Fractal Home
The Mandelbrot Set
Benoît B. Mandelbrot (1924–2010) was a French-American mathematician. He made contributions to a wide range of mathematical problems; of particular interest here is his role as the father of fractal geometry and his involvement in chaos theory. Among other things, he described the Mandelbrot Set and coined the term fractal. He himself strongly influenced the popularization of his work through books and lectures. Mandelbrot spent most of his career at IBM’s central research facility. Later, he became Sterling Professor of Mathematical Sciences at Yale University.
Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line. — Benoît B. Mandelbrot, The Fractal Geometry of Nature
The Mandelbrot Set Machine
The Mandelbrot Set iteration formula for the complex plane, represented by the screen, is: \[ \begin{aligned} z_{n+1} &= z_n^2 + c \end{aligned} \] with \( z_0 = 0 \) and \( c \) as a placeholder for every pixel of the screen. The formula can be split into the real and imaginary components: \[ \begin{aligned} x_{n+1} &= x_n^2 - y_n^2 + c_x \\ y_{n+1} &= 2 x_n y_n + c_y \end{aligned} \] Points are considered elements of the Mandelbrot set (black pixels) if the iteration converges toward a fixed value, which can be practically assumed after reaching the specified maximum number of iteration steps. The iteration is stopped if \( |z_n| > 2 \), since the point \( c \) then no longer belongs to the Mandelbrot set. Colored pixels do not belong to the set; their color indicates the speed of divergence.
Machine commands
Use clicks at a point to zoom in, drag to move the scene, pinch in and out to zoom.