jabolol / julia

A script to generate Julia set image representations effortlessly

Julia set

The Julia set is a fractal defined by the following equation:

$$z_{n+1} = z_n^2 + c$$

where $z$ and $c$ are complex numbers. The Julia set is the set of all complex numbers $z$ for which the sequence $z_n$ does not diverge to infinity. It is named after the French mathematician Gaston Julia who studied the set in the early 20th century.

The Julia set is usually visualized by coloring the complex plane. Each pixel in the image corresponds to a complex number $z$. The color of the pixel is determined by the number of iterations it takes for the sequence $z_n$ to diverge to infinity. If the sequence does not diverge to infinity, the pixel is colored black.

The Julia set is a fractal because it is self-similar. If you zoom in on any part of the image, you will see the same pattern repeating itself.

How is it calculated?

The calcJulia function calculates the number of iterations it takes for the sequence $z_n$ to diverge to infinity. If the sequence does not diverge to infinity, the function returns $0$.

calcJulia :: Complex Double -> Int
calcJulia z0 = iterateJulia z0 0
  where
    iterateJulia z n
      | n == maxIterations || magnitude z > 2.0 = n
      | otherwise = iterateJulia (z * z + c) (n + 1)

The steps of the calculation are as follows:

1. Start with a complex number $z_0$.
2. Calculate the next value of the sequence $z_n$ using the $f$ function.
3. Repeat step 2 until the sequence diverges to infinity or the maximum number of iterations is reached.

The iterateJulia function uses the f function to calculate the next value of the sequence $z_n$. The $f$ function takes one parameter: the current value of the sequence $z_n$. The function returns the next value of the sequence $z_n$.

The $f$ function is defined as follows:

$$f(z) = z^2 + c$$

The $c$ parameter is a complex number. The $c$ parameter is the same for all pixels in the image. Modify this parameter to get different images. It is defined as follows.

$$c = -0.8 + 0.156i$$

How to run

1. Install Haskell.
2. Install the dependencies: cabal install --only-dependencies.
3. Compile the program: cabal build.
4. Run the program: cabal run.
5. Open the julia_set.png file to see the result.