Solutions

 

Ain’t Necessarily So

It ain't necessarily so
The things that you're liable
To read in the Bible
It ain't necessarily so

And the things that you read in Math textbooks ain't necessarily so.

The equation y²=x², seems like it has a simple single solution, y=x. but actually there are two solutions y=x AND y=-x. The second solution is due to the fact that i²=-1. This  suggests that an imaginary axis could be important in resolving this paradox.

Minkowski proposed a method to transform the three dimensions of space into a single dimension. In the two dimensional,  2‑D, version, the other dimension is typically time. This gives rise to inverted triangles fomed by light travelling on a surface in space‑time described by Minkowski space. This is often expressed that reality must be within two light triangles that intersect at the origin. If that space-time surface is rotated about another dimension, then it is suggested based on the conclusion of the first paragraph that this dimension might be the imaginary axis. Thus in 3‑D Minkowski space, there would be dimensions of space, time, AND imagination, while in conventional space, the dimension of space is transformed into length, width, and height, such that there are five dimensions.

Rotating a flat 2‑D flat, Euclidean, surface about an imaginary axis still produces only one solution, while paradoxically there are two solutions. This suggests that the rotation of a flat, Euclidean, surface may not be correct. To resolve the paradox, it  is proposed that the rotation of a hyperbolic surface is required. This still gives rise to a cylindrical 3‑D space, but then there are two solutions,  y=ln(0 ± 2*cosh(x)), if the surface must pass through  the origin.

The rotation of two triangles on a flat surface which intersect at the origin produces two inverted cones whose peaks intersect at the origin. The rotation of two triangles on a hyperbolic surface produces a two-sheet hyperboloid whose sheets also intersect at the origin. However one of the sheets will have the opposite sign of the other sheet and any solution in that sheet will also have a solution that is the opposite sign in the other sheet. It is noted that a change of sign is equivalent to a rotation on the imaginary axis of π. This because of Euler’s Formula and the fact that sin(0)=0 with a rotation of π, is equivalent to sin(π), but cos(0)=1 while cos(π)=-1.

Since cosh(x) is a odd function, that is cosh(x)=cosh(-x), and logarithms are undefined for x<0, then unless there is a rotaion of π when passing though the origin of 0 between the two sheets of the hrperboloid, it will apear to an observer on one sheet of the hyperboloid that there is only one solution, y=ln(cosh(x²)).  The derviative of this solution is tanh(x²).  This repeats with a period of πi.  If x<2/3*π, then the solution is virtually identical to y=x when x>0.  A Nash Equilibrium discontinuity occurs at 5/6*π. The approximation can be used until this discontinuity, at which point the uncertainty  becomes significant and the approximation is no longer valid. Since the absolute can also be stated as a multiple of π, this can be restated that the universe is flat locally, but is hyperbolic universally.