Mandelhaus
Motion: on

A calm doorway into chaos.

Watch the boundary draw itself, then choose your journey: explore the classic Mandelbrot, conjure a Julia from a seed, learn the math, see nature’s echoes, or… discover the hidden room.

Explore Mandelbrot

Butter-smooth zoom with smart rendering and palettes.

Julia from Seed

Type a phrase → get your own Julia set, shareable.

Learn

History, relevance, and approachable math.

Nature

Recursion in coastlines, trees, lungs, lightning…
center: 0.0000 + 0.0000iscale: 2.8
Tip: scroll to zoom, drag to pan. Double-click to zoom in. Shift+double-click to zoom out.
c = 0.0000 + 0.0000iscale: 2.8
Type a seed. Scroll to zoom, drag to pan.

Learn: Mandelbrot, Julia & the beauty of iteration

The Mandelbrot set is the collection of complex numbers c for which the iteration z_{n+1} = z_n^2 + c (starting from z_0=0) does not escape to infinity. Boundary points exhibit infinite complexity—self-similarity without being exactly self-similar.

  • Smooth coloring: Even when a point escapes, using the fractional iteration count (a.k.a. normalized iteration count) yields continuous color gradients.
  • Julia sets: Fix a complex c and iterate z^2 + c for each starting z. Each c produces a distinct Julia set; points inside the Mandelbrot correspond to connected Julias.
  • Distance estimation: With derivatives you can estimate how far a point is from the boundary, handy for lighting/contours.
  • Precision & depth: Deep zooms benefit from perturbation (reference orbit) or high-precision arithmetic.

History highlights: Pierre Fatou and Gaston Julia studied these dynamics in the early 1900s; computers made them visible. Benoit Mandelbrot popularized the set in 1980, showing fractals as a bridge between pure math and natural structure.

Recursion in nature

  • Coastlines: Measured length grows as your ruler shrinks—fractal dimension > 1.
  • Trees & veins: Branching follows self-similar rules for efficient transport.
  • Clouds & mountains: Fractional Brownian motion models realistic roughness.
  • Lightning: Lichtenberg figures form branching patterns akin to diffusion-limited aggregation.
  • Lungs: Bronchial branching maximizes surface area within limited volume.

While not all of these are exact fractals, many exhibit scale invariance across ranges—one reason fractal math shows up in graphics, physics, and even finance.

Room 0: The Boundary Whispers

There is a rumor that if you zoom forever, the set zooms back. That somewhere in the cardioid’s heart there is a point that remembers you. We do not recommend knocking. (But if you must, bring a good palette.)

Entry logs are redacted. The dust on the console was arranged like a tiny Julia.