Counting the same thing twice — page 12
Geometry
The shape of a cell around a random point
Scatter points at random and give each the region nearer to it than to any other. The cells come in every shape from triangles to twelve-sided polygons, and almost nothing about them is fixed by a formula — except one thing: on a closed surface they have exactly six sides on average, every time, because three cells meet at every corner. Two hundred thousand cells measure the rest: 29.5 per cent hexagons, Lewis's law, and small cells surrounded by large ones.
Number
Real fields that factorise uniquely
Among the fields Q(√−d) only nine have unique factorisation, and Gauss guessed as much. Among the real fields Q(√p) he guessed the opposite — infinitely many — and the guess is still unproved. Counting cycles of reduced quadratic forms for the 16,900 primes p ≡ 1 mod 4 below 400,000 finds 79 per cent with class number one, drifting slowly down towards the 75.45 per cent that Cohen and Lenstra's heuristic predicts — and finds the reason the real fields behave so differently: their units are enormous.