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**The real arc of the covenant?**

The diagram to be found at http://theometry.org/arcof_thecov.gif is my attempt at a generalized representation of a transaction taking place as seen over the course of the trans-action. Each straight line in the x-y plane represents how far the transaction has gone (1-x) and what is left to go (x), and how far the transaction has come (y) and what remains for the transaction to become completely finished (1-y). For example the transaction could be exchanging ten camels (1x) for one hundred sheep (1y), or changing one gallon of water (1x) into one quart of wine (1y), or transitioning from moving due east (1x) to moving due north (1y). Note I think you know what I mean, but you could certainly help me by advising me how to make what I am saying mathematically correct. As I am sure it is not.

But what I am sure is that as one draws straight lines from "20% of x gone and 80% of x to go" to "20% of y has arrived and "80% of y to come" over the course of this transaction, a hyperbolic parabola curve (which of course is both literally and figuratively the course of this transaction) emerges. And what also emerges for one's viewing if one is open to it is a "plain" within the x-y plane -- a "plain" with some very weird properties.

For example, if one "lies" at x = 1 and y = 0, and looks into the x-y plane, it is as if one is lying flat on the ground and looking across a flat plain. Whereas if one is "standing" at x=0 and y = 1, it is as if one is wholly looking down on this plain.

While it is obviously correct to explain away this "plain" with its straight diagonals composed of perpendicular line segments (as seen from various points on the horizon of and within this plain) as just an optical illusion, that kind of explanation strikes me a little of being like saying that it's no big deal that an apple falls to the ground. If for no other reason than because the curve of the "horizon" of the "plain" in this diagram looks a lot to me like the path of a heavenly body which is passing by another heavenly body as it goes from being free of (i.e., to coming under) the influence of the other's gravity, and then going away from being under the influence of the other's gravity to becoming free of it again.

Of course, I am likely to be making a mountain out of a mole-hill about this diagram and its curve and its plain, and its diagonal straight lines all of which lie in the x-y plane. OTOH, Isaac Asimov said that when an experiment produces a "funny" result it might be significant. And as the construction of such a curve and such a plain from such a simple algorithm produces such a "funny" diagram, I thought I'd ask you math and science experts about it.

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