The fraction written on every bulb

The shape in every picture of itself derived the Mandelbrot set’s main body from one condition — the map $z \mapsto z^2 + c$ has a fixed point that attracts — and stated in passing that the bulbs round its edge are arranged by the arithmetic of fractions. That sentence is worth a whole essay, because it hides three separate claims. The first is that every fraction $p/q$ between nought and one has a bulb, at the position $p/q$ of a turn round the cardioid. The second is that the fraction is not only the bulb’s address but its behaviour: the periodic cycle that lives in the bulb goes round a fixed point, advancing $p$ places of $q$ at every step. The third is quantitative. The bulb’s size is set by the same two numbers, about $\sin(\pi p/q)/q^2$, so that a reader who knows the fraction knows roughly how large to expect the bulb to be.

None of the three is visible in the escape-time pictures the earlier essays drew. A bulb in a pixel picture is a black blob whose period cannot be read off, whose boundary is blurred by the iteration cap, and whose size depends on the colouring. So this essay draws the bulbs differently. Each outline in the figures is computed directly as the set of parameters at which a cycle of the bulb’s period is exactly neutral, and every claim is checked on hundreds of bulbs rather than on the few that are large enough to see.

The circle of multipliers #

Start from the fixed point. A point $z$ is fixed when $z^2 + c = z$, and its multiplier is the derivative of the map there, $\lambda = 2z$. If $|\lambda| < 1$ nearby points are drawn in and the fixed point attracts; if $|\lambda| > 1$ they are pushed away. Solving the fixed-point equation for $c$ in terms of the multiplier gives

$$c = \frac{\lambda}{2} - \frac{\lambda^2}{4},$$

and the main cardioid is exactly the image of the open unit disc of multipliers under this map. Its outline is the image of the unit circle, $\lambda = e^{2\pi i \theta}$ as $\theta$ runs from nought to one. So every point on the cardioid’s edge carries an angle $\theta$, the angle of its fixed point’s multiplier, and that angle is the coordinate the bulbs are arranged by.

The second figure shows the map from one picture to the other. On the left are the multipliers, a plain circle with the angles $1/2$, $1/3$, $2/3$, $1/4$, $3/4$ and the fifths of a turn marked. On the right are their images, which land on the cardioid’s outline at the roots of the bulbs of periods two to five. The arithmetic is checked rather than assumed: at each of the marked roots, solving for the fixed point and differentiating gives back the multiplier $e^{2\pi i p/q}$ to within $2 \times 10^{-16}$, the rounding error of the computation.

The picture has one feature that the formula makes obvious and the escape-time pictures obscure. The map from multipliers to parameters squashes the circle unevenly. Near $\lambda = 1$, the angle nought, its derivative $\tfrac12 - \tfrac{\lambda}{2}$ vanishes, and the circle folds back on itself into the cusp of the cardioid. Near $\lambda = -1$, the angle one half, the derivative has its largest size, one, and the circle is carried across without compression. The size of that derivative at the angle $\theta$ is $|\tfrac12 - \tfrac12 e^{2\pi i \theta}| = \sin(\pi\theta)$, and that sine will reappear in the bulbs’ sizes. The cardioid stretches the circle most where the angle is a half and least where it is close to a whole turn, and bulbs inherit that stretching.

A cycle born at every root of unity #

When the multiplier is a $q$-th root of unity, $\lambda = e^{2\pi i p/q}$ with $p/q$ in lowest terms, something is born. Applying the map $q$ times multiplies a small displacement from the fixed point by $\lambda^q = 1$, so to first order the $q$-th iterate does nothing, and the fixed point is degenerate for $f^q$. What happens at higher order is that a cycle of period $q$ splits off from the fixed point. On one side of the root the fixed point attracts and the cycle is repelling; on the other side the two exchange roles. The bulb is the region of parameters on the far side, where the new cycle attracts. Every bulb is therefore a region of stability for one cycle, exactly as the cardioid is the region of stability for the fixed point, and its outline is the set of parameters where that cycle’s own multiplier has size one.

That description is the method behind the hero figure. For each fraction, the bulb’s centre — the parameter at which the cycle passes through $0$, so that its multiplier is $0$ and it attracts as strongly as possible — is found by Newton’s method from just outside the root. The outline is then traced by asking, for each angle $t$, for the parameter at which the cycle’s multiplier equals $e^{it}$, and solving the pair of equations $f^q(z) = z$ and $(f^q)'(z) = e^{it}$ together for $z$ and $c$, moving $t$ a little at a time and starting each solve from the last. Every bulb of period up to thirty was traced this way, 277 of them, and at every one of the centres up to period twelve the orbit of nought returns to nought after exactly $q$ steps and no fewer. Those centres are a small subset of all the parameters at which nought is periodic — windows counted like necklaces counted every one of a given period — singled out by being attached to the cardioid itself.

The period-two bulb is a special case with a closed form: a disc of radius $\tfrac14$ centred at $-1$, which is where the computation starts its checks. It is also the only bulb that meets the real axis along a whole interval, and the dark lines that are one point’s orbit followed the real axis on into the doubling cascade that runs from it. The others are not discs, though the larger ones are close to round. The method has one place it must avoid. At the root the cycle collides with the fixed point, the two equations become degenerate, and Newton’s method has nothing to grip, so the tracing stops a few ten-thousandths of a turn short of the root on both sides. The outline’s last gap is invisible at any scale the figures use.

What the cycle does round the fixed point #

A bulb has a period, but it also has a numerator, and the numerator is visible in the cycle. Take the bulb at two fifths of a turn. Its centre is $c = -0.50434 + 0.56277i$, and there the orbit of nought runs through five points and returns. The fixed point of the map has not gone away — it is still there, now repelling, at $-0.4207 + 0.3056i$ — and the five points of the cycle sit round it like the ends of the spokes of a wheel.

The figure numbers the five points in the order the orbit visits them, starting from nought. Read in that order, $0, 1, 2, 3, 4$, they do not go round the fixed point one spoke at a time. Each step jumps two spokes anticlockwise, so that after five steps the orbit has gone twice round and come home. That is the numerator. The fixed point’s multiplier at the root was $e^{2\pi i \cdot 2/5}$, a rotation by two fifths of a turn, and the cycle born there keeps that rotation as the order in which it visits its spokes. The filled Julia set records the same thing at a larger scale: five of its pieces meet at the fixed point, and the map carries each piece to the one two places further round, which is the rotation one c, one picture found in the rabbit’s three lobes, there for the fraction one third.

This is the dynamical meaning of the angle on the cardioid. Before the root, the fixed point attracts and points near it spiral in, turning by about two fifths of a turn each step because that is the argument of the multiplier, and multiplying by a complex number is turning by its angle. At the root that turning becomes exact, and the cycle that branches off inherits it. A fraction $p/q$ describes a rotation by $p$ places of $q$, and the bulb at $p/q$ is the place where the map’s local rotation becomes a cycle that actually performs it.

Turning the dial is reading the fraction #

One example is suggestive and proves nothing. The fourth figure repeats the measurement for five bulbs at once, as dials: for each, the directions from the fixed point to the points of the cycle at the bulb’s centre, numbered by visiting order.

The pattern is exact in every case drawn. At $1/3$ each step moves one spoke; at $3/7$ three spokes of seven; at $3/8$ three of eight; at $5/13$ five of thirteen, the last dial left unnumbered because thirteen labels crowd the circle but the step was measured just the same. Beyond the figure, the same measurement was made on every bulb up to period thirteen, fifty-seven of them, computing the cycle at the centre, sorting its points by direction from the fixed point, and checking that each step of the orbit advances the rank by the same amount, equal to the numerator. All fifty-seven pass.

There is a theorem behind this, and it is one of the foundations of the subject. Adrien Douady and John Hubbard proved in the 1980s that the rotation number of the cycle about the fixed point is exactly the angle of the bulb’s root, for every bulb, and they used it to give every bulb attached to the cardioid its address. The rotation number is the same quantity that organises a quite different picture. The staircase that is flat almost everywhere found the circle map locking onto every fraction in turn, its plateaus arranged in the same order as these bulbs, because a locked orbit of the circle map is a cycle that advances $p$ places of $q$ — the same object, met as a map of the circle onto itself rather than of the plane.

A size set by the angle and the square of the period #

The bulbs in the hero figure shrink quickly as their periods grow, and they shrink more near the cusp than near the one-half bulb. A rough argument predicts both effects and gives a formula.

Work first in the multiplier’s coordinates, where the cardioid is a plain disc. Near the root $\lambda_0 = e^{2\pi i p/q}$, write $\lambda = \lambda_0(1 + \varepsilon)$ for a small $\varepsilon$. The $q$-th iterate of the map near the fixed point turns by $\lambda^q = (1 + \varepsilon)^q \approx 1 + q\varepsilon$, and the cycle that splits off carries a multiplier that starts at one and moves away from one at a rate proportional to $q$ times that again: to first order its multiplier is $1 - q^2\varepsilon$. The bulb is where the cycle attracts, where $|1 - q^2\varepsilon| < 1$, which is a small disc of radius $1/q^2$ in $\varepsilon$, tangent to the unit circle at the root. So in multiplier coordinates every bulb of period $q$ has size about $1/q^2$, whatever its numerator.

The parameter plane is reached through the map $c = \lambda/2 - \lambda^2/4$, which near the root stretches distances by its derivative, of size $\sin(\pi p/q)$ as the multiplier circle showed. Multiplying the two gives the prediction

$$\text{size of the } p/q \text{ bulb} \approx \frac{\sin(\pi p/q)}{q^2}.$$

The square in the denominator is what makes the large bulbs few: the period-three bulbs are about a tenth of a unit across, the period-ten bulbs a few thousandths. The sine is what makes the bulbs near the cusp smaller than those of the same period near the one-half bulb: at $1/10$ of a turn the sine is $0.31$, at $3/10$ it is $0.81$, so two bulbs of period ten differ in size by more than a factor of two because of where they hang.

The size, measured #

The argument is a linearisation. It keeps the first term of everything and assumes the bulb is small enough for first terms to be accurate, which is not obviously true for bulbs of period three or four. The fifth figure tests it on every bulb whose outline was computed for this essay with the numerator at most half the period, 245 bulbs of period up to forty.

Every point lies on or just above the diagonal, over four powers of ten. The ratio of measured size to predicted size runs from $0.983$, at the one-third bulb, to $1.141$, at $17/39$. The one-half bulb is exact, ratio one, because it is a disc of radius $\tfrac14$ and $\sin(\pi/2)/4 = \tfrac14$. Nothing in the computation was fitted: the outlines come from the multiplier equations, the sizes from the outlines, and the prediction from the argument above, and the three agree to within fourteen per cent everywhere they were compared.

That is an unusually good result for an argument that kept only first-order terms, and it is worth saying why it works as well as it does. The linearisation is done at the root, where the bulb is attached, and the bulb’s far side is $1/q^2$ away. The neglected terms are of relative size $1/q$ or smaller, so the error ought to be largest for the small periods. It is, but the small periods include the period-two bulb, for which the formula happens to be exact, and the period-three bulbs, for which the errors happen nearly to cancel. The large periods are where the formula is used, and there the measured sizes settle onto it.

Where the law slips #

Settling onto the formula is not the same as converging to it, and the sixth figure separates the two by following two families of bulbs as their periods grow.

Along the bulbs $1/q$, which crowd into the cusp, the ratio rises from one at $1/2$ to $1.047$ by $q = 20$ and then barely moves, changing by less than a thousandth between $q = 25$ and $q = 30$. Along the Fibonacci fractions $1/2$, $1/3$, $2/5$, $3/8$, $5/13$, $8/21$, $13/34$, $21/55$, which close in on the golden angle $1/\varphi^2$ of a turn, the ratio keeps rising: $1.099$ at $5/13$, $1.139$ at $13/34$, $1.147$ at period fifty-five. The formula is a leading term whose correction depends on where on the cardioid the bulb hangs, and near the golden angle that correction is larger and has not stopped growing at the largest period computed.

The golden direction is special for a reason three gaps and no more made precise: it is the angle worst approximated by fractions, so the bulbs near it are flanked by neighbours of large period on both sides and never by a large bulb. Whether the ratio along the Fibonacci fractions converges to a constant, and if so to what, is not something the linear argument can say and not something forty or fifty-five periods settle. The figure records a measurement, not a limit.

The largest bulb between two #

The first sentence written here about bulbs, in an earlier essay, said that bulbs between the $p/q$ and $p'/q'$ bulbs have period $q + q'$. Read carefully, that is a statement about the largest bulb between them, and the size law turns it into something that can be checked. Between two neighbouring fractions — two whose cross product $p'q - pq'$ is one, as in the Stern–Brocot ordering — every fraction in between has denominator at least $q + q'$, and exactly one, the mediant $(p + p')/(q + q')$, has that smallest denominator. If size falls like $1/q^2$, the mediant’s bulb must be the largest in the gap.

The bars show it at once for the pairs that can be read by eye. Between the one-third and one-half bulbs stands the two-fifths bulb, the tallest bar in that stretch; between one third and two fifths, three eighths; between two fifths and one half, three sevenths. The check behind the figure goes through every pair of neighbouring fractions whose mediant has period at most sixteen, twenty-five pairs, finds the largest computed bulb strictly between them, and asks whether it is the mediant’s. It is, in all twenty-five.

The sine factor could in principle spoil this, since a bulb of larger period nearer the one-half bulb has a larger sine. It never does, because the neighbours of a pair span so narrow a range of angles that the sine barely changes across the gap, while the denominators jump by at least the smaller of the two periods. The Farey arithmetic wins by a wide margin, which is why the rule held on every pair tested rather than merely on most.

What the outlines leave out #

Every figure here is an outline, and an outline is not a bulb. The real bulb is decorated: from every point of its edge at a rational angle hangs a smaller bulb, from that one another, and from each a filament carrying antennae and small copies of the whole set. The computed outlines show only the first level of that structure, the bulbs attached directly to the cardioid, and they say nothing about what hangs from them. The size measured is the size of the stability region for one cycle, not of the limb — the whole tree of decorations that grows from the bulb — and the two differ by a factor that is not small. Nothing drawn here measures the limbs, and the pictures cannot show whether every limb is as tidy as its first bulb.

There is a second omission, about what counts as proved. The rotation number of the cycle is a theorem for every bulb; the size law is not. The argument for $\sin(\pi p/q)/q^2$ is a first-order expansion, and what the figures establish is that it describes the 245 bulbs computed, not that it holds with any particular error for all of them. Rigorous bounds on bulb sizes exist — Jean-Christophe Yoccoz’s inequality bounds the size of a whole limb in terms of $q$ — but they are far weaker than the measured agreement, and the correction near the golden angle, growing at the largest period computed, is not described by any of them.

Still open: the edge between the bulbs #

Every rational angle on the cardioid’s edge has a bulb, and between the bulbs lies every irrational angle. At those points no cycle is born, and what the fixed point does there depends on how well the angle can be approximated by fractions. For most irrational angles the map is conjugate near its fixed point to a rotation, and a disc of points round the fixed point turns rigidly for ever — a Siegel disc. For angles too well approximated by fractions, that disc does not exist; Jean-Christophe Yoccoz identified the exact arithmetic condition, the Brjuno condition, in work for which he received the Fields Medal in 1994.

What is not known is a single complete description of the cardioid’s edge that covers both cases at once and the bulbs between them. Whether the Mandelbrot set is locally connected — whether, near every point of its boundary, it looks like a set built from connected pieces — is open, and it is the conjecture that would make the external description of the set exact, turning the arithmetic of bulbs into a complete map of the boundary. The area of the Mandelbrot set, from two sides met the same obstacle as a number nobody can compute. The angles at which the bulbs hang have a second life outside the set, as the directions from which external rays come in to land at each root, and that arithmetic of binary expansions is where the bulbs’ fractions are written down most completely.

One fraction, three readings #

The bulb at two fifths of a turn could have been named three ways, and the figures show that the names agree. It sits two fifths of the way round the cardioid’s circle of multipliers. Its cycle advances two places of five round the fixed point at every step. And its size, $0.0388$ across in the sense used here, is within two per cent of $\sin(2\pi/5)/25 = 0.0380$. Position, dynamics and size are the same fraction read in three ways.

The size is the reading that goes furthest. It comes from a single linear argument at the root, and it explains why the pictures look as they do: why the large bulbs are the ones with small periods, why the bulbs near the cusp are smaller than those of the same period near the one-half bulb, and why the bulb between two others is the one whose fraction is their mediant. The Farey arithmetic that orders the circle map’s plateaus and the rational approximations of how close a fraction can get orders the bulbs of a picture drawn from one squaring and one addition, and it orders them by size as well as by position.