Think
I ain't no psychiatrist, I ain't no
doctor with degrees
But, it don't take too much high IQ's
To see what you're doing to me.
I also ain’t no doctor with degrees, but
Dr. Nash meet Pythagoras
and Dr. Einstein
The two equations with which most people are familiar are probably:
- c2=a2+b2, Pythagoras’ Theorem for the hypotenuse of a right triangle; and
- E=mc2, Einstein’s formula for energy.
Most people are also aware that the c s in these equations are variables which represent different things:
- the hypotenuse in Pythagoras’ Theorem, and
- the speed of light in Einstein’s formula.
What most people do not realize, is that BOTH equations are actually for triangles: Pythagoras’ Theorem by definition, and Einstein’s formula as an outcome of its Triangle of Energy, E= (mc2)2 = (mvc)2 + (m0c2)2. Further, if m=m0/ϒ, and the terms are simplified, then Einstein’s Triangle becomes ϒ2=1 - v2/c2. Both equations ignore any consideration of the imaginary plane. If they are treated as complex numbers and the imaginary plane has a coefficient of zero, they should be c2=a2+b2+02i and ϒ2=1 - v2/c2+ 02i . Their solution on an elliptical or flat surface is the conventional c=√(a2+b2) and ϒ=√(1 - v2/c2). However on a hyperbolic surface there would be two solutions.
for Pythagoras’s Theorem, and
ϒ=ln(cosh( √(1 - v2/c2))
± sinh( √(1 - v2/c2))), or ln(0 ± 2cosh( √(1 - v2/c2)))
and
ϒ=ln(cosh(-√(1 - v2/c2))
± sinh(-√(1 - v2/c2))), or ln(0 ± 2cosh(-√(1 - v2/c2)))
for the factor of relativistic dilation, which is the Loretz Transform on a flat
Euclidean Surface.
Since cosh(x) is an odd function, that is cosh(x)=cosh(-x), both solutions are the same and there is thus effectively only one solution. If at the origin of (0,0,0) there is also a rotation of the coefficient of the imaginary axis in the transformation of coordinates as a complex number, then according to Euler’s’ formula, the coefficients of the imaginary axis, sin(0) and sin(π) are both zero, but the coefficients of the real axis will be cos(0)=1 and cos(π)=-1. Thus only with a rotation of the imaginary axis by π radians can a function pass though the origin from one sheet of a two-sheeted hyperboloid to the other sheet if this volume is formed by rotating a hyperbolic surface about the imaginary axis. It is proposed that any function passing through the origin from one sheet to the other requires a rotation by π radians at the origin. An input of zero for the natural logarithm, ln, has a value of 1. Thus the range of the solution for in the positive sheet of a two-sheet hyperboloid for Pythagoras’ Theorem is 1<c<1+ln(2cosh(√(a2+b2))) and for the relativistic dilation factor is 1<ϒ<1+ln(2cosh(√(1-v2/c2))). In the negative sheet for Pythagoras’ Theorem it is -1>c>-1-ln(2cosh(√(a2+b2))); and for the relativistic dilation factor it is -1>ϒ-1-ln(2cosh(√(1 - v2/c2))). The midpoint, average, of these ranges is in each sheet is, respectively, √(a2+b2) and √(1 - v2/c2) preceded by a negative sign in the negative sheet and a positive sign in the positive sheet. From the perspective within each sheet, all values would be positive, including this midpoint average.
Assuming that the
universe is a hyperbolic surface as proposed by Mabkhout
It is noted that a range, uncertainty (as in Heisenberg’s Uncertainty Principle), also appears to be an outcome for the solution on a hyperbolic surface. Pythagoras’ Theorem and the Lorentz’s Transform are merely AVERAGES of the ranges of uncertainty required by the hyperbolic solutions. When the uncertainty in the solutions is small, then the values introduced by imagination can be ignored. However at a scale of 2/3 of the absolute, uncertainty becomes too large to be ignored. At 5/6 of the absolute, the average approximations of the midpoint of the ranges are no longer useful.
The values of 2/3 and 5/6 are consistent with n individuals acting as a single system, called a Nash Equilibrium after mathematician John Nash. For a normal random function of the absolute, for example the hyperbolic secant squared, or logistics, distribution, the mean/median, µ, is one half of the absolute. Zero is that mean minus 3 Standard Deviations, σ. Between 0 and µ-2σ there is a small value for the approximation and a large uncertainty. Between µ-2σ and µ+2σ are almost all of the solutions and the uncertainty causing an error from this approximation is small. Only between µ+2σ and µ+3σ, which if µ is ∞/2 and µ-3σ is zero, 1/∞, then µ+3σ is ∞, the absolute, and σ=∞/6, is the approximation large and its error also large. A Nash Equilibrium again is n individuals acting as a system and almost all of those individuals, over 90 percent, will be between µ-2σ and µ+2σ, in other words 2/3 of the absolute. While almost 5% of the individuals will be between zero and µ-2σ, the error will be large, but the approximation will be very small. Only for the almost 5% of the individuals between µ+2σ and the absolute, i.e. 5/6 of the absolute, will the error be large, and the approximation also be large. Thus according to Nash, Pythagoras’s Theorem is correct over 90% of the time and when the values are less that 5/6 of the absolute, this approximation can be used. Pythagoras’ Theorem, like all models, may be wrong, but according to Nash it is useful.
Works Cited
Mabkhout, S. (2012). The infinite distance horizon
and the hyperbolic inflation in the hyperbolic universe. Phys. Essays,
25(1), p.112.
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