Monday, March 30, 2020

Three Maps, One Territory

I now have three distinct ways of describing the same phenomenon, my unified field theory, sin(ln(x))/x.  At this point I think it is probably sin(ln(x))/x + cos(ln(x))/x, granted.

In the first description, the field is a self-interfering sine wave, possibly but not definitely originating as something like f(x) = x + integral(sin(f(x))/f(x)).  I couldn't solve the recursion.

In the second description, the field is the surface area of a geometry which may be a negative 3-sphere, which results in the distance dimension being the arc length of a spiral in a complex plane.  The unreal portions define the instantaneous curvature at a given point.  I arrived at this description trying to solve the recursive function, and deciding it might be easier to consider what a negative closed dimension might behave like; no luck.

In the third description, we are back, in a sense, to the first description; if we think of the field as a rotation in the shape of local space-time, the behavior at a point is defined by the rotation of the field as it arrives at this point; that is, the field causes rotation, but the rotation it causes is rotated by the field between the origin and the point.  Self-interfering wave again, but this time, considering the closed hyperbolic dimension of the rotation itself, perhaps the link to the second description may be a little more apparent.

Or not.

The obviousness of this all is particularly frustrating given the difficulty of trying to convey it.  How is it not readily apparent to anyone who looks that the forces can be described as a series which decreases in both initial impetus, and in the decay of that impetus, of alternating sign? And once this is realized, how is it not eventually apparent that this pattern can be unified in a wave whose amplitude decays proportional to distance, and whose frequency decays proportional to the log of distance?

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