Parabolas

 

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.