Binary expansion
Named by 8 essays across 4 fields — each of them below, with the objects they name alongside it.
The orbit written as a word
Cut the interval in two and record which half each step of an orbit lands in. The orbit becomes an infinite string of two letters, the map becomes the act of deleting the first letter, and questions about trajectories turn into questions about words.
The orbit a computer draws
A chaotic orbit computed in floating point is not the orbit of the point it started from. Sometimes it is the true orbit of a nearby point, which is enough; sometimes the arithmetic simply runs out, and the picture is of the rounding.
A coin in front of every term
Put all plus signs in front of 1, 1/2, 1/3, … and the sum runs off to infinity; alternate them and it settles on log 2. Toss a fair coin for each sign instead, and the sum settles — every time, on a different number. Where it tends to settle has a smooth, flat-topped shape, and at the value 2 that shape takes a height that agrees with one eighth to forty-two decimal places and is not one eighth.
Almost every orbit is fair
Double a number and keep the fractional part, and do it again, and again. Where the orbit goes is written in the number's binary digits, and for almost every starting point it spends exactly a quarter of its time in each quarter of the interval. The proof is a short argument about Fourier coefficients being pushed to infinity — and it leaves room for a set of exceptions with no length at all, which includes every fraction and, for all anyone can prove, every number anyone has ever named.
A coin in front of every power
Toss a coin for each sign of ±1 ± λ ± λ² ± … and the sum lands somewhere. Below λ = 1/2 it lands on a dust with gaps in it, at exactly 1/2 it lands anywhere with equal chance, and above 1/2 it lands on a smooth-looking hill — which for most λ has a density and for the golden value does not, though no picture can tell the two apart.
A third reached only in the limit
No compass-and-straightedge construction trisects every angle. But a quarter, plus a quarter of a quarter, plus a quarter of that, and so on, adds up to a third — and each of those pieces is two bisections away. So an angle can be trisected by bisecting forever, every stage exact and the shortfall shrinking to a quarter each time. The construction never ends, and that is precisely what Wantzel's proof forbids: a construction is a finite thing, and a third of a general angle is only reached in the limit.
The cosines of the powers of two
Start at cos 1 and apply x ↦ 2x² − 1 over and over. The map is chaotic and exactly solvable: the k-th point is cos(2ᵏ), the cosine of two to the k radians. Iterate it in floating point and the computed orbit leaves the true one after 51 steps. Read the binary digits of 1/(2π) instead and the true orbit can be followed for as long as digits can be computed — here 262,080 steps — and at that length it is as fair as any orbit can look. Whether it stays fair is a question about those digits, and nobody can answer it.
Two rays for every fraction
Seen from far away, every point of the Mandelbrot set's outside has an angle, and the curves of constant angle come in to land on the set. At the root of the p/q bulb exactly two of them land, and their angles are written by the staircase under a line of slope p/q. The limbs those pairs fence off share out the whole circle of angles between them, by the identity Σ φ(q)/(2^q − 1) = 1.
Named alongside it
The objects these essays reach for when they reach for this one.
Doubling mapPeriodic orbitAlmost surelyGeometric seriesInvariant measureNormal numberProbability densitySensitive dependenceAngle trisectionBernoulli convolutionCantor setCharacteristic function