[EM] Improvements in Band-Rectangle districting

[Michael Ossipoff]

I'd defined "diameter" as the maximum distance across a district, and
said that minimizing maximum or average diameter in a band's districts
should be part of the way of drawing the bottom edge of a band.

But, due to the irregular left and right edges that a band can have, I
suggest an easier approach:

Chose the vertical position for the horizontal line defining a band's
bottom edge such that:

1. The band contains a whole (integer) number of district populations,

and

2. When the band is divided into districts, by vertical lines, the
following quantity will be as close as possible to 1:

The number of bands in the district, multiplied by the vertical
dimension of the band, divided by the arithmetic mean of the lengths
of the band's top and bottom lines.

But, because the top band can have an irregular top edge, and the
bottom band can have an irregular bottom edge, then, for those two
bands, I'd replace "vertical dimension" with
"arithmetic mean of the band's greatest vertical dimension and its
least vertical dimension.

Of course, for brevity and simplicity, that 2nd version of #2 should
just replace the 1st version, as the general rule.

Someone could object that the band's left and right edges will
sometimes be irregular too, and so wouldn't it be better to average
the band's greatest and least widths? Maybe better, but not necessary.
It's simpler and easier to specify and measure the band's top and
bottom lines, and doing so won't have a significant,
consistently-directed, effect on the bands' districts' diameters.

By the way, the web articles that I've looked at seem obsessed with
"compactness". There apparently are many different definitions of it.
But it isn't necessary to strictly maximize it. For one thing, the
important thing is that the district diameters aren't too
unnecessarily great, because that would make campaign travel
unnecessarily difficult. But it isn't necessary to strictly minimize
the greatest district diameter in the state, or the sum of the
district diameters. It's enough to use a rule that makes a reasonable
effort to keep the diameters from getting too great. Other than that,
a simple way of achieving an automatic rule, with no human input, is
the important thing.

Band-Rectangle's districts tend, overall, to have vertical and
horizontal dimensions that don't differ by an unreasonable factor,
resulting in generally, overall, reasonable district diameters.

Mike Ossipoff