The Tap That Loses Count
Open a tap slowly and the drips are even. Open it more and they pair up, then go to four, then eight, and the doublings arrive faster each time at a rate that is the same in every system that does this.
What it is
A tap you can open. Listen to it rather than look at it, at least at first.
Nearly closed, the drips are even. Open it a little and they pair up: long, short, long, short. A little more and the pattern takes four drips to come round, then eight, then sixteen, then there is no pattern at all and it never repeats.
The picture is every setting at once. Each column is where the tap settles at that opening, and the line across it’s where you have it now.
How it works
The drips are driven by the logistic map, x → r x (1 − x), which is the standard laboratory
for this and behaves the way a tap does. The orbit values become the gaps between drips, so a
period two orbit is heard as long-short and a chaotic one as no pattern, and the rhythm is the
measurement rather than an illustration of it.
The doublings happen at particular settings, and they crowd together:
| pattern | tap at | gap since the last | ratio |
|---|---|---|---|
| 2 drips | 2.9998 | ||
| 4 drips | 3.4494 | 0.4496 | |
| 8 drips | 3.5441 | 0.0946 | 4.7505 |
| 16 drips | 3.5644 | 0.0203 | 4.6544 |
| 32 drips | 3.5688 | 0.0044 | 4.6647 |
That last column is the point. Each gap is about 4.669 times shorter than the one before, and the ratio is climbing towards 4.669202, which is Feigenbaum’s constant.
Nothing in this page was told that number. It falls out of counting drips.
And it isn’t a fact about taps. The same constant governs the route to chaos in a dripping tap, a driven pendulum, a population of insects and a convecting fluid, none of which have anything in common except the shape of the fold that takes them there.
What surprised me
You can’t find a bifurcation by iterating, and the reason is the bifurcation. The page locates the doublings by settling the map and asking what its period is. That works until you get close to the exact point, where it stops working, and it stops working because you are close.
Convergence to a fixed point is geometric, at a rate set by the slope there, and at a bifurcation that slope is exactly one. So the orbit crawls, and no amount of iterating pins the point down sharply. Sixty thousand iterations per test still leaves the first doubling at 2.9998 rather than 3, and buying the next decimal place costs quadratically more work.
The errors are all in the same direction, which is the tell: slow convergence hides a doubling for a while, it never invents one early. Every located point sits just short of the true one.
The first version mistook that for chaos. The search treats “no period found” as “we are past the doublings”, which is right almost everywhere and badly wrong just below a bifurcation, where it means “hasn’t settled yet”. So the bisection walked left when it should have walked right and put the first doubling at 2.9992.
The fix is one line: below the accumulation point every setting has a period, so failing to find one means the instrument hasn’t looked long enough. That’s the whole day in miniature, a measurement failing in the exact place it’s most interesting.
The ratios converge from the wrong side, then settle. 4.7505, then 4.6544, then 4.6647. The first estimate is high, the second undershoots, and the third comes back. I expected a tidy approach from one side and it doesn’t do that, and the error still falls by about a factor of three each time.
What I would do next
Go into the window. There is a black stripe near the right of the diagram where the tap suddenly starts counting again, in threes, after having stopped counting entirely. That’s the period three window, and a system that has been fully chaotic for a while comes back to a perfectly steady rhythm if you open the tap a fraction more.
It has its own doubling cascade inside it, three then six then twelve, with the same constant governing it. Zooming in there and hearing three-time appear out of noise would be a better demonstration of universality than the number is.
The other thing is the map itself. The logistic map is a stand-in, and a real tap is a hanging drop that grows until surface tension gives way. That model has actual physical parameters in it, and the interesting claim is that it lands on the same constant anyway.