~~~~~~~~~~ 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.