The same thing twice — page 18
Dynamics
A chaotic map that always comes back
Arnold's cat map stretches a square by the golden ratio squared and wraps it round, and any two nearby points fly apart: a difference of a million-millionth fills the square in twenty-seven steps. Apply the same map to a picture made of pixels and it dissolves into noise within a few steps — and then, after exactly thirty steps on a 61 × 61 grid, every pixel is back where it started. Chaos and exact return are both true, and the Fibonacci numbers decide how long the return takes.
Geometry
Where charges settle on a sphere
Put equal electric charges on a sphere and let them push each other apart until nothing moves. Four settle at the corners of a tetrahedron, six of an octahedron, twelve of an icosahedron — but eight do not make a cube, and twenty do not make a dodecahedron. The cube loses to two squares turned half a corner apart, because charges on a sphere prefer triangles; and however many charges there are, exactly twelve more of them have five neighbours than have seven.
Number
The largest prime in a typical number
Count a number's prime factors and the answer is about log log n. Ask instead how big the largest one is, and the answer is a power of n with a law of its own: half of all large numbers have a prime factor above n^0.6065, and the share whose primes all stay below n^(1/u) is Dickman's ρ(u), a function defined by its own past. The same law governs the longest cycle of a random shuffle, and there it arrives at once; for the numbers themselves, at ten million, it is still visibly on its way.