About Mandelbrot Deep Zoom

What is the Mandelbrot set?

The Mandelbrot set is the set of complex numbers c for which the iteration z_n+1 = z_n² + c (starting at z₀ = 0) does not escape to infinity. Points inside are typically black; points outside are colored by how quickly the iteration escapes. The boundary between inside and outside contains endlessly detailed fractal structure — the visual subject of deep-zoom exploration.

What this explorer does

Mandelbrot Deep Zoom lets you pan, pinch, scroll, and tap your way along the set’s boundary. Switch to Julia mode to explore Julia sets with the same deep-zoom pipeline, adjust the shape parameter c on a parameter map, morph shapes over time, or bridge from any Mandelbrot point with Explore Julia from this point.

Calculations and rendering happen entirely on your device. There is no account system, backend API, or server-side fractal computation in this static deployment.

Deep zooming

As you magnify, floating-point precision in ordinary GPU math runs out quickly. This app switches rendering strategies by depth:

• GPU fast path — standard escape-time iteration in a WebGL 2 fragment shader at modest zoom levels.
• Emulated double path — double-single arithmetic in the shader when zoom exceeds roughly 2×10⁴.
• Perturbation / reference-orbit path — a high-precision reference orbit is computed on the CPU (BigInt fixed-point coordinates). The GPU shader perturbs around that orbit for deep magnification beyond roughly 3.5×10⁵ zoom.
• Extended perturbation — additional handling for extreme depths beyond roughly 10²⁷ zoom.

Iteration counts scale automatically with zoom depth. This is practical deep-zoom rendering — not unlimited mathematical precision, but techniques that extend usable magnification far beyond standard float shaders.

Smart and Fixed zoom paths

• Smart Track — autopilot repeatedly steers toward high-detail boundary regions while zooming in or out.
• Fixed center lock — holds an exact center while only changing scale; pairs with mathematically stable deep anchors (period-2 tip, main cusp, and others).

Color palettes, Breathe, and Fractal Echo

Choose from palettes including Aurora, Electric, Ember, Monochrome, Cosmic, Psychedelic Flow, Breathe, and Cosine reference. Optional Acid Flow animates exterior colours with a smoothed phase clock (disabled when reduced motion is on). The Breathe palette adds a slow inhale–exhale color pulse that continues even while zoom is paused.

Fractal Echo composites fading copies of earlier frames on top of the current image, leaving tunnel-like afterimage trails during zooms and breathing motion.

Music-reactive visualization

Load a local MP3, WAV, or other supported audio file. The Web Audio API analyzes frequency and beat energy on your device — nothing is uploaded to a server. Two visual sync modes are available:

• Breathe Sync — maps bass and beats into the Breathe palette pulse.
• Full Spectrum — also maps mids and highs into hue, shimmer, and brightness across palettes.

A default demo track is available on the main page, or you can load your own local audio file.

Controls and interaction

Basic Controls provide experience presets (Normal, Cinematic, Breathe, Fractal Echo, Music Pulse, Deep Lock) and quick actions. Advanced Controls expose fine-tuning for view quality, iterations, autopilot behavior, music settings, echo parameters, and coordinate tools.

• Tap — center and dive ~2×
• Double tap — fast dive ~4×
• Drag — pan
• Pinch / scroll wheel — zoom
• Keyboard — Space (auto zoom), R (reset), D (discover), +/− (zoom), Escape (close panels)

Normal exploration does not rewrite the browser URL. Refresh returns to the default home view (with UI preferences restored from local storage). Tap Share in the bottom controls to copy a plain homepage link or a readable filter link (experience look, color theme, and Acid Flow).

Technical notes

The app is a single static HTML page with inline CSS and JavaScript, a WebGL 2 canvas, and an optional 2D canvas overlay for Fractal Echo. Preferences are stored in localStorage. See also llms.txt for a machine-readable summary.