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Software Requirements

SoftwareRequirements


Android iOS Windows MacOS
with best with Chrome Chrome Chrome Chrome
support full-screen? Yes. Chrome/Opera No. Firefox/ Samsung Internet Not yet Yes Yes
cannot work on some mobile browser that don't understand JavaScript such as.....
cannot work on Internet Explorer 9 and below

 

Credits

Dieter Roess - WEH- Foundation; Tan Wei Chiong; Loo Kang Wee; Francisco Esquembre ; Wolfgang Christian

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Sample Learning Goals

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For Teachers

This is a simulation of a Mandelbrot Set, along with its corresponding Julia Set.

The graph on the left shows the Mandelbrot Set, while the graph on the right shows the corresponding Julia Set of the complex number c, denoted by the white dot on the Mandelbrot Set.

The Mandelbrot Set is a set in the complex plane formed by the recursive equation:

{\displaystyle z_{n+1}=z_{n}^{2}+c}

 

, where z and c are complex numbers.

 

For a Mandelbrot Set, z(0) is defined to be 0, while c is an arbitrary point in the complex plane. The Mandelbrot Set is defined by the set of values of c where the absolute value of z|z| does not escape to infinity when run through the above iteration.

 

For example, take c = 1. The iteration goes as follows:

 

z(1) = z(0)^2 + 1 = 0^2 + 1 = 1

z(2) = z(1)^2 + 1 = 1^2 + 1 = 2

z(3) = z(2)^2 + 1 = 2^2 + 1 = 5

z(4) = z(3)^2 + 1 = 5^2 + 1 = 26

z(5) = z(4)^2 + 1 = 26^2 + 1 = 677

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As this shows, the absolute value of z very quickly inflates to infinity in just a few iterations. Therefore, c = 1 is not part of the Mandelbrot Set.

 

Now let us look at c = -1. The iteration will go as follows:

 

z(1) = z(0)^2 + 1 = 0^2 + (-1) = -1

z(2) = z(1)^2 + 1 = (-1)^2 + (-1) = 0

z(3) = z(2)^2 + 1 = 0^2 + (-1) = -1

z(4) = z(3)^2 + 1 = (-1)^2 + (-1) = 0

z(5) = z(4)^2 + 1 = 0^2 + (-1) = -1

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Since the iteration oscillates back and forth between z = 0 and z = -1, the absolute value of z for c = -1 does not tend to infinity. We say that |z| is bounded by a finite value.

Therefore, c = -1 is part of the Mandelbrot Set.

The shape formed by the Mandelbrot Set is also a fractal, where zooming in to the boundary of the shape formed by the set reveals repeating patterns of equal or increasing complexity as the magnification increases.

On the other hand, a Julia Set is formed by fixing the value of c as an arbitrary complex number rather than fixing z = 0. The same recursive equation is run, and the conditions that form the set remain unchanged, but the pattern that emerges changes depending on the value of c that is set.

The degree of divergence or convergence to zero is indicated by colour shading. Its gradation can be changed by a slider, labelled internal convergence, which produces interesting colour schemes.

The colours of the two sets can be changed with the combo box. Available colours are:

- Mixed Colours

- Red

- Green

- Blue

 

Research

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Video

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 Version:

  1. http://weelookang.blogspot.sg/2016/02/vector-addition-b-c-model-with.html improved version with joseph chua's inputs
  2. http://weelookang.blogspot.sg/2014/10/vector-addition-model.html original simulation by lookang

Other Resources

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