How to use

tested on Debian:11.6 and Ubuntu:22.10 only.

Clone the repository

git clone https://github.com/malletgaetan/fractol.git

Build the binary

cd fractol
make

Execute the binary

./fractol "mandelbrot"

How to compute mandelbrot

naive vs optimized approach

If you don't have enough, better solutions would be:

• use SIMD instructions
• use threads
• use your graphics card

here is a nice repository that compare these solutions.

Intuition

If by iterating a formula

Zn = (Zn-1)^2 + C # Z and C being complex numbers

if the distance of the point Z to the center is closer than 2, the point is considered in the Julia set, thus colored in black.

To generate it, the Z complex number will represent a pixel on the screen, with x the real part of the complex and y the imaginary part. Then we'll cherry pick a C, a different C would generate a different set of Julia. So basically, a

Z + C

combo would be in the set of Julia, if after iterating a certain number of time the suite, the point Z would remain at the maximum distance of 2 from the center (0,0).

Complex numbers arithmetic

In our case, we'll only square and add complex numbers so:

addition

Z + W = Z.r + Z.r + Z.i + W.i

multiplicaton

Z * Z = Z.r^2 + 2 * Z.r * Z.i + Z.i^2

example:
Z = 2 - 5i

Z * Z = 2^2 + (2 * 2 * 5) + (5^2 - i^2)
since i^2 == -1 we have:
Z * Z = 2^2 + (2 * 2 * 5) - 25
Z^2 = -21 -20i