Convergence rate
Named by 25 essays across 8 fields — each of them below, with the objects they name alongside it.
Getting pi by dropping needles on the floor
Throw a needle at a lined floor enough times, count how often it crosses a line, and pi falls out. There is no circle anywhere in the experiment.
How long until every one turns up
Draw at random from six equally likely kinds until all six have appeared. The wait is not six draws, and it is not sixty; it is fourteen point seven, and the number is a harmonic sum wearing a hat.
The sum that fits in one square
Half, then a quarter, then an eighth, forever. Adding infinitely many things sounds like it should give infinity, and the picture that says otherwise is a square with a corner left uncut.
How far from the average a thing can be
Knowing only an average and a spread — nothing about the shape, nothing about the number of outcomes, nothing about symmetry — the chance of landing three standard deviations out is at most one in nine. And there is a distribution that lands there exactly that often, so the bound cannot be improved.
The average settles and the wobble does not
Two theorems are usually met a page apart and sound as though one is a sharper version of the other. They are the same sums looked at through two different magnifying glasses: divide by the number of them and everything collapses to a point, divide by its square root and a shape appears.
A map that shrinks everything
One extra hypothesis — that every distance is shortened by at least a fixed factor — turns the existence of a fixed point into its uniqueness, an algorithm for finding it, and a bound on the error after any number of steps.
Counting what has no formula
There is no expression that gives the nth prime, and yet the number of primes below a bound is predictable to within a fraction of a per cent — by a function that is not a formula for the primes but an integral of the wrong-looking quantity.
How fast the bell arrives
The limit theorem says a standardised sum approaches the bell curve and says nothing about when. The rate is one over the square root of the number of terms, the constant in front is made of the third moment, and both are visible.
![The same map, written in the basis of its own eigenvectors. Three panels: the map [2, 1, 1, 2] on the standard grid, the diagonal stretch by 3 and 1 it becomes on the eigenvector grid, and the two put back together.](/f/the-same-map-written-in-the-basis-of-its-own-eigenvectors-fd61d7c3.svg)
The same map in a better basis
Measured along its own invariant directions, a linear map stops shearing and becomes two independent stretches. Nothing about the map has changed; the grid it is described against has.
The error that does not care how many dimensions
A grid gets rapidly better in one dimension and hopelessly worse in twenty. Random points get better at the same slow rate whatever the dimension, which is why a method that is bad everywhere ends up being the only one that works.
Sampling where the answer lives
Monte Carlo error cannot be made to fall faster than the square root, so the only thing left to attack is the constant in front of it. Drawing points where the integrand is large, and dividing by how often they were drawn, leaves the answer alone and can shrink the noise many times over.
Points too even to be random
Independent random points clump, and the clumping is what makes the error fall only as the square root. Points chosen to be evenly spread rather than independently beat that rate, and the price is that nothing about them is random at all.
A tail too small to be a whole number
If e were a fraction with denominator q, then q! times e would be a whole number. It splits into a whole part and a tail, the tail is squeezed strictly between nothing and one, and there is no whole number there.
How fast the staircase arrives
The slope at a crossing decides whether an orbit reaches it. The same number decides how fast — and when the slope is zero the arithmetic changes kind, from a fixed factor per step to a doubling of the correct digits.
Too many orders to list
The rule is an average over every order the players could have arrived in. At seven players that is five thousand orders and at twenty it is more than there are seconds in the age of the universe — so the average is sampled, and the error falls at a rate that can be measured.
No single input can move it far
Independence was never the hypothesis doing the work. A quantity built from many separately drawn inputs concentrates whenever changing one of them moves it only a little — and that covers quantities which are not sums of anything and have no formula at all.
The series everything else is measured against
A geometric series is not one series among many. It is the yardstick: a total exists if its terms eventually fit under one, so a single comparison settles infinitely many questions — and the test built from it says nothing at all in exactly the place where the interesting cases are.
The repair at the boundary
Where the geometric yardstick says nothing, compare a series with itself at doubled spacing. That one move turns every 1/n^p back into a geometric series, reads the threshold off at p = 1, and then produces an infinite hierarchy of boundaries with no slowest divergent series anywhere in it.
Almost no number is one
Sums of two squares look common — a sixth of all numbers up to a million are one. The fraction is falling to nothing, at a rate so slow that no computation will ever make it obvious, and the constant in front of it has been computed to fifty places and identified with nothing.
The one that hardly ever comes up
Make the kinds unequally likely and the tidy decomposition into stages fails, because a stage's rate now depends on which kinds turned up rather than on how many. What replaces it is an alternating sum over every subset — and the rarest kind turns out to be nearly the whole answer.
Pinned between two sequences
The ring dissection makes the answer obvious and proves nothing. Archimedes' method proves it and makes nothing obvious — it never exhibits the area at all, it rules out every other value — and the recursion that drives it computes π by hand with one square root a step.
A double root halves the error instead of squaring it
Near an ordinary root, Newton's method squares its error at every step and a handful of steps reach full precision. Near a double root it only halves the error — twenty steps where five would do, and a ceiling of about eight correct digits that no number of steps can break through. Doubling the step repairs the double root and ruins the simple one.
A cubic method that is Newton's in disguise
Halley's method, from 1694, uses the second derivative as well as the first and cubes the error at every step where Newton's squares it. It is also, exactly, Newton's method applied to a different function — p divided by the square root of p′ — and that single fact explains why its basins are calmer, why it walks out of the trap that holds Newton for ever, and why its boundaries are still fractal.
What a jump costs evenly spread points
Scrambled Sobol' points integrate a smooth function with an error that falls like N to the power −3/2. Put a jump in the integrand along a tilted line and the exponent halves, to −3/4, because about √N of the N boxes the points fill are cut by the jump and each of those is left to chance. A jump along an axis costs almost nothing, the cost grows with the length of the jump and the dimension, and integrating across the jump before sampling gives the whole rate back.
What it costs to find the point that stays put
Brouwer's theorem promises a point a map leaves where it is, and Sperner's corridor walks to it. On a gentle map the walk is short; twist the map about its fixed point and the corridor follows every turn, so its cost grows like the square of the grid. Halving the square by winding number finds the same point in a number of readings proportional to the grid's side whatever the map does — and no method that only reads the map can do fundamentally better.
Named alongside it
The objects these essays reach for when they reach for this one.
ApproximationExpectationVarianceConvergenceHarmonic seriesIterationLimitMonte CarloSamplingGeometric seriesIntegralDerivative