Saturday, 27 February 2010

Fractals with Octave: Original Function Studied by Gaston Julia

The story so far

After another long pause, here is the sixth instalment of this series on fractals with Octave. In this article, we have a look at the complex series that started is all, the one that Gaston Julia was interested in.

Once Upon A Time: The Original Fractal Function

The classic Mandelbrot set we all know and love is derived from quite a simple series, described in the very first article of this series. Gaston Julia was interested in a slightly more complex series, as explained by Paul Bourke:

zn+1=z4 + z3/(z-1) + z2/(z3+4z2+5) + c

The initial condition is:

z0=0

The Mandelbrot Set

We can produce a Mandelbrot set of that equation with the mandelbrot function that we created in previous articles.

octave:1> Mj=mandelbrot(-1.4+1.05i,1.4-1.05i,320,64,
> @(z,c) z.^4.+z.^3./(z-1)+z.^2./(z.^3.+4*z.^2.+5).+c;
octave:2> imagesc(Mj)

And here is the result:

Original Julia Series, Mandelbrot Set

We can then zoom on the bottom right corner of that image.

octave:1> Mjz=mandelbrot(0.25-0.65i,0.45-0.8i,320,64,
> @(z,c) z.^4.+z.^3./(z-1)+z.^2./(z.^3.+4*z.^2.+5).+c;
octave:2> imagesc(Mjz)

Original Julia Series, Mandelbrot Set x10

And again.

octave:1> Mjzz=mandelbrot(0.378-0.725i,0.398-0.74i,320,64,
> @(z,c) z.^4.+z.^3./(z-1)+z.^2./(z.^3.+4*z.^2.+5).+c;
octave:2> imagesc(Mjzz)

Original Julia Series, Mandelbrot Set x100

And again!

octave:1> Mjzzz=mandelbrot(0.3863-0.7314i,0.3883-0.7329i,320,64,
> @(z,c) z.^4.+z.^3./(z-1)+z.^2./(z.^3.+4*z.^2.+5).+c;
octave:2> imagesc(Mjzzz)

Original Julia Series, Mandelbrot Set x1000

The Julia Set

Now, let's have a look at the Julia set for the same series and for c=0.3873-0.7314i, which is the point towards which we zoomed on in the Mandelbrot set.

octave:1> Jj=julia(-1.2+1.6i,1.2-1.6i,240,64,0.3873-0.7314i,
> @(z,c) z.^4.+z.^3./(z-1)+z.^2./(z.^3.+4*z.^2.+5).+c;
octave:2> imagesc(Jj)

Original Julia Set, c=0.3873-0.7314i

Let's zoom into that picture.

octave:1> Jjz=julia(0.2+0.1i,0.4-0.05i,320,64,0.3873-0.7314i,
> @(z,c) z.^4.+z.^3./(z-1)+z.^2./(z.^3.+4*z.^2.+5).+c;
octave:2> imagesc(Jjz)

Original Julia Set, c=0.3873-0.7314i, x10

And again.

octave:1> Jjzz=julia(0.366+0.09i,0.386+0.075i,320,64,0.3873-0.7314i,
> @(z,c) z.^4.+z.^3./(z-1)+z.^2./(z.^3.+4*z.^2.+5).+c;
octave:2> imagesc(Jjzz)

Original Julia Set, c=0.3873-0.7314i, x100

And again!

octave:1> Jjzzz=julia(0.366+0.09i,0.386+0.075i,320,64,0.3873-0.7314i,
> @(z,c) z.^4.+z.^3./(z-1)+z.^2./(z.^3.+4*z.^2.+5).+c;
octave:2> imagesc(Jjzzz)

Original Julia Set, c=0.3873-0.7314i, x1000

The End

This is the last article in this series. I hope you've enjoyed it, even if it took me a very long time to finish it. Have a play with the Octave functions described throughout the articles, modify them, improve them, there's a lot of fun stuff to do and amazing pictures to create with fractals.

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