line has gone
• But the endpoints are never removed – there
must be something left!
More Complex Fractals