If anyone does not know what Chaos theory is and has never plotted out a fractal on their computer then here is a bit of science fun. You don't need to be a math wiz - that's the point really, producing fantastic complexity out of very simple looking equations.
Chaos is actually 'sensitive dependence on initial condition' sometimes called the 'Butterfly Effect' - if a butterfly flaps it's wings in Europe it causes a Hurricane in India.
It refers to simple looking equations that should produce simple output and normally do...but not always. It turns out that any equations with an x squared or higher (called Non Linear equations) can produce fantastically complex graphs when plotted. The scientist knows all the rules of the system and still he cannot predict what it will do beyond the start.
Here's a simple example. This is the logistic difference equation,,,very simple. Biologists would write it as Ynext=R*Y*(1-Y) and for them it is used to model animal populations. It reads as follows :- next years population (Ynext) is this years population (Y) multiplied by a constant rate of breeding (R) and then multiplied by 1-Y. (A mathematician would know it as the equation of a parabola and write it as y = ax(1 - x) - Same thing).
To use it you choose a value for R (say 2) and then feed in the starting population - say 0.2 (where 0 represents extinct and 1 is the maximum population conceivable).
Ynext=2*0.2*(1-0.2) = 0.32
Then simply feed the result (0.32) into the equation again
Ynext=2*0.32*(1-0.32) = 0.4352
Now keep running it. The idea is that it will settle down to a fixed value after a few repetitions (called 'iterations').
Ynext=2*0.4352*(1-0.4352) = 0.49160192
Ynext=2*0.49160192*(1-0.49160192) = 0.499858945
Ynext=2*0.499858945*(1-0.499858945) = 0.5
Ynext=2*0.5*(1-0.5) = 0.5
Done. Answer = 0.5
Try it for yourself. You can use a simple calculator or set it up in a spreadsheet (or download the skeleton for one I did earlier if you have Excel - HERE). Start with R at about 2 Then try again with R=3 and again with R=3.5. Below R=3 it behaves as expected. As R is chosen higher strange things happen. Instead of settling down it carries on oscillating ip and down...Past 3.5 the system becomes Chaotic....ie apparently random.
That's where Chaos theory started. A simple little equation producing apparently random results.
Now here is the same type of thing done as a plot rather than just figures. Take a triangle (any triangle). On each of the 3 sides draw another triangle which is 1/3 the size of the existing side.
Now carry on repeating.
Click HERE to use a little application to do this fast
This is a fractal and is a very disturbing shape. Although it looks harmless when considered it is weird. The length of the line drawing the shape is INFINITE....every time you add another 3 triangles you increase the line by 4/3...so we get 1*4/3*4/3*4/3*4/3*4/3....to infinity So you have an infinitely long line but in an enclosed area which never gets bigger.
It's called the Koch Curve and used to drive mathematicians into nervous breakdowns.
Here's some other fractals to finish.....
Sierpinski Triangle
Logistic Equation Interactive tutorial
Complete Tutorial on Fractals
Pictorial Tutorial - more advanced
The Mandelbrot Set (Daddy of all fractals)
More links to topics on Fractals
Regards
Chris
Chaos is actually 'sensitive dependence on initial condition' sometimes called the 'Butterfly Effect' - if a butterfly flaps it's wings in Europe it causes a Hurricane in India.
It refers to simple looking equations that should produce simple output and normally do...but not always. It turns out that any equations with an x squared or higher (called Non Linear equations) can produce fantastically complex graphs when plotted. The scientist knows all the rules of the system and still he cannot predict what it will do beyond the start.
Here's a simple example. This is the logistic difference equation,,,very simple. Biologists would write it as Ynext=R*Y*(1-Y) and for them it is used to model animal populations. It reads as follows :- next years population (Ynext) is this years population (Y) multiplied by a constant rate of breeding (R) and then multiplied by 1-Y. (A mathematician would know it as the equation of a parabola and write it as y = ax(1 - x) - Same thing).
To use it you choose a value for R (say 2) and then feed in the starting population - say 0.2 (where 0 represents extinct and 1 is the maximum population conceivable).
Ynext=2*0.2*(1-0.2) = 0.32
Then simply feed the result (0.32) into the equation again
Ynext=2*0.32*(1-0.32) = 0.4352
Now keep running it. The idea is that it will settle down to a fixed value after a few repetitions (called 'iterations').
Ynext=2*0.4352*(1-0.4352) = 0.49160192
Ynext=2*0.49160192*(1-0.49160192) = 0.499858945
Ynext=2*0.499858945*(1-0.499858945) = 0.5
Ynext=2*0.5*(1-0.5) = 0.5
Done. Answer = 0.5
Try it for yourself. You can use a simple calculator or set it up in a spreadsheet (or download the skeleton for one I did earlier if you have Excel - HERE). Start with R at about 2 Then try again with R=3 and again with R=3.5. Below R=3 it behaves as expected. As R is chosen higher strange things happen. Instead of settling down it carries on oscillating ip and down...Past 3.5 the system becomes Chaotic....ie apparently random.
That's where Chaos theory started. A simple little equation producing apparently random results.
Now here is the same type of thing done as a plot rather than just figures. Take a triangle (any triangle). On each of the 3 sides draw another triangle which is 1/3 the size of the existing side.
Now carry on repeating.
Click HERE to use a little application to do this fast
This is a fractal and is a very disturbing shape. Although it looks harmless when considered it is weird. The length of the line drawing the shape is INFINITE....every time you add another 3 triangles you increase the line by 4/3...so we get 1*4/3*4/3*4/3*4/3*4/3....to infinity So you have an infinitely long line but in an enclosed area which never gets bigger.
It's called the Koch Curve and used to drive mathematicians into nervous breakdowns.
Here's some other fractals to finish.....
Sierpinski Triangle
Logistic Equation Interactive tutorial
Complete Tutorial on Fractals
Pictorial Tutorial - more advanced
The Mandelbrot Set (Daddy of all fractals)
More links to topics on Fractals
Regards
Chris
