A precision tool for exploring the Mandelbrot set, Julia sets, and parameter space — built for people who care about the mathematics.
FractalSet renders the standard escape-time Mandelbrot and Julia sets with configurable iteration depth, palette, and shading mode. Coordinates are exposed as exact decimal values; every view is shareable via a permanent URL. No dependencies, no sign-in required — just the mathematics.
Every view exposes the center coordinates and zoom level as precise decimal values. Copy and paste into papers or code. The URL encodes the full state for permanent sharing.
Set the maximum iteration count from 50 to 2000+. Low counts reveal coarse structure quickly; high counts expose the finest boundary detail for research-quality images.
The Julia set explorer lets you map the Mandelbrot parameter space interactively — drag a point across the Mandelbrot set and the corresponding Julia set renders live, illustrating the fundamental theorem of the filled Julia set.
Switch between Mandelbrot, Julia, and Burning Ship (the Michelitsch–Rössler variant with |Re(z)| + i|Im(z)| folding). More fractal types in development.
Toggle between smooth normalized-iteration coloring, integer bands, angle shading, and orbit-trap modes to visualize different mathematical properties of the orbit.
Every view generates a shareable URL encoding coordinates, zoom, iteration count, palette, and fractal type. Use in papers, lectures, GitHub repos, or alongside a proof to direct readers to the exact region under discussion.
A point c is in the Mandelbrot set if and only if the orbit of 0 under z → z² + c remains bounded.
Escape radius: The bailout is , the standard choice. At this radius the orbit is guaranteed to diverge to infinity.
Smooth coloring: The default shader uses the normalized iteration count (Linas Vepstas / standard method) to produce a continuous value, eliminating integer banding artifacts.
Precision: The viewer uses 64-bit IEEE 754 double-precision arithmetic for the standard renderer, reaching useful zoom depths around . Deep zoom mode uses an extended fixed-point representation for zooms beyond the double-precision limit.
Julia set connectivity: The fundamental dichotomy theorem (Fatou / Julia / Douady-Hubbard) is directly visible in the parameter space explorer — pick a inside the Mandelbrot set and the corresponding filled Julia set is connected; pick one outside and it is a Cantor set.
Burning Ship: The Michelitsch–Rössler (1992) variant renders the absolute-value folding of the complex plane, producing the characteristic ship and flame structures.
FractalSet is particularly useful in complex analysis, dynamical systems, and topology courses where students need to build geometric intuition for iteration and convergence.
Open the Julia explorer and drag along the boundary of the main cardioid to show how the Julia set transitions from connected to Cantor as c crosses the Mandelbrot boundary.
Julia Explorer →Adjust iteration count from 50 to 500 to show how higher depth resolves the boundary more finely — a concrete example of computational approximation of a mathematical limit.
Mandelbrot Viewer →Use the coordinate article to walk students through the relationship between pixel positions on screen and points in the complex plane, building intuition for the Argand diagram.
Read the Guide →Zoom into any Mandelbrot boundary point to motivate the discussion of Hausdorff dimension 2 — the boundary is so complex it 'fills' 2D space.
Read the Article →No installation. No account required. All mathematics runs in your browser.
