Geodetically Stable Linear Orthogonal Redistricting Revisited

A surprisingly small number of efforts to develop ‘algorithmic’ political redistricting schemes have been advanced, with the goal of circumventing the exceedingly dubious political redistricting designs of various governmental agencies. And even though these projects have been undertaken by some of the most illustrious theorists, there has been shown an odd reluctance to put very much serious effort into them. Well, I have volunteered to take a deep dive into the depths of this daunting, yet very significant issue. So here we go.

We begin by looking at the (mostly) spherical earth from the perspective of the familiar North and South Poles, with the (‘imaginary’) lines-of-longitude and lines-of-latitude. These form ‘square-looking’ shapes near the equator, and ‘trapezoidal-looking’ shapes near the poles. This particular phenomenon will turn out to have no effect upon the development of our redistricting method. Nonetheless, our algorithmically derived districts will consist of lines drawn parallel to the lines of latitude and longitude.

We will accept that upon the earth there exists some ‘region’ (n.b. important term) for which there will be elected some predetermined number of delegates or officials. And this region must be divided up into ‘districts’ having (very nearly equal) population quotums. And the people in each of the various districts will elect one delegate per district. So let’s create a simple prototype method.

What we want to accomplish is to divide up regions into ‘districts’ (n.b. important term) in which each district elects one delegate. To do this, we first establish the most-northern, most southern, most western, and most eastern points of a region in question. And then draw a square-looking, (or in some cases) trapezoid-looking figure between them, having lines parallel to lines-of-longitude and lines-of-latitude. The simplest case is where, by reason of having a small population, a region only deputizes just ‘one delegate’, i.e it constitutes just one single district. ‘One delegate’ is rendered in quotes here, because the minimum will turn out to be two delegates, for reasons that will presently become clear.

Suppose some region that, by reason of having a larger population, is entitled to having four delegates – or twice as many as the two that the previously mentioned region had. To enable this the region must be divided, or ‘partitioned’ (n.b. important term) into two geographical ‘parts’ (n.b. important term). This division would be a line drawn from north-to-south (_NS_), or from west-to-east (_WE_), the line positions being established so as to divide the rectangle-looking parts into geographical areas containing equal population numbers.

Here we encounter a basic problem. Should the line run _NS_, or _WE_? Some people will insist in should run _NS_, others will insist it should run _WE_. The simplest solution is to partition the region in both ways, dividing the region into four parts, two of which will always overlap. This will mean that each voter will actually ‘belong’ in two overlapping parts, which may now be accepted as ‘districts’, and they can cast one vote in each one. So each voter will cast a vote in two districts – they can choose one poling place in which to vote in both districts – although each candidate may contend for only one district. This will result in the election of four delegates. Twice as many as the region that deputized only two delegates; note that the number of elected delegates will be even in both cases. This number will always have to be even.

We have discussed regions entitled to 2 delegates, and regions entitled to 4 delegates, but some will no doubt be entitled to 6 delegates. The most obvious approach is to divide such regions into 3 parts along one ‘axis’ (say, _WE_), but to also divide them into 3 parts along the other ‘axis’, providing 6 districts and 6 delegates.

What about the more challenging case of a much larger region entitled to 46 (2*23) districts. It will require 23 _WE_ districts, and 23 _NS_ districts. The basic ‘configuration’ (or ‘tessellation pattern’) will (in all but one respect) be identical for both sets of districts, but the rectangles created, which must enclose equal numbers of people, will be of different proportions. Let’s try it. First, we must decide whether to do our _WE_ partitioning first, or to do our NS initial partitioning first – this choice is unimportant because the end result will be the same. Let’s do the _NS_ initial partitioning first. We will simply draw straight continuous lines within our rectangle from north to south – and these must divide it into (‘long’) rectangles each of which contains the same number of people – so they will not be equally spaced. Let’s continue.

We eventually need 23 rectangular districts within our regional rectangle, but making an actual rectangle out of 23 rectangular districts will be a challenge since our rectangular region must be comprised of some product of some integral number of district rectangles (or, it would surely appear). But that would require that 23 would need to be a product of two integers, which it obviously cannot be. Let us ignore the 3 inconvenient ‘extra’ districts for the moment, and notice that =/ 20 = 4*5 /=. Having 20 districts presents no problem, so let us pretend for now we have only the 20. What we probably should do is just draw 3 straight lines _NS_, to create 4 ‘long’
rectangular _NS_ spaces, each containing the same number of people. These initial 3 lines will be called ‘primary lines’ (n.b. important term). For now, we want 20 districts, so we create smaller rectangles, each containing an equal number of people, by individually dividing up the 4 _NS_ rectangles into 5 smaller rectangles, each enclosing an equal number of people. The 4 _WE_ lines that create these 5 smaller rectangles are called ‘secondary lines’ (n.b. important term). We now have 20 rectangles enclosing equal numbers of people. Note that unlike the primary lines, the secondary ones cannot be ‘straight and continuous’. The 20 rectangles now become 20 districts.

However, we really want 23 rectangles. All we do is select the 3 ‘narrowest’ of the 4 spaces created by our primary lines, and ‘shoehorn’ an extra _WE_ line into each of them, to obtain a total of 23 rectangles in our region. We will have to perform a population-proportional (increased) width adjustment on each of our 4 ‘long’ rectangular _NS_ spaces with an additional _WE_ rectangle to keep the number of people equal in all of the rectangles. The 3 (originally) ‘narrowest’ of the 4 _NS_ spaces must be ‘widened’ so as to each contain =/ (6/23)*(the total ‘population-wise’ _WE_ ‘width’ of the regional square) /=, and the remaining one must be ‘narrowed’ to =/ 5/32*(the regional ‘population-wise’ _WE_ width) /=, in order to ensure equal population in each rectangular district.

Then, we repeat this operation, except that the initial primary lines go _WE_ instead of _NS_, and we obtain 46 districts, with each voter participating in two districts. This may sound rather detailed, or even complex, but compared with other algorithmic methods it is dirt simple, and it affords vastly higher stability and general reasonableness from the perspective of the voter. In fact the average voter might be able to work it out merely by trial and error successive approximation.

In addition to ‘shoehorning’ extra districts into secondary partitionings, we can also ‘subtract’ them as well, to obtain good results. Here is a preliminary list of possible combinations, where the ‘first’ figure represents primary lines, the next represents the secondary ones, and the additional added figures allow for districts that are not integer products in number. It seems clear that there should be a rule that the number of _WE_ districts should never exceed twice the number of _NS_ ones, and vice versa, so as to maintain reasonable length/width proportions. Here is a (preliminary) list:

1*1=1, 1*2=2, 1*2+1=3, 2*2=4, 2*2+1=5,
2*3=6, 2*3+1=7, 2*4=8, 3*3=9, 3*3+1=10,
3*4-1=11, 3*4=12, 3*4+1=13, 3*5-1=14, 3*5=15,
4*4=16, 4*4+1=17, 3*6=18, 3*6+1=19, 4*5=20,
4*5+1=21, 4*5+2=22, 4*5+3=23, 4*6=24, 5*5=25,
5*5+1=26, 5*5+2=27, 4*7=28, 4*7+1=29, 5*6=30,
5*6+1=31, 5*6+2=32, 5*7-2=33, 5*7-1=34, 5*7=35,
5*8-4=36, 5*8-3=37, 5*8-4=38, 5*8-5=39, 5*8=40,
5*8+1=41, 5*9-3=42, 5*9-2=43, 5*9-1=44, 5*9=45,
5*9+1=46, 5*10-3=47, 5*10-4=48, 5*10-5=49, 6*8+2=50
[…]

Because districting is actually just a special form of proportional representation, based upon social proximity it can promote good political justice. And it need not depend (at least heavily) upon the participation of typically corrupt parties or election managers and individual ballot ‘editing’.

What are you minimizing?

Variance?

Boundary length?

Variiance of what? Boundary lengths of what? Hyper-reticence to the max, here!

I am not preoccupied with ‘minimizing’ things. I am only proposing a very good method of providing high quality reapportionment.