Julia sets, and the map of all of them

The rule $z_{n+1} = z_n^2 + c$ has two numbers in it — the starting point $z_0$ and the constant $c$ — and that means there are two different questions you can ask of it.

The Mandelbrot set answers: vary $c$, always starting from $z_0 = 0$ — which constants produce bounded orbits? That picture lives in the parameter plane: each pixel is a different rule.

$$M = \{\, c : \text{the orbit of } z_0 = 0 \text{ under } z \mapsto z^2 + c \text{ stays bounded} \,\}$$

A Julia set answers the opposite question: freeze one particular $c$, and vary the starting point $z_0$ instead. Which beginnings survive under this one fixed rule? Now the picture lives in the dynamical plane:

$$K_c = \{\, z_0 : \text{the orbit of } z_0 \text{ under } z \mapsto z^2 + c \text{ stays bounded} \,\}$$

(Properly, $K_c$ is the filled Julia set and the Julia set is its boundary — the rendered pictures show the filled set, so this guide says "Julia set" the way explorers do.) Every point of the parameter plane names an entire Julia set of its own — the Mandelbrot set is, in a precise sense, a map of all Julia sets.

A neighbourhood in the parameter plane. Seahorse Valley in the parameter plane. Every pixel here is a whole potential Julia set — pick one c and hold it fixed… c ≈ −0.745 + 0.113i · span 0.05 · open this view in Mandala →

…and the Julia set it names. The Julia set for that exact c. Seeds from near the boundary give the richest sets — and they resemble where they came from. Julia, c ≈ −0.745 + 0.113i · span 3.2 · open this view in Mandala →

The map tells you what you'll find

The correspondence runs deeper than bookkeeping, and the reason starts with a question the Mandelbrot chapter left hanging: why must the orbit start at $z_0 = 0$? Because $0$ is the map's critical point — the one place where the derivative $\frac{d}{dz}(z^2+c) = 2z$ vanishes. A century-old theorem of Fatou and Julia says the critical orbit is the diagnostic orbit:

The Julia set of $z \mapsto z^2 + c$ is connected — one unbroken shape — exactly when the orbit of the critical point $0$ stays bounded. If the critical orbit escapes, the Julia set shatters into a totally disconnected dust (a Cantor set) — no piece of it touches any other.

But "the critical orbit of $c$ stays bounded" is precisely the definition of $c \in M$. So the Mandelbrot set is not just an index of Julia sets — it is the answer sheet: choose $c$ from inside $M$ and its Julia set is connected; choose from outside and it is dust. There is nothing in between.

The inheritance goes further than connectedness. The richest Julia sets come from seeds near the boundary of $M$, and they visibly inherit the character of their neighbourhood: a seed from Seahorse Valley gives a Julia set full of seahorse curls; a seed from a spiral region gives spirals. (This is no coincidence — and at one kind of boundary seed, the Misiurewicz points of the Mandelbrot chapter, it is a theorem of Tan Lei: there the Mandelbrot set and the Julia set are provably asymptotically similar.)

The dendrite at c = i. On the boundary itself, the Julia set thins to a branching filament with no interior at all. Julia, c = i · span 3.4 · open this view in Mandala →

Exactly on the boundary, stranger things live. The worked orbit of the Mandelbrot chapter showed $c = i$ is a Misiurewicz point — bounded, but only just: its Julia set is the dendrite above, a shape with branches but no interior at all, the connected case pushed to its skeletal limit.

A field guide of seeds

Some Julia sets are famous enough to have names. Every row is one click away in the explorer — enter the seed via Locations → "go to a coordinate", or Alt-hover it on the Mandelbrot map:

Seed $c$	Name	Where it lives, what you get
$-1$	Basilica	Centre of the period-2 bulb (exact): two-lobed, orbit beats $0 \leftrightarrow -1$.
$\approx -0.1226 + 0.7449i$	Douady rabbit	Centre of the biggest period-3 bulb: three ears repeating everywhere.
$-\tfrac34$	San Marco	The exact pinch-point between cardioid and period-2 bulb: domes and spires on a razor's edge.
$\approx -1.7549$	Airplane	A period-3 minibrot's centre out on the real antenna: swept-wing crosses.
$i$	Dendrite	Misiurewicz point (exact): pure branches, no interior.
$0.3$	(dust)	Just outside $M$ on the real axis (the critical orbit escapes slowly): a Cantor dust that almost holds together.

The pattern to notice: seeds at bulb centres give sturdy, thick Julia sets (their orbits are superattracting); seeds at boundary pinch-points and Misiurewicz points give the filigree; seeds outside give dust — finer and more structured the closer to the boundary you choose them.

Try it in the explorer

Mandala keeps both planes one gesture apart. In the parameter plane, hold Alt (or long-press on touch) and a small inset previews the Julia set for the $c$ under your pointer, live; click to commit and the view opens that Julia set for real. In the Julia plane, the same inset flips around and becomes a locator showing where your current seed sits on the Mandelbrot map. Every escape-time family here has both planes — Burning Ship Julia sets are just as real as Mandelbrot ones, and far less travelled.