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*
eb*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 |
|
|
Hyperbolic |
a*eb*x |
-∞ to ∞ |
a* to ∞ |
if a |
ln(x/a)/b |
-∞ to ∞ |
0 to ∞ |
ln(x) |
|
-∞ to a* |
if a |
* 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(ex+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 85th 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(
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.
