Bulbs Know Fractions
Every bulb on the Mandelbrot set repeats with a period equal to the denominator of where it is attached, and between any two bulbs sits their Farey mediant.
What it is
The main cardioid with the bulbs hanging off it, each labelled with the fraction of the way round it’s attached and the measured length of the cycle its orbits settle into. Those two numbers are always the same. It cycles through the bulbs, tap to step.
How it works
Iterate z → z² + c for three thousand steps to land on the attractor, then look for the smallest shift that brings the orbit back to where it was. That’s the period, measured rather than derived.
Locating the bulbs needs the cardioid’s own coordinate. Any c can be written as λ/2 − λ²/4 for some multiplier λ, and the cardioid’s interior is exactly |λ| < 1. So walking λ out past the unit circle at angle 2π·p/q steps cleanly out of the cardioid and into the bulb rooted there, at any angle, with no geometry to guess.
| angle | root | measured period |
|---|---|---|
| 1/2 | −0.7500 | 2 |
| 1/3 | −0.1250 + 0.6495i | 3 |
| 2/5 | −0.4818 + 0.5317i | 5 |
| 3/7 | −0.6064 + 0.4124i | 7 |
| 3/8 | −0.3536 + 0.6036i | 8 |
| 4/11 | −0.2919 + 0.6253i | 11 |
| 5/12 | −0.5580 + 0.4665i | 12 |
The ordering is arithmetic too. Between the p/q and r/s bulbs, the largest bulb sits at the mediant (p+r)/(q+s):
| between | and | largest between them | measured period |
|---|---|---|---|
| 1/3 (period 3) | 1/2 (period 2) | 2/5 | 5 |
| 1/4 (period 4) | 1/3 (period 3) | 2/7 | 7 |
| 2/5 (period 5) | 1/2 (period 2) | 3/7 | 7 |
So between the period-2 and period-3 bulbs there’s no period-4 bulb. There’s a period-5 one, because 2 + 3 = 5.
What surprised me
Twice, and both times it was about measurement rather than about the set.
My first attempt at locating bulbs used a guessed outward normal, the direction from a point inside the cardioid to the root. It worked well enough to get every period right and it was still wrong, because stepping the other way along it didn’t land back inside the cardioid.
That taught me something I hadn’t thought about. Near a bulb root, “outward from the boundary of the set” doesn’t mean “out of the set”. Outward is into the bulb, which is as interior as anywhere. The set has no outside there in the direction you’d expect one.
The second was sharper. Testing that a point just inside the cardioid has period 1, I got no answer at all. Not the wrong period, none. The orbit hadn’t converged after three thousand iterations.
Convergence goes like |λ|ⁿ, and just inside the boundary |λ| is a hair under 1, so a point at |λ| = 1 − 10⁻⁵ needs something like 10⁵ iterations before the cycle is visible.
The period is a perfectly well-defined property of every point in the interior, and it becomes unmeasurable as you approach the boundary. Arbitrarily slowly convergent, from both sides, and worst exactly at the roots where the interesting structure lives. The same thing bit me on the outward side, because right at a root the period-q cycle is parabolic, so I have to step well into a bulb, by about 0.35/q², to see the q at all.
The bulbs shrink like 1/q², so that leaves a workable but narrow band: far enough in to converge, not so far as to leave. Every number in this day’s tables comes from inside that band, and the fractions and the periods only look like clean arithmetic because the measurement was carefully placed.
What I would do next
Newton’s method on f_c^q(0) = 0 to find each bulb’s superstable centre exactly, where the cycle contains zero and converges fastest. The band problem disappears completely if you can aim at the one point in each bulb that converges quickest.