Fractal Guide
What is a Multibrot Set?
The Multibrot set generalizes the Mandelbrot set by varying the power in the recurrence formula. Increasing the power to 3, 4, 5โฆ reveals fractals with beautiful rotational symmetry.
Contents
1. Definition of the Multibrot
The Multibrot set is defined by z(n+1) = z(n)^d + c, where d is an integer โฅ 2. When d = 2, it's the standard Mandelbrot set.
Increasing d dramatically changes the set's appearance. d = 3 gives 3-fold symmetry, d = 4 gives 4-fold symmetry, and so on โ the fractal exhibits d-fold rotational symmetry.
2. Power and Symmetry
d = 2 (standard Mandelbrot) has the iconic cardioid and circular shape. d = 3 becomes roughly triangular, d = 5 roughly pentagonal.
All Multibrot sets exhibit fractal self-similarity near their boundaries. Higher-power sets appear rounder overall, converging toward the unit circle as d โ โ.
3. Mathematical Background
The Multibrot's properties change with d. The d-fold rotational symmetry derives from z^d's symmetry. Topologically, all Multibrot sets are known to be connected and compact.
Each Multibrot set for a given d serves as a 'catalog' of corresponding Julia sets. As with the Mandelbrot set, choosing c inside the set yields connected Julia sets.
4. Rendering and Exploration Tips
Rendering uses the same escape time algorithm as the Mandelbrot set. However, computing z^d becomes more expensive for larger d, so rendering slows as the power increases.
In Tool Palette's Fractal Gallery, you can adjust the power d with a slider and observe shape changes in real time. Start from d = 2 and gradually increase to see how symmetry evolves.
Explore the Multibrot Set
Adjust the power and explore the Multibrot in the Fractal Gallery.
Go to Fractal Gallery