Mandelbrot Fractal

The Mandelbrot Set is a set of complex numbers \( c\) such that \( f(z) = z^2 + c \) does not diverge when recursively iterated from \( z_0 = 0 + 0i \). The Mandelbrot Fractal is obtained when a coloring scheme is applied based on the number of iterations taken for \( z \) to diverge.

Description

The X-coordinate of the default configuration (press 'reset' to obtain) ranges from -2.5 to 1 whereas the Y-coordinate ranges from -1 to +1. The complex number \( c \) is equal to \( x_0 + i y_0 \), where \( x_0 \) and \( y_0 \) are coordinates within their respective ranges of X and Y. \( f(z) \) is recursively iterated from \( z_0 = 0 + i0 \). It diverges if \( |z| > 2 \). The magnitude of \( z \) is checked at every iteration, until a maximum number of iterations (100 in this case). If \( |z| \) is never greater than 2, then it is highly probably that it never diverges, and is a part of the Mandelbrot set.

Elements of the Mandelbrot set are colored black, whereas elements that diverge are coloured on the basis of number of iterations they take to diverge. This coloring scheme gives rise to the Mandelbrot Fractal, which is highly self-similar, detailed and hypnotic in nature.

1. An offline, full-screen, interactive version of the Mandelbrot Fractal is available here. Requires Java to run.
2. This simulation, despite being highly optimized, is computationally heavy, and may lag on lower-end computers.
3. If the region being simulated has a greater proportion of elements belonging to the Mandelbrot set, then it will take longer to generate.

Developed by ChanRT | Fork me at GitHub