Redistricting With 'Quadripoints'

It is obvious to me that redistricting should not be performed by ‘commissions’ (obvious nests of corruption!). So some algorithmic (mathematically precise) method is called for. Here’s one:

We are clearly reconciling geography with population, so that’s a bit tricky. For a large enough area we would use ‘sailor’s math’ (measuring distance in terms of angles on the earth’s surface to account for earth curvature), but the basic principles will still apply. First we frame the region to be partitioned within a rectangle. Now. We can draw a line East to West which neatly divides the North and the South into regions of equal population. Or alternatively we could draw one from North to South which neatly divides the East and West into equal regions of equal population. If we 'overlay these two lines we will have four rectangles. However for example, the the NE and the SW rectangles could have huge populations, while the NW and the SE rectangles could have little or even no population.

But we could draw 15 East to West lines that divide the North and South into 16 equal population regions. An do just the same with 15 lines from North to South to produce 16 equal East to West population regions. Wherever these perpendicular lines intersect there will exist a point we will call a ‘quadripoint’. (And we will then have 256 rectangles.)

(Just to get technical, the number of quadripoints (Pq) will correspond to the number of rectangles (Rn) according to: =/ Pq = (sqrt(Rn) - 1)^2 /=.) So we will have 225 quadripoints.

Now. Each of these 225 quadripoints will form the point of intersection of four (northeast (NE), southeast (SE), southwest (SW), and northwest (NW)) rectangles, and without too much difficulty we can find that particular quadripoint whose four rectangles are the most equal in terms of population.

Then we can repeat this process (perhaps with 4 rather than 16 regions) with each of our four new essentially equal-population rectangles, and then have 16 essentially population-equal rectangles. And these would be our districts.

Are you familiar with the Shortest Splitline Algorithm? It has a similar purpose.
https://rangevoting.org/GerryExamples.html

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The name and the description of the “Shortest Splitline Algorithm” are misleading. When there is an uneven population density, then the shortest line to cut an area into districts is not necessarily a straight line, but can also be a curve. Compactness redistricting usually leads to a shorter total splitline than the so-called “Shortest Splitline Algorithm”.

Perhaps it is ‘just me’, but frankly I do not understand what you are talking about. Warren D. Smith has done great work analyzing the ‘shortest splitline algorithm’ method of redistricting, which I believe could work, despite the ‘issues’ you appear to be announcing (but not, it seems, explaining). One objection I have to it is that the districts that it would produce are simply incredibly odd-looking. This is part of why I am suggesting “redistricting with ‘quadripoints’,” which produces neat-looking rectangular districts.

How can a shortest line be a curve on an essentially planar surface?

Well, this is what districts look like now. (Image source: http://www.theseventhstate.com/?p=2387 )


Can’t get much more odd-looking than that.

For an example of a case where curved boundaries minimize perimeter, consider a state with a circularly shaped city with a high population density with small area, surrounded by a region of low population density, but large area. The regions have equal total population. The state has 2 districts. The district boundary that minimizes perimeter is likely to be the city limits.

The deviousness of how politicians have divvied up states and geographical political units is certainly mind-blowing.

But do note that with the shortest splitline method, there could actually be ‘thousands’ of border dwellers with tiny protuberances of land, each claiming that their little nub ought rightfully be declared to be the end point of a splitline. But with the quadripoint method, only four points, the northernmost, southernmost, easternmost and westernmost points need to be established.