Julia sets
A Julia image fixes the function and varies the starting point. Every pixel asks how its orbit behaves under repeated application of the same complex rule.
Filled Julia set and boundary
The filled Julia set contains starting points with bounded orbits. Its boundary is the Julia set proper, where arbitrarily nearby points can have sharply different futures. Coloring the escaping exterior reveals equipotentials and orbit speed.
The filled Julia set contains all initial values z₀ with bounded orbit under z↦z²+c; its boundary is the Julia set. Outside points can be colored by escape time, while interior points may reveal periods, potential or distance. A renderer should not conflate these categories: the black interior, colored exterior and boundary contour express different statements about the same orbit.
Connected or dust-like
For the quadratic family, parameters c inside the Mandelbrot set produce connected filled Julia sets; parameters outside produce disconnected sets. Near different bulbs, their geometry reflects the attracting cycle organized by that part of parameter space.
For quadratic polynomials, the critical orbit of zero determines connectedness. If c lies in the Mandelbrot set, the filled Julia set is connected; outside it, the set breaks into a Cantor-like dust. Near the boundary, extremely thin connections are easy to lose numerically. A coarse preview should therefore not make a final judgment about whether a Julia set is connected.
One family, radically different worlds
Small changes to c can turn a smooth-looking island into dendrites, spirals or dust. The Mandelbrot set acts as a directory for these changes, but a Julia set is not simply a crop of the Mandelbrot image. It lives in a different plane and assigns meaning to pixels differently.
Small parameter movements can shift from compact dendrites to spirals, dusts or thick basins. The variety is not arbitrary: it follows the position of c relative to hyperbolic components and their boundaries. A good gallery therefore stores the complex parameter as precisely as crop and zoom. A beautiful Julia rendering without c is scarcely more reproducible than a photograph without location or focal length.
Sources and further reading
This article summarizes the following specialist sources in original wording. Accessed and editorially reviewed 12 August 2026.
- Julia SetWolfram MathWorld
- Mandelbrot SetWolfram MathWorld
Continue the idea
The Mandelbrot set
What the Mandelbrot set contains, how z² + c creates its boundary and why deep zoom reveals related rather than identical copies.
Julia, Fatou and complex dynamics
The early twentieth-century theory behind Julia sets, Fatou sets and the later Mandelbrot set.
Complex numbers as a plane
A visual introduction to complex numbers, magnitude, angle and the squaring operation behind Mandelbrot and Julia images.