The shape in every picture of itself

Pick a complex number $c$. Start at zero and repeat one line:

$$z \mapsto z^2 + c.$$

Either the numbers stay in a bounded region forever, or they run off to infinity. There is no third outcome. Colouring the plane by which happens gives this.

Every visible feature of that picture — the bulbs, the filaments, the specks of black in the apparently empty regions — is produced by a squaring and an addition. Nothing else is in the rule.

What the iteration does #

The arithmetic is worth doing by hand once, because the rest of the essay is about consequences of it and the consequences are easier to believe if the mechanism is concrete.

Squaring a complex number doubles its angle and squares its length. So $z \mapsto z^2$ alone is simple: anything of length under one spirals in to zero, anything over one runs away, and the unit circle stays put while its points get shuffled around it by angle-doubling — which is the doubling map of the last field, on a circle instead of an interval.

Adding $c$ each time breaks that symmetry. Now the contraction and the escape are competing with a fixed push, and which wins depends on $c$ in a way no formula answers.

Try $c = -1$: the orbit is $0, -1, 0, -1, \ldots$, a two-cycle, bounded forever, so $-1$ is in the set. Try $c = 1$: $0, 1, 2, 5, 26, 677, \ldots$, gone. Try $c = 0.4$: $0, 0.4, 0.56, 0.7136, \ldots$ — creeping up, and eventually past the point of no return. Those four cases are what the generator asserts, because a picture of an iteration should be checked against the iteration done by hand.

Why the orbit of zero, and not any orbit #

The definition looks arbitrary. Every $c$ gives a whole map, with infinitely many orbits; why is the set defined by what happens to one particular starting point?

Because zero is the critical point of $z^2 + c$ — the place where the derivative vanishes, the point at which the map folds the plane over on itself. There is a theorem, and it is the reason the definition is not arbitrary: for these maps, if any attracting cycle exists, the orbit of the critical point converges to it. The critical orbit finds every attracting cycle there is, so watching it watches everything.

That is why one orbit is enough, and it is also why the set is a picture of a family rather than of a system. The interval maps of the last field had one bifurcation diagram summarising every parameter’s behaviour on one axis; this is that diagram with a plane of parameters instead of a line.

The bands are not decoration #

Everything outside the set escapes. The contours say how fast.

A point far from the set escapes in two or three steps. A point just outside takes hundreds, because it spends a long time being nearly captured before the fixed push wins. The bands are level sets of that count, and their crowding towards the boundary is a measurement: escape time goes to infinity on approach to the set, which is what makes the boundary a boundary rather than an edge.

Drawing them as contours rather than as coloured pixels is deliberate. This site’s rule for figures is that raster carries texture and never information — anything a reader would measure gets drawn as a line. Here the reader measures the boundary, so the boundary is where the lines are.

That caption states the one honest limitation of every picture of this set. What is drawn black is not “the set” but “the points that had not escaped by step $N$”, and those are different sets for every $N$. Increasing $N$ shrinks the black region, always, and the true set is the intersection over all $N$ — a limit that no computation reaches.

The cardioid is not drawn, it is derived #

The largest feature of the set has a formula, and deriving it is the one place in this subject where an equation answers a question about the picture.

A point $c$ is in the main body when the map $z \mapsto z^2 + c$ has an attracting fixed point. Being fixed means $z^2 + c = z$; attracting means $|2z| < 1$, since the derivative is $2z$. Write $z = \tfrac12 e^{i\theta}$ for the boundary case $|2z| = 1$ and solve the first equation for $c$:

$$c = z - z^2 = \tfrac12 e^{i\theta} - \tfrac14 e^{2i\theta}.$$

That curve, traced as $\theta$ runs once round, is a cardioid — and it is exactly the outline of the largest region in the figure. The same argument applied to two-cycles gives a circle of radius $\tfrac14$ centred at $-1$, which is the largest bulb, and applied to $k$-cycles gives the rest of the bulbs.

So the big shapes are stability regions, one per cycle length, and the reason there are infinitely many of them is that there are infinitely many cycle lengths. The period of each bulb can be read off its position round the cardioid, and the arithmetic that governs which period sits where is the Farey arithmetic this collection met in the mediant — bulbs between the $p/q$ and $p'/q'$ bulbs have period $q + q'$.

What is not derived by any of this is the boundary itself. Every bulb has a formula; the set of points that are in no bulb does not.

What is actually known #

The set has a reputation for being mysterious, and some of what is said about it is proved.

It is connected. Douady and Hubbard proved in 1982 that the set is a single connected piece — every visible island is attached to the mainland by a filament too thin to draw. That is not obvious from any picture, and the picture actively suggests otherwise.

It is contained in the disc of radius two. If $|z|$ ever exceeds $2$ with $|c| \le 2$, the orbit escapes, and the proof is two lines: once $|z| > \max(|c|, 2)$, each step makes $|z|$ larger by a definite factor. That is why every escape test on the drawing above uses the same threshold, and why the test is exact rather than a heuristic.

Its boundary has dimension two. Shishikura proved in 1991 that the boundary — a curve, in the loose sense — has Hausdorff dimension exactly $2$, the dimension of the plane it sits in. A one-dimensional object being as crinkled as a two-dimensional one is the precise version of what the zoom above shows informally.

Its area is unknown. Nobody knows the area of the set in closed form. Numerical estimates put it near $1.506$; the error bars have not closed, and no method that would close them is known.

Not quite self-similar #

The word “fractal” attaches to this set immediately, and the sense in which it applies is narrower than the pictures suggest.

Zoom anywhere on the boundary and eventually a small copy of the whole set appears — a baby Mandelbrot, complete with its own cardioid and bulbs. There are infinitely many, at every scale, everywhere.

They are not exact copies. Each is surrounded by different filigree, distorted differently, and the deviation is measurable rather than aesthetic. So the set is not self-similar in the way a rectangle that eats itself is, where the copy is the original scaled by a fixed ratio. It is quasi-self-similar: copies appear at every scale, and no two are the same. The contrast is worth holding beside the square that cannot shrink, where an exact scaling argument settles a question outright — exact self-similarity is a tool, and approximate self-similarity is only a description.

That distinction matters because the exact kind is what makes a scaling constant computable. The Feigenbaum constant exists because the interval cascade repeats itself by a fixed ratio; the Mandelbrot set’s baby copies have no single such ratio, which is one reason it has no comparable constant.

Two ways a point can be inside #

The black region is not homogeneous, and the difference between its parts is the open question of the subject.

Inside a bulb, the orbit of zero converges to an attracting cycle — the bulb’s period — and the behaviour is completely understood. These are the hyperbolic components, and the cardioid, every bulb, and every baby cardioid at every scale is one.

Whether that is all of them is unknown. A point could in principle be in the set without being in any hyperbolic component: its critical orbit bounded but attracted to nothing, wandering forever inside a region with no cycle to fall into. Such components have never been found and nobody has proved they cannot exist.

The density of hyperbolicity conjecture says they do not — that the hyperbolic components are dense, that the black region is exactly the closure of the bulbs. It is the central open problem here, it is equivalent to several other statements that look unrelated, and it has been open since the pictures were first drawn.

It has a one-dimensional twin, which is the version worth carrying: on the real line it says that for a dense set of parameters, the logistic map has an attracting cycle. Read against the bifurcation diagram, that says the smear is full of windows too narrow to see — that every apparently chaotic band has order arbitrarily close by. Nobody has proved it in either dimension.

Where the last field is hiding in this one #

The connection between this picture and the previous essays is not an analogy. It is the same objects.

Restrict $c$ to the real axis and $z^2 + c$ becomes a real map of an interval, conjugate to the logistic map by a change of variable. The real slice of the Mandelbrot set — the segment from $-2$ to $0.25$ — is exactly the parameter range of the logistic cascade, and its features line up.

The main cardioid’s real span is where the logistic map has an attracting fixed point. The largest bulb is the two-cycle. The chain of shrinking bulbs down the negative real axis is the period-doubling cascade, and the point where they accumulate, $c \approx -1.401155$, is the Feigenbaum point in the other coordinate system. The visible island near $c = -1.75$ is the period-three window.

So the horizontal line through the middle of the hero figure is the bifurcation diagram, and the two-dimensional set is what happens when the parameter is allowed to leave the real line. The change of variable is explicit — $c = \tfrac{r}{2}\left(1 - \tfrac r2\right)$ carries the logistic parameter to the quadratic one — so the correspondence is an identity rather than a resemblance, and every feature counted in the cascade essay can be found here at a known place.

Holding c still instead #

Everything above varies $c$ and starts the orbit at zero. Doing it the other way round — fixing $c$ and asking which starting points stay bounded — gives a different picture for every $c$, and the two families are tied together by a theorem.

That assertion is the interesting one, because it means the Mandelbrot set is a catalogue of Julia sets rather than an object of the same kind. Each point of the plane is one Julia set, and the black region marks the ones that are connected. Move the parameter across the boundary and the corresponding Julia set shatters into dust — a discontinuous change in a picture, produced by a continuous change in a number, which is the next essay’s subject.

What the drawing costs, and what it hides #

The escape test is cheap and the honest picture is not, and the gap between them is where every misleading rendering of this set comes from.

The cap decides the black. A point in a bulb of period 40 has an attracting cycle, so it never escapes — but a point near it, outside the set, may take thousands of steps. At a cap of ninety they are the same colour. Every published picture of this set at every resolution is showing a region strictly larger than the set, and the excess is largest exactly where the structure is most interesting.

The sampling decides the filaments. The connecting filaments are thinner than any sample grid, so a picture drawn by sampling can never show them, and the set therefore always looks disconnected in exactly the way the theorem says it is not. The figures here sample a $300 \times 300$ grid; a filament of width $10^{-40}$ is not going to appear on one.

Neither is a bug to fix. Raising the cap and the resolution moves both thresholds and removes neither, because the object has structure below every threshold. That is what dimension two for a boundary means, stated as a fact about rendering.

The right response is the one this collection takes with every computed figure: say what the drawing is a drawing of. It is a drawing of “had not escaped in $N$ steps, sampled on a grid of $n$ points” — a well-defined object, computable, close to the set in a sense that can be made precise, and not the set.

The one thing the picture is for #

It is worth being clear about what this figure is evidence of, since it is a computation and computations can lie.

It is not evidence that the boundary is complicated at every scale — that is a theorem, proved without pictures. It is not evidence about which points are in the set — that requires unbounded computation for any point not settled by an assertion.

What it is evidence of is the claim in the first paragraph: that a rule this short produces this. Nobody expected it — and the strongest evidence for that is the reception. The set was drawn before it was understood — Brooks and Matelski in 1978, Mandelbrot in 1980 — and the theorems came afterwards, prompted by the pictures. That order is unusual, and it puts this alongside the four-colour theorem and the Feigenbaum constant as results the machine reached first.

The next rung holds the parameter still and varies the starting point instead, which turns one picture into a different picture for every $c$ — and ties the two families together with a theorem the generator above already checks.