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. x 2 +1, 2x 2 +1

“Chaos often breeds life, when order breeds habit” Henry Adams Early Chaos Ilya Prigogine showed, Complex structures come from simpler ones. Like order coming from chaos.

“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 The Prize of King Oscar II of Norway

Chaos in the Real World There are plenty of chaotic systems in the real world: –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 Simple fractal Formed by cutting out equilateral triangles Has dimensions Sierpinski & Pascal

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 dimensions

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 20 th iteration, a 1km long piece of paper would cover less area than a pin point Jurassic Park Fractal

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