Universality
Named by 12 essays across 3 fields — each of them below, with the objects they name alongside it.
A constant that does not care which map
The gaps between successive period doublings shrink by a factor. Measure that factor for the logistic map and you get 4.669. Measure it for a completely different map and you get 4.669, and nobody expected that.
The rule that computes
One of the 256 elementary rules can run any program. Not simulate one, not approximate one — a machine that can compute anything computable, built from a lookup table with eight rows and nothing else.
Fourteen fractions that list the primes
Change the Collatz rule so that the multiplier depends on the remainder modulo some other number than two, and the resulting maps can compute anything a computer can. John Conway proved it in 1972, which means no method can decide, for every such map, whether its orbits come down — and fourteen fractions, applied in order, turn out to be enough to print every prime.
The graph that coin tosses always make
Take infinitely many vertices and toss a coin for every pair to decide whether they are joined. The result is random in every detail — and, with probability one, it is always the same graph. The same graph can be written down without any coins, by joining two numbers when one binary digit of the larger is a one.
![A strand of the logistic map's diagram between r = 3.8284 and 3.8572, upside down. The bifurcation diagram of the logistic map cropped to r in [3.8284, 3.8572] and x in [0.44, 0.565], with the value axis inverted.](/f/a-strand-of-the-logistic-map-s-diagram-between-r-3-8284-36b84e6c.svg)
Every window is the whole diagram again
Zoom into one strand of the period-three window of the logistic map and the whole bifurcation diagram is there again — a single value, a fork, a cascade of doublings, chaos, bands and windows — upside down and ninety-seven times narrower. The doublings inside it shrink by Feigenbaum's 4.669, exactly as the originals do, because near the top of its hump the map's third iterate is itself a one-humped map, turned over.
Windows counted like necklaces
The logistic map has one window of period three, two of period four, three of period five, five of period six, nine of period seven — and the sine map, which shares no algebra with it, has exactly the same numbers, in exactly the same order along the parameter. The counts are 1, 1, 1, 2, 3, 5, 9, 16, 28, 51, and they are the number of ways to thread a necklace of beads in two colours.
A flow that doubles like a parabola
Otto Rössler's three equations have one nonlinear term and a single knob. Turn it and the orbit closes after one loop, then two, then four, then never — the logistic map's cascade, in a flow in three dimensions. Record each peak of the orbit against the one before and the reason appears: the points lie on a single rounded hump. The gaps between doublings shrink by ratios 3.49, 4.54, 4.59, climbing towards Feigenbaum's 4.669, and the chaos contains a period-3 window that repeats the whole cascade.
One constant for each shape of top
Feigenbaum's 4.669 is the same for every smooth one-humped map — provided the hump is rounded like a parabola. Make the top flatter, like the graph of x to the fourth, and the doublings crowd together at 7.285; make it sharper, like x to the power 1.25, and they crowd at 3.260. Measured across twenty shapes of top, the constant is a smooth rising curve with 4.669 as one point on it, and the spatial constant falls as it rises.
Eigenvalues that keep their distance
Fill a large symmetric matrix with independent random numbers and its eigenvalues do something no list of independent numbers does: they refuse to crowd together. The reason is a count of conditions in a two-by-two matrix, and its consequences — a semicircle, gaps that vanish at nought, a spectrum almost as rigid as a lattice — reach from heavy nuclei to the zeros of the zeta function.
Doubling without an attractor
Feigenbaum's 4.669 holds for every map that folds and loses area. A map that keeps area — a frictionless pendulum sampled once a swing — has no attractors at all, yet its periodic orbits double just the same: a period-1 orbit at a = 3, period 2 at a = 4, then 4.1205, 4.1344, 4.1360 and on. The intervals shrink by 8.721 each time, not 4.669, and the orbits by 4.018, not 2.503. Let the map leak a little area and the cascade starts with one constant and finishes with the other.
Two constants at the flat point
Turn up the circle map until its graph goes flat at one point, then hunt for the setting at which an orbit turns by the golden ratio. The settings that lock to 1/2, 2/3, 3/5, 5/8 close in on it, each gap 2.8336 times smaller than the last, and the orbit's returns to its start close in by 1.2886 at every step. Below the flat line the same two numbers are the golden ratio's own. Change the map to a different one with the same kind of flat point and both numbers come back unchanged: they belong to flatness, not to the sine.
The silver mean at the flat point
At the golden-mean rotation number, the circle map with a flat point has two universal constants, 2.8336 and 1.2886. Repeat the measurement at the silver mean, √2 − 1, and two different numbers appear — 6.7992 and 1.5868 — shared by every map with the same kind of flat point. Below the critical line the silver gaps shrink by (1 + √2)², the square of the silver ratio, exactly as a rigid rotation predicts; on it they do not. Every rotation number with a repeating continued fraction carries its own pair, and a number whose expansion ends in ones inherits the golden pair, whatever came first.
Named alongside it
The objects these essays reach for when they reach for this one.
Period-doublingRenormalisationBifurcationFeigenbaum constantLogistic mapPeriodic orbitScalingCircle mapComputationRotation numberSelf-similarityUndecidability