Take almost any simple equation that feeds its own output back into itself, and turn one knob toward chaos. It will double its rhythm — one beat, then two, then four, then eight — and the rate at which the doublings come faster is always the same number. Not roughly. Exactly. And the equation barely matters.
In 1975 a physicist at Los Alamos named Mitchell Feigenbaum was working out, by hand on an HP-65 calculator, where a simple population model breaks into chaos. The model is the logistic map: take a number x between 0 and 1, and replace it with r·x·(1−x), over and over. For small r the number settles to one value. Turn r up and it can't settle — it flips between two values, then four, then eight, the period doubling again and again, faster and faster, until the doublings pile up and the whole thing dissolves into chaos.
Feigenbaum measured how much faster each doubling arrived than the last. The ratio kept landing near the same number — about 4.669 — and it stopped changing. Then he tried it on a completely different equation, one built from a sine wave instead of a parabola. He got the same number. Same digits. He famously called his wife and said he'd found something, and that it was going to make him famous.
It did. That number — 4.6692016091…, now called Feigenbaum's constant δ — turned out to be universal: it governs the route to chaos not just for the logistic map but for an entire class of systems, including real physical ones, water heated from below, dripping faucets, oscillating circuits. The specific equation is almost irrelevant. Only one feature of it matters, and at the very end of this page you'll get to break the constant by changing that one feature.
None of this is mine to assert. Everything below recomputes itself in your browser. Let's start with the picture.
This is the bifurcation diagram of the logistic map. Horizontal axis: the knob r, from 2.8 up to 4. Vertical axis: the values the system eventually settles into. Read it left to right and you watch the period double — one curve splits into two, two into four — and then shatter into the speckled chaos at the right. The brighter a region, the more often the system visits it. Drag a box on the diagram to zoom into the cascade; the same forking structure repeats inside itself, forever.
Now the measurement. There's a clean way to pin each doubling: find the value of r at which the cycle of length 2n is superstable — the value where the cycle passes exactly through the peak of the map. Call it Rn. Feigenbaum's δ is the limit of the ratios of the gaps between them:
δ = limn→∞ (Rn − Rn−1) ⁄ (Rn+1 − Rn)
Press the button. The page locates each Rn by bracketed bisection — solving f2ⁿ(½) = ½ — and reports the ratio as it goes. Watch it climb toward 4.669. Then switch the map: the logistic parabola and the sine wave produce different accumulation points and different Rn — but the same δ. That equality is the whole miracle.
| n | Rn (superstable) | gap | ratio → δ |
|---|
The logistic map's doublings pile up at r ≈ 3.5699; the sine map's at r ≈ 0.8656. Totally different machines. They speed into chaos at the same rate.
If the equation barely matters, something must. It's this: the order of the maximum — how sharp or flat the single hump of the map is at its peak. The logistic parabola and the sine wave both have a quadratic peak (locally it looks like a parabola, like x²). Every map with a quadratic peak shares δ = 4.6692. Change the peak's shape and you change the constant.
Here is one family that lets you dial it: f(x) = r·(1 − |2x−1|p). At p = 2 this is the logistic map (the algebra collapses to 4r·x(1−x)). Slide p and the hump goes from a sharp spike toward a flat-topped plateau — and Feigenbaum's constant moves with it: 4.669 at p = 2, 7.285 at p = 4, 9.296 at p = 6. The diagram morphs; the number breaks. Both are recomputed live as you drag.
The reason the equation washes out is renormalization. Look at the map, then look at the map applied twice and zoomed into the region around the peak: as you keep doubling and zooming, the rescaled maps converge to a single universal shape — a fixed point of the "do-it-twice-and-rescale" operation. δ is an eigenvalue of that operation, so it can't depend on where you started, only on the symmetry class (the order of the peak). The second Feigenbaum constant, α = 2.5029078750…, is the matching spatial scaling — how much you have to zoom each time.
This is not hand-waving. In 1982 Oscar Lanford III gave a rigorous proof of the existence of that fixed point and the universality — and it was a computer-assisted proof, one of the early famous ones, carried out with controlled interval arithmetic to bound every rounding error. So the deepest fact on this page is itself a thing a machine had to check, which is a sentiment this whole site is built on: never trust; verify.
One more thing, because it's too good to leave out. The logistic map is a disguised version of the most famous object in mathematics. The map x → r·x(1−x) is the same dynamical system as z → z² + c — the iteration that draws the Mandelbrot set — restricted to the real number line. Run the change of variables and the logistic window r ∈ [1, 4] becomes exactly the real segment c ∈ [−2, ¼] of the Mandelbrot set. The period-doubling cascade you zoomed into in Instrument 01 is literally the row of circular bulbs marching down the spike on the left of the Mandelbrot set — each bulb a doubled period — and they shrink by Feigenbaum's δ. The accumulation point lands at c ≈ −1.401155, the Mandelbrot set's own Feigenbaum point.