Mandelbrot Set vs Julia Set: How the Two Fractals Are Linked
The Mandelbrot and Julia sets are generated by the exact same formula, z → z² + c — the difference is which variable is fixed. Compare the two fractal explorers and see how every point on the Mandelbrot set corresponds to a unique Julia set shape.
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⚡ Quick answer
Both fractals iterate the formula z → z² + c, but they fix different things: the Mandelbrot set fixes the starting point (z₀ = 0) and varies the parameter c across the whole complex plane, colouring each c by whether the orbit escapes. A Julia set fixes c at one specific value and instead varies the starting point z₀ across the plane. Every point you click inside the Mandelbrot set corresponds to a connected Julia set; every point outside corresponds to a Julia set that shatters into "fractal dust".
📊 Mandelbrot Set Explorer vs Julia Set Fractal Explorer — Interactive Complex Dynamics
| Mandelbrot Set Explorer | Julia Set Fractal Explorer — Interactive Complex Dynamics | |
|---|---|---|
| Formula | z → z² + c, iterated from z₀ = 0, plotting escape time as a function of c | z → z² + c, iterated from a variable z₀, for one fixed c, plotting escape time as a function of z₀ |
| What varies across the image | The parameter c (one point per pixel) | The starting value z₀ (one point per pixel); c stays constant for the whole image |
| Number of distinct sets | Exactly one Mandelbrot set — the master map | Infinitely many — a different Julia set exists for every possible value of c |
| Relationship between them | Acts as an "index" or "table of contents" to every Julia set | Each individual Julia set is one "entry" looked up by picking a point c on the Mandelbrot map |
| Connectedness rule | Always a single connected shape | Connected ("fractal dust"-free) if c is inside the Mandelbrot set; disconnected dust if c is outside it |
| Interactive hook on this site | Zoom into regions like Seahorse Valley or Elephant Valley | Click anywhere on the Mandelbrot set to instantly generate and view the matching Julia set |
| Best for learning | How a 2D parameter scan can map the entire behaviour of a dynamical system at a glance | How a single fixed parameter shapes the fate of every possible starting condition |
The Mandelbrot set looks like a completely different object from a Julia set at first glance, but they are two views of the exact same iteration: zₙ₊₁ = zₙ² + c. The Mandelbrot set answers the question "for each possible c, starting from zero, does the sequence stay bounded or fly off to infinity?" — it fixes the starting point and sweeps c across the whole complex plane. Every pixel in this site’s Mandelbrot explorer is really an entire orbit, summarised down to a single colour representing how quickly (or whether) it escapes.
A Julia set flips the roles: pick one specific c and ask "for each possible starting point z₀, does the orbit stay bounded?" The astonishing result, discovered by Gaston Julia and Pierre Fatou around 1918, is that the Mandelbrot set is literally a map of every Julia set’s shape: if your chosen c lands inside the Mandelbrot set, the corresponding Julia set is one connected blob; if c lands outside, the Julia set explodes into disconnected "fractal dust". This site’s Julia Set explorer lets you click directly on the Mandelbrot set to set c and instantly see which of those two outcomes you get — the clearest possible demonstration of the link between the two fractals.
Want to explore more? Browse the full library of 1000+ interactive, browser-based simulations, or see other side-by-side comparisons.