This page is the second part of a description of a model of the nature of the universe. Specifically, this part deals with the speed of light as a “geometry” or as one of the eleven component dimensions that make up the universe.
I strongly recommend that you start with part one before reading this section. 

A Geometry for D_½

The whole problem was, now that I knew D_½ represents the speed of light, to find some corroboration on paper, and further, to establish a geometry for D_½, that agrees with what we already know about the propagation of electromagnetism. The first part, corroboration, was a cinch. A simple reworking of the famous Einstein equation shows this is true.

1. Start with E = mc²
2. Substitute the known D - values for E and m, namely, that E = D₂ and m = D₄
3. Rewrite the D - values in their alternate form, as powers of i:

4. Now, solve algebraically for c:

Voila!

Unfortunately, that’s the easy part. A dimension demands a geometry, and if we’re looking for D_½, claiming it to be the speed of light, it had better make sense insofar as it representing the propagation of electromagnetic waves. Admittedly, I did not know how to begin.

I considered the abstract notions of what kind of phenomena might result from the interaction of D₁ with D₂ to form D_½, and then of D₂ with D₄ to form D_2/4 reducible to ½. In the first case, the interaction of a photon with time defined c, but didn’t help to describe it. In the second, the photon’s relation to matter is well known to yield c² (a concept that must be held aside for later), but this wasn’t any help either. So I started thinking about the absurd number I had in front of me: i^½  

i^½ means the square root of i, or the square root of the square root of -i. A few steps took me to the inevitable quadratic:

i^{½ = x}

So I plugged in my coefficients, solved for the 4 x-values, and got:

x = i^½ (given), x = - i^½, x = i^3/2, x = -i^3/2  

Apart from the interesting fact that i^3/2, another proposed dimension, solves this equation, this did not provide anything particularly useful.

And yet, I thought, this stupid number, i^½, must have a value, other than what we get from the circular reasoning above, so what is it? And then I got clever.

I borrowed a magic trick from the universe. If the universe can use an abstraction like time to create a spatial dimension, why can’t I do the same thing? By using my notation for time, that is, i, I decided to create a 3-dimensional graphic representation using only two variables! How? Let’s review some intermediate algebra.

Among the families of numbers, the largest, and most commonly overlooked, is the set of complex numbers. The complex numbers is the only set that actually includes all the numbers, both real and imaginary. The number 5, for instance, which is an integer, or whole number, is also rational, real and complex, because it can be written as 5 + 0i, a standard form given as a + bi, where a is always the real part, and b is the coefficient of the imaginary part.

Likewise, the number -3i would be written in standard form as 0 - 3i. Often, these numbers are represented graphically. To do this, the real part is considered the abscissa, or x-value, and the imaginary coefficient is considered the ordinate, or y-value. On such a graph, 5 + 0i would be plotted on the x (real) axis, at the point (5,0), 0 - 3i would be plotted on the y (imaginary) axis, at the point (0,3).

A point lying on neither axis, like 2 + 4i, would be plotted in the first quadrant at (2,4), and would be drawn as a vector from the origin to that point.

Now we can take this idea one step further. One way to evaluate complicated expressions is to avoid all calculation, and read the value directly off a graph. A graph has the advantage of being able to created from relatively few known values, with all the others then interpolated based upon a recognized pattern. Therefore, if we want to know the value of i^½, let’s draw the graph of y = i^x, and read the corresponding y-value at the point where x = ½.

Since we already know 5 of the values we need, it ought to be fairly simple to infer the overall behavior of the graph. Rewriting what we know as a table of values, we get this:

One thing can be seen immediately. We are not going to be able to plot these y-values on a simple, one-dimensional y-axis. While the x-values are all real numbers, the y-values will have to be plotted on two axes, one real and one imaginary. Our equation in two variables, y = ix, will produce a 3-dimensional graph. At this point, rather than try to draw this graph, I am going to describe it. It ought not to be too much of an ordeal as long as you follow closely.

1. Orientation of the axes. The x-axis is in its conventional position (horizontal). The vertical axis, ordinarily called y, is now labeled y (real part). A third axis, at right angles to both of these, that is, going directly in and out of the page, we give the designation y (imaginary part).

2. Plotting points.
The first point (0, 1 + 0i): No movement along the x-axis. One unit up along the y (real). No movement either in or out of the page.

Second point (1, 0 + 1i): One unit to the right on the x-axis. No movement either up or down. One unit out of the page. (Out, or toward us, is positive on the y (imaginary) axis.)

Third point (2, -1 + 0i): Two units to the right on the x-axis. One unit down along the y (real). No movement either in or out of the page.

Fourth point (3, 0 - 1i): Three units to the right on the x-axis. No movement either up or down. One unit into the page.

Fifth point (4, 1 + 0i): Four units to the right on the x-axis. One unit up along the y (real). No movement either on or out of the page.

3. Draw the graph. Connect the points with a smooth curved line. The result is a symmetrical spiral, or coil, that is a constant one unit away from the x-axis, and wraps around the x-axis once in four x-units or ¼ time in a single x-unit.

A note on the radius of this spiral: It is a constant one unit from the x-axis, but only at the 5 plotted points is this 1 unit readable as an entirely real, or entirely imaginary quantity. At all other points, this distance, which we know must be 1, is expressible only as a complex number on the in the form a + bi since it has both real and imaginary components.

The next step is to read the required value off the graph. We are trying to establish the numerical value of i^½, sow e need to find the corresponding y-values where x = ½. At x = ½, the spiral graph is both above the x-axis and out of the page. This tells us that the number we are looking for has both a real and an imaginary component. Since x = ½ lies midway between x = 0 and x = 1, and is symmetrical about the x-axis, it is reasonable to say that it will have completed one-eighth of a rotation at that point. If we were to view the situation from a perspective on the x-axis, we would see something like the illustration below, in which the x-dimension appears collapsed, and only the y (real) and y (imaginary) dimensions are visible.

This situation is immediately recognizable as the unit circle from which the basic trigonometric (circular) functions are derived. The one-eighth rotation forms an angle of p/4 radians (45°).

The lengths of the legs of the isosceles triangle are by definition sin p/4 and cos p/4 (equivalent), or numerically,

And we have discovered something else, something much more important, because it addresses the original question, which was, a geometry for D_½. If we replace the numerical quantities with the trigonometric expressions from which they were obtained, we get the following:

i^½ = sin p/4 + i cos p/4

By now you can probably translate this equation into D-terminology, but I’ll spare you the trouble:

The geometry of D_½ consists of a moving sine wave. This is in perfect agreement with the physics of the propagation of electromagnetism.

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