Journey into the Infinite Beauty of Complex Dynamics
Mandelbrot Set
Julia Sets
Orbit Traps
The Mandelbrot Set
The Mandelbrot set is the set of complex numbers c for which the iterative function z = z² + c does not diverge when starting from z = 0. Discovered by Benoit Mandelbrot in 1980, it reveals infinite complexity at every scale.
Zoom: Use mouse wheel or pinch to zoom in/out
Pan: Click and drag to move around
Self-Similarity: Zoom in anywhere to discover infinite detail
Fractal Dimension: The boundary has dimension ~2, between a line and a plane
Connected Set: Despite appearances, the Mandelbrot set is a single connected shape
1.0x
Zoom Level
0.0
Center X
0.0
Center Y
256
Iterations
Julia Sets
Julia sets are closely related to the Mandelbrot set. For each point c in the complex plane, there's a unique Julia set. Points in the Mandelbrot set have connected Julia sets, while points outside have disconnected "Fatou dust" Julia sets.
Parameter c: Each c value creates a different Julia set
Algorithm: z = z² + c, starting with z = (pixel position)
Animation: Watch the Julia set morph as c changes
Connection: Points in the Mandelbrot set produce connected Julia sets
Click the "Set from Mandelbrot" button, then click on the Mandelbrot tab and click any point to set c!
-0.70
c Real
0.27015
c Imaginary
256
Iterations
Connected
Set Type
Orbit Trap Rendering
Instead of coloring based on escape iteration, orbit traps color pixels based on how close the orbit comes to specific shapes (like circles or lines). This creates unique and artistic visualizations of the same mathematical structures.
Circle Trap: Colors based on distance to origin during iteration
Cross Trap: Colors based on proximity to x or y axes
Line Trap: Measures distance to a diagonal line
Point Trap: Distance to a specific point in the complex plane