The same thing twice — page 15
Analysis
Three sines tie a knot
Let a point move with a different sine wave in each of the three directions of space, the three frequencies whole numbers with no common factor, and it traces a closed curve that is usually knotted. With frequencies 2, 3 and 7 the curve is the knot 5₂. The trefoil, the simplest knot of all, can never be made this way: the half-turn that shifting time by half a period performs forces a condition on the knot's polynomial that the trefoil fails.
Dynamics
How smooth the disguise is
An unlocked circle map is a rigid rotation in different coordinates, and whether those coordinates are smooth is decided by arithmetic. Solving for the change of coordinates means dividing, harmonic by harmonic, by numbers that come arbitrarily close to nought — and the golden rotation number keeps them far enough away that the result is not merely smooth but analytic, while a number lying close to a fraction makes it buckle.
Probability
Two constants that do not walk at random
Pollard's factoring method trusts x² + c modulo a prime to repeat as soon as a random function would, after about √p steps. For c = 1 and c = 3 it does. For c = 0 and c = −2 it runs twelve to eighteen times longer on average, with almost no tail and enormous cycles, because those two maps are multiplication in disguise and their cycles are set by the order of 2 rather than by chance. Every other constant walks at random — including in the one respect in which x² + c is plainly not random, that it is two-to-one.