alecksphillips / fractals

Experiments with generating escape time fractals with julia

Fractals

Experiments with generating escape time fractals in julia

In general escape time fractals are generated by plotting the set of points in the complex plane where iteration of some function $f_c(z)$, $$z_{n+1} = f_c(z)$$ does not diverge to infinity.

Depending on the function $f_c$, different fractals are generated.

Colorings can be added e.g. based on the number of iterations is takes a point to escape (see histogram colouring).

Alternatively, colours can be selected from a map (such as an image) using an orbit trap.

Mandelbrot set

$$f_c(z) = z^2 + c$$ iterated from $z_0 = 0$

Julia set

If $c$ is held constant and $z$ varied, $$f_c(z) = z^2 + c$$ the corresponding Julia set is generated. Differing values of $c$ give different Julia sets

Burning ship fractal

$$f_c(z) = (|Re(z)| + i|Im(z)|)^2 + c$$ iterated from $z_0 = 0$

Buddhabrot

Probability distribution of the orbits of points that do escape to infinity under iteration of the function: $$f_c(z) = z^2 + c$$

Using different numbers of maximum iterations for each channel in an RGB image:

Animations

We can animate the julia sets by varying $c$ with time, e.g. moving the point with a lissajous curve:

Similarly, using an image for an orbit trap and using the same trajectory of $c$: