Mandelbrot set

Settings

Choose a color palette:

Mouse & Touchscreen Controls

Mouse

Left mouse button + drag - simply move the fractal in any direction
Mouse wheel up - zoom in
Mouse wheel down - zoom out
Zoom center - wherever the cursor is when you scroll

Touchscreen (phone, tablet)

One finger - free movement across the fractal
Two fingers - pinch to zoom in / zoom out (classic pinch-zoom)

The “Fullscreen” ⛶ button is positioned at the upper-right corner of the fractal display.

Basic Concepts

Mandelbrot set - the set of points $c$ in the complex plane for which the recursive relation $z_{n + 1} = z^{n} + c$ , with $z_{0} = 0$ holds. The Mandelbrot set is one of the most famous and beautiful fractals. It is known far beyond mathematics thanks to its colorful visualizations. The exact area of the set is unknown, but it is approximately estimated as ≈ 1.5065918849. The center of mass of the set lies on the x-axis at about the point -0.28676842048.

History

At the beginning of the 20th century, mathematicians Pierre Fatou and Gaston Julia explored the mysterious world of complex functions. Their work laid the foundation for future discoveries, but at the time no one could imagine the astonishing images hidden within the formulas. Decades later, in the computer age, Benoît Mandelbrot was the first to “bring to life” these abstract ideas. On the screen appeared a set that today bears his name-an infinitely intricate figure, where each fragment repeats itself at new scales. Mandelbrot saw in this not merely a mathematical object, but an entire phenomenon. He called such structures fractals and devoted to them his book “Les Objets Fractals: Forme, Hasard et Dimension.” The Mandelbrot set became a symbol of how order and chaos can intertwine, giving rise to endless diversity of forms. Its images still captivate: with every magnification, new patterns emerge, like an infinite gallery contained within a single formula.