Explore the generalization of Mandelbrot set - z_{n+1}=z_n^p+c
The Multibrot set (polynomial iteration family) is a generalization of the Mandelbrot set, defined as the set of all complex numbers c for which the iteration formula z_{n+1} = z_n^p + c does not diverge. When p=2 it becomes the classic Mandelbrot set, p=3 corresponds to the Tricorn set, and other p values produce fractals with varying shapes. The power p can be any real number, including non-integers, creating infinitely diverse fractal structures.
For each point c on the complex plane, we start with z_0 = 0 and repeatedly apply the iteration formula z_{n+1} = z_n^p + c. Complex power operation uses the formula z^p = e^{p(ln|z| + i·arg(z))}, where arg(z) is the argument of the complex number (principal value range -π to π). If after sufficient iterations |z_n| still does not exceed 2, the point is considered to belong to the Multibrot set (displayed as black). If |z_n| exceeds 2, the point escapes to infinity, and we color based on escape speed (iteration count).
The Multibrot set demonstrates rich phenomena in complex dynamics. As the power p changes, the fractal's connectivity, symmetry, and boundary complexity undergo significant changes. Integer powers produce rotational symmetry (p-fold symmetry), while non-integer powers break symmetry, creating unique asymmetric patterns. This fractal family is an important tool for studying complex polynomial iteration, chaos theory, and fractal geometry.
Try different power values to observe changes in fractal morphology. Start with p=2 (classic Mandelbrot), then gradually increase or decrease p. Explore boundary regions where the richest details exist. Non-integer p values (like 2.5, 3.7) produce particularly interesting patterns. Increasing iteration count reveals finer edge details but reduces rendering speed.