In the previous post we discussed the Mandelbrot set. There is another beautiful algebraic fractal, called Julia set 𝓙. It is named after the French mathematician Gaston Julia, who has discovered. It is obtained by the same formula as the Mandelbrot Set:
zn+1 = zn2 + c
The difference is that z0 = x + yi is the current point (x, y) in the complex plane and c is a fixed complex number, while for the Mandelbrot set z0 = 0 and c = x + yi.
Both sets are closely related. There are a lot of different Julia sets, corresponding to different values of c. Their shapes and properties depend on where the point is located, related to the Mandelbrot set 𝓜. Inside 𝓜, the corresponding Julia set 𝓙 is connected. Outside 𝓜, it falls apart into a set of isolated points, called Fatou set 𝓕. Further away, it turns into Fatou dust – infinite number of points with zero area, similarly to the Cantor dust.
For simple plotting, you can use the following code:
"Julia set
'Define the function
JuliaSet(z; c) = $Repeat{z = z^2 + c @ i = 1 : n}
#hide
'Set the plotting parameters
PlotStep = 1','PlotWidth = 600','PlotHeight = 400
#show
'Plot for'c = -0.4 + 0.59i'and'n = 200'iterations
$Map{abs(JuliaSet(x + 1i*y; c))) @ x = -1.5 : 1.5 & y = -1 : 1}
When you run it inside Calcpad, you will get the following result:
To obtain better and colorful images, we will use different approach this time. We will modify the JuliaSet function in a way to return directly the number of iterations for which the result overpasses 2. The source code is provided bellow:
"Julia set
J(z; c; n) = abs($Repeat{z = if(abs(z) < 2; z^2 + c; 4) @ i = 1 : n})
JuliaSet(z; c) = $Find{J(z; c; x) - 2 @ x = 1 : n}
#hide
PlotStep = 1','PlotWidth = 600','PlotHeight = 400','Precision = 10^-6
#show
'Plot for'c = -0.4 + 0.6i'and'n = 100'iterations
$Map{JuliaSet(x + 1i*y; c)) @ x = -1.5 : 1.5 & y = -1 : 1}
It runs a little bit slower, but the result is worth the wait:
You can download Calcpad for free and experiment by yourself with different values of c. For example, the lightning image bellow is a special case of Julia set for c = i and is called “dendrite” fractal.
Calcpad 