Mandelbrot set
Settings
Mouse & Touchscreen Controls
- 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
- One finger - free movement across the fractal
- Two fingers - pinch to zoom in / zoom out (classic pinch-zoom)
Basic Concepts
The Mandelbrot set is the set of points c in the complex plane for which the recurrence relation zₙ₊₁ = zₙ² + c remains bounded, starting with z₀ = 0. 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.