She’s Like A Rainbow
She's like a rainbow
Coming, colours in the air
Oh, everywhere
She comes in colours
Rainbows are pretty, but they are illusions.
To paraphrase Lady MacBeth, out damned parabola, out I say.
There are two forms of growth: geometric, also known as
parabolic, and exponential. In geometric growth the assumption is that the rate
of increase is continuous AND constant. This means that in the presence of a limit
to growth, that eventually the growth will exceed that limit and the growth
will become imaginary. By contrast exponential growth is continuous but
NOT constant. It will approach the limit but will not exceed it and will never
become imaginary.
A parabola is the midpoint between an ellipse (a circle is
a perfect ellipse) that has an eccentricity less than 1, and a hyperbola that
has an eccentricity greater than 1. A parabola is a formula/curve with an eccentricity
exactly equal to one. However I would suggest that it is metastable to
be exact, and the universe is thus proabaly stably hyperbolic as proposed by Mabkhout (Mabkhout,
2012).
This may mean that parabolas, Gravity’s Rainbow, and gravity itself, are illusions.
They are useful illusions, but illusions, nonetheless.
Just as there are those individuals that believe in a Flat
Earth, because Pythagoras’ Theorem which only applies on a flat surface, gives
good results, compared to the correct spherical formula. (Actually those individuals
probably do not even know that there is a Pythagoras’ Theorem!). The reason
it gives good results is that when the spherical surface is very large, compared
to the values in Pythagoras’ formula, then there is an imperceptible difference
between a flat and spherical surface. However commercial airplane pilots will tell you
that you that the Great Circle Distance is much more useful than Pythagoras’ formula
over distances between continents.
Similarly Newton proposed a Law of Gravity. (An aside. It
is only called a Law because of the History of Science. Today it would be called
a Theory). Newton was proven incomplete by Einstein. Newton’s Law assumes a
constant mass, and Einstein’s Theory shows that the relative mass of an object increases as that object approaches the speed of light. But
when an object is moving at very low speeds compared to the speed of light,
there is virtually no difference between Newton’s Law and Einstein’s Theory. But
Einstein’s Theory was still applied in a flat universe and therein lies the rub.
In a non-flat universe, the increase in relative mass might not follow Lorentz’s
adjustment, but the difference might be imperceptible at the speeds commonly encountered.
However if the universe has a hyperbolic shape, then gravity
itself might be an illusion where two or more objects approach a common center
on that curved, hyperbolic, surface. In this case, just as Einstein’s Theory is
more correct than Newton’s Law, Newton’s Law continues to be used because it is simpler
to apply, even if Gravity is an illusion, Newton’s Law and Einstein’s Theory might
be “wrong” but in the words of George Box, they might be useful.
If the universe has an absolute, then the distribution of
objects in that universe can be expected to follow a distribution within that
absolute. For an arbitrary absolute, for example π, objects within that absolute
should be expected to follow a random distribution. One such normal random distribution
is the logistics, also known as the hyperbolic secant squared, distribution, whose
Probability Density Function, PDF, is (1/(4*s))*sech2((x‑µ)/(2*s)),
where s and µ are parameters of that distribution. Its Cumulative
Distribution Function, CDF, is also a hyperbolic trigonometric function, ½*tanh((x‑µ)/(2*s))+½.
If objects are uniformly distributed, then the mean, and the median, of that
normal logistics distribution, µ, is at half of the absolute, or in this
example, π/2. When x is at that mean, median, there should be 50% of all objects,
a PDF of 0.5, which requires that in this example s=0.5. Its variance, σ2, is s2π2/3
or 0.822.
Another normal distribution is the Gaussian distribution. Its
Probability Density Function, PDF, is 1/(σ*√(2*π))*e(-0.5*((x-µ)/σ)^2),
where its parameters are σ
and µ. Its Cumulative Distribution Function, CDF, is 0.5*(1+ERF((x-µ)/(σ*√(2)))),
where ERF is the standard error function.
As shown in the figure below when the PDF of the normal logistics function
at the mean, median is 0.5, the PDF of the Gaussian is 0.44. If the PDF of the Gaussian is 0.5 at the mean, median,
then the s parameter of the logistics distribution must instead be 0.44 and its
PDF is then 0.57 instead of 0.5. Neither the logistics
nor the Gaussian distribution are zero at value of x of zero or the absolute. The Gaussian distribution
on a flat surface has almost the same values as the Gaussian distribution on a hyperbolic
surface.
Also shown in that figure is a parabola with a coefficient
of 1 which has a value of 0 at an x of 0. This does
NOT have a value of 50% at the median, mean, µ.
In fact it becomes very large near the absolute. A
parabola can be made to take on a value of 50% at the median. The reflection of the adjusted parabola, which
creates a discontinuity, can be used at the mean, median. This will result in a value
of zero at the absolute. However a parabola and its reflection is not as simple as a logistics distribution. A logistics distribution is also smooth and does not create
a discontinuity at the mean, median. However using a simple parabola can highlight
how the integral of the PDF, the CDF can be viewed.
The CDF, integral or area under the curve, of the PDF of a simple parabola
is identical to the formula for the area of a triangle.
However, as shown, it has a height of 1.57,
π/2, rather than a
height of 1 at the absolute, if the absolute is assumed in this example to be
π, which should be the CDF and is approximately the value
of the CDF of the normal logistics distribution on all surfaces and the normal Gaussian distribution on a flat or hyperbolic
surface. However if the CDF of a parabola, the formula for the area of a triangle,
is translatedon the y axis, reduced, by a value of .29, it becomes almost identical to the
CDF of the normal distributions near the mean, median.
It continues
to approximate the CDF of the normal distributions up to a distance of
π/6
from the mean, median, of
π/2.
At
this distance the slope changes to become approximately half of the previous
slope and this continues for a distance of
π/6 from the last change.
At this point, the slope again changes to become approximately half of
the previous slope. This continues to an
x of the absolute and an
x of zero, the absence of the
absolute.
The fact that the formula for the hypotenuse of a triangle on
a flat surface must be adjusted suggests that the correct formula should not be on
a flat, surface. The logistic
distribution is consistent with, and uses, hyperbolic trigonometric functions. It has parameters of s=0.5 and µ=the
absolute divided by 2. It is observed that
an s of 0.5 is also consistent with the mean of a single choice of that absolute. It is also suggests that the Gaussian distribution
was an attempt to derive a normal distribution on a flat surface, when it
should have been derived for a hyperbolic surface. It is also observed that the 68/95/99 rule of a Gaussian
distribution on a flat surface corresponds to a 52/85/100 rule for a logistics distribution
on the hyperbolic surface of the universe ( 52% percent of the values fall within ±1/3 of the mean, median; 85% of
the values fall within ± 2/3
of the mean,median; and 100% of the values fall within ± 3/3 of the mean, median. Rather than an arbitrary variance, when the choice
parameter, s, is equal to 0.5, the variance, σ2, has a fixed value of 0.822467 and the
square root of the variance, also known as the Standard Deviation, has a fixed
value of 0.9069.
The 68/95/99 rule is for the multiples of the standard deviation of a Guassian distribution on a flat surface. For a logistics distribution on a hyperbolic surface, when 100% of the values fit within the range of the absolute,
- 70.6% are within the mean, median, ± σ;
- 94.3% are within the mean, median, ±2σ; and
- 99.0% are within the mean, median, ±3σ.
It is suggested that the surface of the universe is
hyperbolic. It is suggested that the distribution
of objects follows a logistics distribution.
It is suggested the parameters s, variance, and standard deviation must all take on nonzero values in reality.
At the mean, median, the dominance is 100% and it remains
this value for any outcome. However
- at
the mean, median, ± 0/3 of the mean, median, there is 25% certainty;
- at the mean, median, ± 1/3 of the mean, median, there is 52% certainty;
- at the mean, median, ± 2/3 of the mean, median, there is 85% certainty; and
- at the mean, median, ± 3/3 of the mean, median, there is 100% certainty.
If the distances commonly encountered are
less than 1/3 of the range of the absolute, then there is no appreciable difference
between the results for a flat or hyperbolic surface.
Parabolas may be an illusion. We may aspire to live on circles. But we appear to live on a hyperbola.
Mabkhout, S. (2012). The infinite distance horizon
and the hyperbolic inflation in the hyperbolic universe. Phys. Essays,
25(1), p.112.
