Shapes

 

Aquarius/Let the Sunshine In

Harmony and understanding
Sympathy and trust abounding
No more falsehoods or derisions
Golden living dreams of visions
Mystic crystal revelation
And the mind's true liberation
Aquarius
Aquarius

The “Age of Aquarius” is also the “Age of 'The Twilight Zone'”

“There is a fifth dimension beyond that which is known to man. It is a dimension as vast as space and as timeless as infinity. It is the middle ground between light and shadow, between science and superstition, and it lies between the pit of man's fears and the summit of his knowledge. This is the dimension of imagination. It is an area which we call ‘The Twilight Zone’”

Viewers of the old TV series, or visitors to Walt Disney World’s “Twilight Zone” Tower of Terror might recognize this intro. Have you ever thought about what this means? The first three dimensions are the dimensions of space: length, width, and height. The fourth dimension is that of time. Minkowski developed a transform to combine the three dimensions of space into a 2-D chart of space‑time, which his student Einstein would later make much more famous. Mathematicians would call the fifth dimension i, the imaginary number which is the square root  of minus 1, √-1,   If you rotate the surface of space-time around the axis of imagination you will develop a three-dimensional volume of the “Twilight Zone”.  So what should the surface be?

A function, as opposed to an equation, is one which has only one solution for each input value. There are four families of functions:

·       Linear, f(x) = y = a *x +b;

·       Power, f(x) = y = a * xⁿ +b;

·       Exponential, f(x) = y = a* e^b*x; and

·       Logarithmic, f(x) = y = a + b * ln(x)

The logarithmic function is the inverse, f^-1(x), of an exponential function, therefore it is convenient to combine these into one surface, hyperbolic, which means that there are three surfaces. Some other hyperbolic functions are the hyperbolic sine, hyperbolic cosine, hyperbolic tangent, etc., and all of these can be expressed as exponentials. Similarly the regular circular trigonometric functions: sine, cosine, tangent, etc., can all be expressed as series of power functions, and it is conventional to call their surface spherical. Because it is NOT possible to express linear functions as either hyperbolic or circular trigonometric functions, and both spherical and hyperbolic surfaces are also curved, it is convenient to call a linear surface flat. It is also conventional to call flat surfaces Euclidean and both spherical and hyperbolic surfaces non-Euclidean.

 The surfaces, their functions, inverses, domains, and ranges are:

	Function	Domain	Range		Inverse	Domain	Range
Flat	a*x+b	-∞ to ∞	-∞ to ∞		(x-b)/a	-∞ to ∞	-∞ to ∞
Spherical	a*xn +b	-∞ to ∞	-∞ to ∞		n√((x-b)/a)	-∞ to ∞	-∞ to ∞	(x-b)/a >0 or else imaginary solution  when n is even
Hyperbolic	a*eb*x	-∞ to ∞	a* to ∞	if a is  positive	ln(x/a)/b	-∞ to ∞	0 to ∞	ln(x) is not defined for negative numbers
			-∞ to a*	if a is negative

* The starting point of the range varies with the hyperbolic function. For example  cosh(x) has a range of 1 to ∞, but cosh(x)=½(e^x+e^-x) which requires that for it a is 1.

As shown in the table above, a real number as an input to the inverse function becomes an imaginary number as an output on a spherical surface. A real input added to a zero imaginary number is indistinguishable from a complex number whose coefficient of the imaginary component is zero. It is suggested that the input should allow real or complex numbers without the inverse requiring a change in the case of number for its output solution. Additionally spherical surfaces are finite in reality, which is a consequence of the circular identity, cos²+sin²=1. Circular trigonometric functions are infinite and repeating, but they repeat with a period of 2π. Hyperbolic surfaces have two domains, and those domains can accommodate either real or complex numbers as inputs. This is because of the hyperbolic identity cosh²‑sinh²=1 and the fact the hyperbolic trigonometric functions are infinite and repeating, but only in imaginary surfaces, planes. That is they have a period of 2πi.

The universe appears to be infinite, which rules out its surface being spherical. It must be either flat or hyperbolic. But if the surface is flat then the variance, σ², of that surface must be zero and if there is any transition  between domains it must be at ½, 50%, of σ. However observations of traffic, fluid, etc. suggest that there is a transition, but it occurs not at 50% of σ, but closer to 85% of σ. This is consistent with design recommendations for a road at the 100th highest volume, the dividing line between Level Of Service, LOS, C and D of traffic, the parametric rail capacity of 85%, a speed limit being set at the 85^th percentile of a speed study, etc. The transition point, if the surface is hyperbolic, would be 5/6 σ² or 83.3% of σ. It is noted that the Nash Equilibrium is also at 5/6 σ².

If the surface is hyperbolic and the input is complex with a coefficient of zero for the imaginary axis, then σ=1.0579, which is not appreciably different than the solution on a flat surface of σ=1. Since the absolute is by definition twice its mean/median, this means that the absolute on a hyperbolic surface is a random number because that absolute has a mean/median and a variance.

This is not appreciably different than acknowledging that the distances on the surface of the Earth are round by saying ”Distances are locally flat, but globally spherical.” The corresponding statement for space would be “Space is locally flat, but globally hyperbolic.”  This does mean that Pythagoras’ Theorem is c=ln(cosh(a²+b²)), not
c=√(a²+b²) and that the relativistic dilation factor is γ=ln(cosh(1-v²/c²), not γ=√(1‑v²/c²). Melkhout( (Mabkhout, 2012) did propose that the universe has a hyperbolic shape and that by assuming this and solving its Einstein Field Tensors in hyperbolic space the need for Dark Matter and Dark Energy vanishes. Mahout also points out that a hyperbolic shape explains the rotation of galaxies, is consistent with both the age and size of the universe at large scales and  Planck Energy and Planck length at small scales, and a hyperbolic shape that is almost flat is consistent with a period of initial but short inflation after the Big Bang. It is proposed that not only is the shape of universe hyperbolic, but the universe is one sheet of a two-sheeted hyperboloid. The Big Bang is the point of intersection between these two sheets.

As to the song lyrics, remember that while the song was from the musical Hair, it was a big recording hit for the singing group the “Fifth Dimension.”

Mabkhout, S. (2012). The infinite distance horizon and the hyperbolic inflation in the hyperbolic universe. Phys. Essays, 25(1), p.112.