Finally! (Hopefully) Geodetic Linear Orthogonal Redistricting (GLOR) Perfected

~~~~~~~~~~ Pt. 1: The Geographic Coordinate System ~~~~~~~~~~

This is the third thread in which I have tried to perfect this type of algorithmic redistricting. There are countless blind alleys with this. One of the biggest challenges is that the earth is – aside from several nuances – a sphere. It would be pleasant if we could completely ignore this, and just assume that it was flat – or ‘planar’ – but we cannot. Surprisingly, the spherical nature of the earth has no significant effect upon this new Geodetic Linear Orthogonal Redistricting (GLOR). Yet we must be familiar with the Geographic Coordinate System in order to perceive that, so I will begin with a description of that first. I promise that this Geodetic Linear Orthogonal Redistricting (GLOR) will finally solve the algorithmic redistricting problem extremely beautifully. I will finish my description of that solution in Pt. 2 as soon as I finish fixing a few of my friend’s computers. In about two days, I think. It will work simply and almost perfectly.

It would be easier if the world was flat. But if it was perfectly flat we would not be able to sit up in the morning. Of course, if it was perfectly spherical, we we still would not be able to sit up. We will need to discuss a small bit of spherical geometry, but we should not aim for the sort of perfect descriptions that are typical in physics or pure mathematics, since we must account for many practical considerations. I once worked as a so-called ‘frequency coordinator’ designing line-of-sight microwave repeater paths across North America, and if a microwave signal paths needed to pass through mountainous terrain where there had occurred forest fires, I had to account for 20 year’s growth of trees that could eventually present ‘blockage’ to the signals. The people who launch satellites on rockets have to deal with much more complex aspects, but at least none involving tree growth. The geometry we need here, based on the Geographic Coordinate System, is extremely simple, but there is still the significant challenge of its need to be comprehensible to the voting public. So I will use a few simplified terms. And there is no need at all to be overprecise about all aspects. For example, the earth bulges somewhat at the equator because of centrifugal force caused by its rotation, but it’s very tiny and irrelevant here.

Let us begin with a few ‘bowdlerized’ geometry terms and concepts. The earth’s geometry can be defined by its ‘axis’, a straight line which it spins around, causing night and day, and its diameter – or radius if you prefer: -// d = 2*r //-. This line intersects with one point on the the spherical earth that we term the true north pole, and also an opposite point that we term the south pole. Now for my description of the ‘great circle’ idea. A great circle is just a circle which has the same diameter as the earth, and also whose center is located at the earths center. The equator is an example of this. We now must define ‘longitudinal great circles’ to have the special property of intersecting with both the north and south poles. There could exist an infinite number of these, and if we have enough of them they could, for all practical purposes, be utilized to ‘trace out’ the shape of the earth’s surface. They do resemble the outer features of the slices of oranges and lemons. I will just call them ‘longitudinal lines’ – or ‘lon-lines’. I actually think of all lon-lines as being parallel, even though the always meet at the north and south poles, but here I will just call them ‘para-angular’. I recall there was once a strange argument among mathematicians about whether parallel lines ever meet, but we don’t need to worry about that.

There exists an arbitrary, albeit highly conventionalized, very special lon-line that, naturally, intersects the north and south poles, but also the Royal Observatory, in Greenwich, London, England, and is dubbed the ‘prime meridian’, and it is considered to be located a 0 degrees in the ‘360 degree system’. All of the ‘angles of rotation’ ('ang-rot’s) of other lat-lines – east-west about the axis – are measured in relation to this prime meridian. And please remember that every lon-line is para-angular to every other. I should mention here that the east-west ends of all political districts will be comprised solely of arc-portions of various lat-lines. Note that north-south lon-lines can also be measured in terms of the angles of rotation (ang-rots) concept, since the earth’s rotation is not relevant to this mathematical concept.

The center of the earth lies on the axis, half-way between the north and south poles. If we draw any straight line from this center that is perpendicular to the axis, it will intersect with the equator, and with this, we can define the equator, which is a great circle, and also a ‘line of latitude’ or ‘lat-line’. Every lon-line intersects with the equator. Now, let’s begin at the point where the prime meridian lon-line intersects with the equator lat-line, and draw an arc whose vertex is earth’s center, with one side intersecting with the the point of intersection of the equator and the prime meridian, and the other rotated northward along the prime meridian by 10 arcdegrees. Then let’s draw this same angle northward from the equator, whose vertex is the earth’s center, for many other of the lines of longitude around the equator. We will then be able to construct a smaller lat-line circle that intersects with every point formed by the 10 degree northward rotation along all of the longitudinal lines. This will be a line of latitude. And we can also construct an even smaller one 20 degs. north of the equator, until we reach 90 degs., at which point the diameter of our line of latitude will be zero, and it will have reached the north pole. Finally, they can be subdivided into arc-portions of 10, 20, and so degs. from say, west to east. When they reach 360 degs. they will be back at Greenwich again. Notice that these lines of latitude never intersect with each other, and that other than the equator, they are always smaller than great circles. And all these lon-lines and lat-lines tend to form `NS by `EW sections that appear different at the poles than they do at the equator. I should mention here that the north-south ends of all political districts will be comprised solely of arc-portions of various such lat-lines.

I read once that the ancients knew about spherical trigonometry, and that some of the sources of the Piri-Reis map, which were either ancient or extremely ancient, and in any case more ancient than the modern “discovery” of the science, were perhaps drawn using it and were discovered in the library of Constantinople in 1513.*

How this relates to solving gerrymandering I can’t say, but I do believe the crux point might be placed in the measuring of efficiency gap when there are more than 2 parties, rather than in the exact delineation of the borders, topography and baseline curvature notwithstanding.

This post has piqued my interest and I’m going to circle back to read it more closely when it’s not so late. Mapping is the best!

*I am not a cartographer or a cartographic historian, but I hope to be one some day, when I’m old, but only if that still involves watercolors at that point in the future.

In reply to the @Sara_Wolf comment – Geodesy, in spite of its broad usefulness and great antiquity, is a rather hoary old subject loaded with odd terms and concepts developed by ancient spherical cartographers. There’s an odd aspect to it in that ‘distance’ can be measured in ‘angles of rotation’, i.e. ‘angular distance’ – the angle formed by a vertex at earth’s center by two lines drawn to two points on the surface, and can also be measured in terms of ‘arc length’ – the length of shortest arc upon the surface. And these two measurements are directly proportional. At some point in a the far past I began studying spherical geometry, but I lacked the requisite masochism to go very far into it.

~~~~~~~~~~ Pt. 2: Geodetic Linear Orthogonal Redistricting (GLOR) Described ~~~~~~~~~~

Let us begin with some of the most easy-to-envision examples. These will sit north of the equator – in the northern hemisphere – and will be located right next to the equator. Here, the `NS by `EW sections will closely resemble squares, although they are actually three dimensional constructions. It’s worth noting that if they had been located at the north or south poles they would resemble triangles, but let us just consider sections located next to the equator for now. When contemplating constructions within this region, there is relatively little significant harm in treating small regions as truly ‘flat’ or planar, so let us examine some basic ideas with that in mind. Later, we will have a means to deal with the fact that while lat-lines are always equidistant, lon-lines are relatively far apart at the equator, but they always meet at the poles.

Suppose there is an island nation with 59049 inhabitants – the fact that 59049 equals 3^10 will keep everything neat and simple for now. It has a council of 9 delegates, each of whom represents a district comprised of 6561 inhabitants. -// 9*6561 = 59049 //-. What we want is an algorithmic method of dividing the island – the ‘region’ – into partitions – which will come to be called districts – into areas that each contain 6561 inhabitants. Furthermore, these partitions will have four sides, with the east and west sides being in a sense parallel with the lines of longitude – or ‘lon-lines’ – even though the always meet at the north and south poles – so let us call call them ‘para-angular’, although we should really think of them in terms of arc length along the surface. And the north and south sides of these partitions will be parallel with ‘lines of latitude’ or ‘lat-lines’. So these partitions – if near the equator – appear to be square or rectangular, even though they are actually three dimensional.

It seems somehow ‘organic’ to describe some methods that will not work before describing what will. There is no complicated math involved, but making things simple can be very daunting at times. Since we are near the equator we can initially pretend that the earth is flat. Cadastral land surveyors mapping out small plots of real estate do this routinely.

Importantly, for every method described in this entire discussion, we will define the regions to be partitioned by constructing a ‘rectangle’ comprised of the most eastern and western lon-lines, and the most northern and southern lat-lines that ‘frame’ the geographic region being discussed. And we will consider this ‘rectangle’ to be the ‘region’ itself.

For the first unworkable method, suppose we simply partition the region into nine identical sections, so it will somewhat resemble a tic-tac-toe board. This is a big fail since obviously some sections could contain 20,000 inhabitants while others could contain no inhabitants at all. Another poor method would partition regions into nine sections defined only by lat-lines, adjusting the width of the lat-lines so as to have 6561 in each section. The obvious problem here is that the sections would tend to be relatively long and narrow east-west bands. We could apply essentially the same method by partitioning with lon-lines, creating relatively long and narrow north-south bands.

Importantly, the latter two methods are deficient in terms of ‘regionality’. We should strive to have the sections – which will come to be called political districts – be as close to being square as possible so as to maintain a ‘sense of community’. Long narrow sections do not fulfill this goal – they lack regionality.

There are two rather complementary methods that will actually work. Remember we need nine sections, each containing 6561 inhabitants. Using the first-described method, we partition the region using lon-lines into three long narrow sections, each containing 19683 inhabitants. Importantly, these sections will be deemed ‘primary partitions’, and this procedure will be termed a ‘primary partitioning’.

We cannot now simply complete this project by simply drawing straight lat-lines over the just-above lon-line partitioning, since that would immediately give rise to the ‘tic-tac-toe board’ conundrum of having unequally inhabited sections.

However, we can treat each primary partition individually; we can simply divide each primary partition of 19683 inhabitants individually, using lat-line segments, into three ‘secondary partitions’, each containing 6561 inhabitants. It is important to remember that the primary and secondary partitions are of different natures.

There is still a potential issue to deal with. People will question why the primary partitions here are lon-lines while the secondary ones are lat-lines. We could just as easily use lat-lines to draw the primary partitions and lon-lies to draw the secondary partitions. This could constitute an issue in some regions – for example, in the USA the State of Connecticut has most of its population near its southern shore, while the State of Massachusetts has more people near its eastern shore. I think the simplest solution is to use ‘alternating’ lon-line primary partitions for, say, a first election, followed by lat-line primary partitions for the next election, and so on. Obviously, each inhabitant would reside simultaneously in two distinct sections – in other words, districts – at the same time. Obviously, candidates must choose just one district in which to contend.

Its important to note that all partitionings must be based upon square numbers; 1, 4, 9, 16, 25, and so on. Of course, relatively few regions will be entitled to a number of districts that happens to be a square number. This can be dealt with by inserting – or removing – a reasonable number of secondary partitions within the primary ones. For example, a ‘1’ based partitioning could be split into two parts, and one of those parts could be treated as a secondary partition and split into two sections, yielding three equally populated sections. Beyond that, we can 'leap to a ‘4’ based partitioning into which we can insert 4 more secondary partitions for a total of 8. Then we can leap to ‘9’, and so on. I have concluded that, in the interest of maintaining regionality, the ratio of the number of secondary partitions within any primary one to the total number of primary partitions should never exceed 2 to 1, and this rule is always maintainable. The question arises as to which primary should a new secondary be inserted. I would always choose the primary having the smallest arc portion, since this will cause its arc portion to ‘widen’ due to its having to contain a larger number of inhabitants.

Why must all partitionings must be based upon square numbers? Why not start with, say, 12? That would force us to begin with a 3 by 4 rectangle, and it would create a brand new north-south vs. east-west asymmetry. One of those orientations would get 3 and the other one would get 4, and we would have a whole new contest, very much like the one we needed to resolve for the primary vs. secondary partitioning issue.

There do exist various possible rules pertaining to the determining the best point at which to make the leap from one square number based partitioning to the next. One consideration is that secondary partitions can be removed as well as inserted into primary ones. And we could always wait until the primary to secondary partition 2 to 1 ratio rule is about to be exceeded. After a lot of trial and error I have decided that the most reasonable and explicable rule is to simply make the leap at the point when the next larger square number based partitioning is reached.

We do not have to worry about the fact that the lat-line arc lengths between the lon-lines diminish as we approach the north and south poles. This is because all of the relevant rules that could create issues are based upon population totals, and not upon geographic areas.

I do hope this exposition was reasonably comprehensible and comprehensive.