The Shape That Shatters at the Border

The same rule, read the other way

Last time we ran one rule, z → z² + c, and watched it draw the Mandelbrot set. The recipe was: pick a constant c, start the running number z at zero, iterate, and colour c black if the number stayed bounded. Sweeping c across the plane drew the black bug.

There is a second knob on that machine, and last time we left it alone. The rule has two inputs, the constant c and the starting value z. The Mandelbrot set froze the start at z = 0 and swept the constant. A Julia set does the opposite: freeze the constant at some chosen c, and sweep the starting point instead.

Now the question is asked of every possible seed. Fix, say, c = −1. For which starting points z does z → z² − 1 stay bounded, and for which does it fly off? Colour the plane of starting points by that answer and you get a shape, black where the seed stays penned in, bright where it escapes. That shape is the Julia set for c = −1. Pick a different c and you get a different shape. There is a whole Julia set hiding behind every point of the plane.

Six worlds, one dial

The panel above is that dial. The gallery chips drop c onto six famous spots, and the big square redraws the Julia set for whichever constant is loaded. Each one is a different fractal grown from the same arithmetic, the only thing changing is the constant added at each step.

• The circle (c = 0). With nothing added, the rule is just z → z². Anything inside the unit circle shrinks to 0; anything outside blows up. The boundary is a perfect circle, the one Julia set that is smooth. Every other value of c roughens it into a fractal.
• The basilica (c = −1). Now 0 bounces forever between 0 and −1, a 2-cycle, and the circle buckles into a chain of bulbs joined at pinch points.
• Douady's rabbit (c = −0.123 + 0.745i). Here the orbit of 0 falls into a 3-cycle, and the set sprouts pairs of ears at every junction, all the way down.
• The dendrite (c = i). The orbit of 0 lands exactly on a repelling cycle and never leaves. The set has no inside at all, just an infinitely branched tree of hairs.
• The Siegel disc (c = −0.3905 − 0.5869i). The orbit of 0 circles forever without ever settling or escaping. Deep inside sits a disc the dynamics rotate by an irrational angle, like a record.
• The spiral (c = 0.285 + 0.535i). A constant near the map's edge; the set curls into slender arms while only just holding together.

Every one of these sets is symmetric under z → −z, which you can see in the two-fold rotational symmetry of each picture. That is not decoration, it is arithmetic: (−z)² = z², so a seed and its mirror through the origin always share the same fate.

Connected, or dust, and nothing between

Here is the surprising part. Look at the six sets and one of them, the dendrite, has no solid interior, while another, near the map's edge, threatens to break apart. You might expect a smooth range in between, sets that are mostly whole with a few pieces flaking off. There is no such range.

Gaston Julia and Pierre Fatou proved this in 1918, working entirely by hand, decades before anyone could render a single one of these images. A Julia set is either one connected piece or a totally disconnected dust of infinitely many separate points, a Cantor dust. It is never partly one and partly the other.

So which do you get? The answer is decided by a single orbit. Take the constant c, start at z = 0, and iterate. If that one orbit stays bounded, the Julia set is connected. If it escapes, the Julia set is dust. Module 02 draws exactly that orbit in the complex plane for whatever constant is loaded: watch it spiral quietly inside the bailout ring for a connected set, or bolt past it for a dust.

And "the orbit of 0 stays bounded" is a phrase we have met before. It is the exact definition of the Mandelbrot set. Which means:

A Julia set is connected precisely when its constant c lies inside the Mandelbrot set.

The map was an atlas all along

That is why the little Mandelbrot map in the corner of the panel is clickable. Click a point inside the black body and its Julia set holds together; click just outside and the same set shatters into dust; the dot shows where you are. The boundary of the Mandelbrot set is the precise seam between the two outcomes.

This reframes the whole picture from last time. The Mandelbrot set is not one fractal sitting alongside the Julia sets. It is a catalogue of every one of them. Each of its points is an entire Julia set: the interior points index the connected worlds, the exterior points index the dusts, and the infinitely detailed boundary is where they change from one to the other. The link, connected inside and dust outside, was proved by Adrien Douady and John Hubbard in the 1980s, the same work that showed the Mandelbrot set itself is connected.

So the black bug you fell into last time was quietly holding all of these shapes at once. Every point you passed on your way down was a whole other world, waiting to be grown.

Nothing here was looked up. Every Julia set, the atlas, and every orbit and verdict is recomputed from z → z² + c in your browser as the page loads. I checked the engine offline first (18 checks, all passing): the circle case, the two-fold symmetry, the periods of the six gallery orbits, the escape steps of the dust cases, and the connectedness rule itself, that 0's orbit is bounded exactly when c is in the Mandelbrot set.