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.