Small cases lie — page 7
Number
A cube root that looks like chance
The continued fraction of √61 repeats after eleven terms, and every square root's must. The cube root of 2 obeys no such law. Computed exactly, by a method of Lagrange's that never leaves the whole numbers, its terms run 1, 3, 1, 5, 1, 1, 4 … and throw up a 534 at the thirty-fifth place and a 7,451 at the 571st — and in every statistic anyone has measured, they look like the terms of a number picked at random.
Topology
Six sticks tie a trefoil, and five cannot
Build a knot from straight sticks joined end to end and ask for the fewest. A trefoil takes six, and the six corners can be whole-number points in a box ten units wide. Five sticks can cross one another five times in a picture, as often as a cinquefoil needs, and still tie nothing — the five crossings always twist three one way and two the other. The fewest sticks is a measure of how knotted a knot is that no diagram shows directly.
Topology
Almost every long loop is knotted
Close a random walk into a loop and ask whether it is knotted. With ten steps almost never; with a hundred, more than one time in five it can be proved knotted by a single number; with two hundred and fifty, more than half. The chance of staying unknotted falls exponentially with length — Frisch, Wasserman and Delbrück guessed it for polymer rings around 1961, and it was proved in 1988 — because a knot needs only one small tangle somewhere, and a long loop has room for many.
Computation
The fewest moves to draw it
Every construction with ruler and compass is a sequence of moves — draw this line, draw that circle — and it is natural to ask for the shortest. Nobody can answer that by cleverness alone, but a machine can answer it by trying everything: the midpoint of a segment takes four moves and no fewer, the square on it five, a third of it five. Take the ruler away and the compass pays for it: six circles for the midpoint, seven for the square.
Discrete
A labelling every tree seems to have
Number the points of a tree 0 to n − 1 and write on each edge the difference of the numbers at its ends. The labelling is graceful if the edges then carry 1 to n − 1, each exactly once. Every tree anyone has ever checked — every one of the 551 trees on twelve points, and every tree up to about thirty-five — has such a labelling, and no one knows why. A graceful tree also tiles a complete graph by rotation, which is why the question was asked.
Dynamics
Two numbers in one jagged record
A measured record comes with no rule, so its dimension has to be estimated from a finite stretch of samples — and two things go wrong that never trouble a graph built from a formula. The popular estimator depends on the units the record is written in, and the dimension, which describes the record up close, turns out to be independent of its memory, which describes it from far away. For a self-affine path the two are tied by D = 2 − H; for a record, they are two numbers.
Geometry
Five where four were promised
In one unknown, a polynomial with three terms has at most two positive roots, whatever its degree. The natural guess for two equations in two unknowns, each with three terms, was two times two: four. It is wrong. Bertrand Haas found two such equations in 2002 whose curves cross five times in the positive quadrant, all five crowded within a tenth of one corner, and five turned out to be the most there can ever be.
Number
Euler's sixty-five convenient numbers
For some n, whether a prime can be written as x² + ny² is settled by its remainder on division by 4n alone, the way a prime's remainder on division by four settles whether it is a sum of two squares. Euler found sixty-five such n, from 1 to 1,848, called them convenient, and used the largest to prove that 18,518,809 is prime. Every one of them is a number whose class group has no element of order more than two — and whether the list is complete is still not known.
Number
Perfect but for one factor
In 1638 Descartes wrote to Mersenne with an odd number whose divisors add up to exactly twice the number — provided one of its factors, 22021, is counted as a prime. It is not; it is 19² × 61. Nearly four centuries later that number is still the only odd one of its kind known, and a search of every odd number below ten million finds no other. It is the closest anyone has come to an odd perfect number, and what it shows is how an odd perfect number would have to be built.
Analysis
A series that waits on π
Add 1/(n³ sin² n) for n = 1, 2, 3, … and the terms are mostly tiny, except where n is almost a multiple of π and sin n is almost nought. Ten million terms add to 30.3145, four-fifths of it from the single term at n = 355. Whether the sum is finite depends on how closely fractions can approach π — on a number called its irrationality measure — and the best proof available says only that the measure is below 7.1, where the series needs it below 2.5.
Algebra
The shape a random ball grows into
Give every road of the square grid a random travel time and ask what can be reached from one point in time t. The region is ragged, and rescaled it converges to a fixed convex shape — but which shape is unknown for every natural law. Computing it shows a curve within a few per cent of a circle for continuous travel times, a flat side where fast roads percolate along a diagonal, a time per step that is still drifting at a hundred and twenty-eight steps, and fluctuations that grow like the distance to the power one third rather than one half.
Discrete
The longest wait for a prime
Near a number x the primes are on average log x apart, and the longest stretch without one below x is much longer: about the square of log x, if the primes behave like random numbers with the right density. Sieving to thirty million finds twenty-four record gaps, all climbing in the shape of that square and all below it. A random model of the primes gets the scale right and the details wrong, and the correction for what it gets wrong predicts gaps larger still — larger than any anyone has found.
Dynamics
A growth rate no step contains
Start with 1, 1 and at every step either add the last two numbers or subtract them, choosing by a coin. The sequence wanders, shrinks, even returns near zero — and grows, almost surely, like 1.13198824 to the power of the step. That number is a Lyapunov exponent of a product of random matrices, and it cannot be found by averaging anything about a single step: the two steps do not commute, and the rate depends on which way the vector is pointing when each one is applied.
Dynamics
Fractions repeat and roots look random
Almost every number between 0 and 1 has base-φ digits with the frequencies Parry's measure predicts. Which particular numbers do? Every fraction, provably, does not: its expansion repeats from the first digit, with a period equal to the period of the Fibonacci numbers modulo its denominator. And √2 − 1 and 1/π, computed exactly to twelve thousand digits, match every predicted frequency to within sampling error — which proves nothing about them at all.
Geometry
How many pairs can be one apart
Place n points in the plane and count the pairs exactly one unit apart. A square grid with its spacing as the unit gives two pairs per point; the same grid with the unit chosen as a length it realises in many directions — the square root of 5, of 25, of 65, of 325 — gives three, five, eight. Counting points on circles caps the total at a constant times n to the four-thirds. Between that cap and the grids the truth has not moved since 1984, and at every size anyone can draw, the two look alike.
Number
Class groups drawn at random
The class number of an imaginary quadratic field measures how badly unique factorisation fails there, and it looks arbitrary: 1, 2, 4, 3, 5, 12, 1. Count how often 3 divides it and the answer is not one time in three but noticeably more — heading, by a conjecture of Cohen and Lenstra, for 44 per cent. The reason is that class groups behave like finite groups chosen at random with each group weighted by one over its number of symmetries, so groups with few symmetries, like the cyclic group of order 3, turn up more often than the naive count suggests.
Dynamics
The word a line spells in a cube
A ball bouncing in a cube hits three kinds of wall, and the order spells a word in three letters. On a square table the word had n + 1 different blocks of length n; in the cube it has $n^2 + n + 1$ — three, seven, thirteen, twenty-one — for a direction in general position. What general position has to exclude turns out to be more than any rational relation among the three speeds.
Geometry
Three sides and four are proved
Among all shapes of a given area, the disc has the smallest lowest eigenvalue. Among triangles it is the equilateral one, among quadrilaterals the square — both proved by sliding chords to an axis. For five sides the regular pentagon wins every computation, every nudge raises its value by the square of the nudge, and there is still no proof.
Geometry
The trisectors on a sphere almost agree
Draw Morley's construction on a sphere and the triangle the trisectors cut out is not equilateral — but its sides agree to within a tenth of a percent even when the angles add to 222°. Every step of the rotation proof survives on the sphere except one, and that one is the angle sum.
Number
How many primes a typical number has
A typical number near N has about log log N different prime factors, and the count is spread around that in a bell curve whose variance is log log N as well. Both facts are theorems. Neither is visible at any size anyone can count: up to ten million the average is right and the spread is less than half what the limit says.
Dynamics
The column rule 30 will not explain
Start rule 30 from one live cell and read down the middle. The column has passed every test of randomness tried on it, and three plain questions about it are open: whether it ever repeats, whether its 1s make up half of it, and whether its n-th cell can be had without n steps of work. What can be measured says something about each — and the diagonals beside it are periodic, with periods that double going inwards.
Number
Is the partition count even half the time?
The number of partitions of n is even for 50.0% of the n up to half a million, its runs of one parity are as long as a coin's, and nothing proves that the share is a half — the best theorems only show there are at least about √n of each. Modulo 5 and 7 the zeros carry Ramanujan's congruences and something more: an excess that follows whether 1 − 24n is a square.
Analysis
The best nodes have no formula
Interpolating through n points magnifies any error in the data by at most the Lebesgue constant of the points. Chebyshev's points keep it near (2/π)·log n; stretching them to the ends of the interval brings it within two hundredths of the best possible; and the best possible points, characterised in 1978 by having every bump of the error curve the same height, have never been given a formula.
Computation
A mate is rare until order ten
Euler asked whether a Latin square can have an orthogonal mate, and for every order but two and six the answer is yes for some square. For a square chosen at random the answer is different. Exactly 6 of the 56 reduced squares of order five have a mate and none of order six does; of squares drawn at random, about one in a hundred of order seven has one, one in three hundred of order eight, one in seventy of order nine — and three in five of order ten.