I was thinking about telephone area code splits. How do they decide where to draw the line, if there is no ‘natural’ boundary (mountains, water, a highway, a county line)? How would I do it?
I’d start with a list of the smallest possible units (branch exchanges) and a matrix showing the number of calls between each unit and each other unit during the past year. Find the largest entry in the matrix, and merge those two units, unless the resulting unit would be more than 3/5 of the total. Repeat until no merger is possible.
Why 3/5? Perfect halves are too good to hope for (if a ‘natural’ division is desired), but the exercise has been somewhat wasted if one of the ‘halves’ is 2/3 of the total. (A limit less than 2/3 can result in a three-way split. If the territory has three natural cores, I wouldn’t want to split one of them in half.)
3/5 is the simplest rational between 1/2 and 2/3 – but is it the best criterion? Maybe the golden ratio is somehow ideal, or some other number that I haven’t thought of. What ratio presents itself as special?
I chose to look at the cases where three units remain and merging the two smaller does not affect the ratio between the biggest and the smallest. Let the sizes of the units be 1, x, y (1 ≤ x ≤ y). The criterion is met when (1+x)/y = y/1. This defines a segment of a parabola, bounded by the lines 1=x and x=y; the endpoints are (1, √2) and (τ, τ) where τ = (1+√5)/2, the golden ratio.
If there are two units and the ratio of their sizes is 1:y, then the greater must be y/(y+1) of the whole. Thus y = √2 gives 2−√2 = 0.585786437627 . . . = {1, 1, 2, 2, 2, . . . } and y = τ gives τ−1 = 0.61803398875 . . . = {1, 1, 1, 1, 1, . . . }. (The brace notation here is shorthand for continued fractions.) The simplest number between them is –guess what?– {1, 1, 2} = 3/5.
Rather than a fixed maximum size, might the limit be that a merger must not increase the ratio between the largest and smallest units? That rule would forbid any merger if all units start at about the same size, which is clearly wrong; but such a rule may come into effect when the largest unit reaches 1/3 of the total.