Physical Equation that contains the Golden Ratio

Dec 28, 2020
Different views / interpretation of the Golden Ratio
  • Mathematics - most irrational number, nothing special about it, end of story
  • Physics - no application whatsoever, (for now)
  • Arts - found in paintings, ancient structures, music, etc.
  • Biology - found in sunflower, nautilus shell, etc.
  • Mystic Arts / Others - creation, Penrose tiling, E8, optimum fractal, galaxies, DNA, human body, etc.

The Search for Patterns

“And if you gaze long into an abyss, the abyss also gazes into you.” — Friedrich Nietzsche

The Golden Ratio is insignificant on its own. Why is it common in nature? Now that’s a more interesting question. It cannot be denied that the Golden Ratio is observed in nature but for some reason, it is difficult to comprehend its importance. It’s like the air that we breathe, we know it’s there because its keeping us alive, otherwise we will be in a different place, but we cannot see or touch it.

We use patterns to describe nature and if we look hard enough, we can even create a mathematical equation for the pattern. This does not mean that the pattern follows the equation. It’s the other way around, the equation follows the pattern.

Keep in mind that the equations we use to describe the patterns are mental constructs, it’s all in our mind. We create these mental constructs to make sense of what we see. Nature can work fine without the equations.

Let’s say there are 5 cats and 2 dogs. The cats and dogs are real but the numbers “5” and “2” are not. It’s only a numbering system that was invented to give meaning to what we see. A child can look at the cats and dogs but without the concept of numbers, he will not be able to group them as “5 cats and “2 dogs” or perform arithmetic operations.

As mentioned before, this ratio is insignificant on its own. So saying that the Golden Ratio is 𝑥^2−𝑥−1= 0 and try to fit this in the physical world is a futile exercise.

While mathematics is used to manipulate equations and to some degree can be applied to describe the physical world, it should not be the starting point for physical observation. Note that mathematics belongs to the realm of the abstract and not exactly physical reality.

To make things clearer, the Golden Ratio (phi) is just part of a physical equation and not the equation itself. Same as (pi) is not the equation for a volume of a sphere but part of the equation.

To give a physical meaning to this ratio, the first step is to find an equation that describes a physical phenomenon which includes this ratio and then use this ratio to describe other phenomenon.

Now comes the tricky part…. How can this equation be derived? Well it should not be derived using “AXIOMS” because we are dealing with a physical phenomenon. Another way is to use a proven scientific method and that is to “GUESS” it -> Compute the consequences of the GUESS -> Compare with observations (R. Feynman).

Below shows two (2) tests of the Golden Ratio

First Test of the Golden Ratio - Planetary Rotation

Golden Spiral & Golden Angle
Wikipedia: In geometry, a Golden Spiral is a logarithmic spiral whose growth factor is φ, the Golden Ratio. That is, a Golden Spiral gets wider (or further from its origin) by a factor of φ for every quarter turn it makes.

Wikipedia: In geometry, the Golden Angle is the smaller of the two angles created by sectioning the circumference of a circle according to the Golden Ratio; that is, into two arcs such that the ratio of the length of the smaller arc to the length of the larger arc is the same as the ratio of the length of the larger arc to the full circumference of the circle. It measures approximately 137.5077640500378546463487 ...°

Below is the Physical Equation that contains the Golden Ratio. The values of “Rd” (x-axis) can be derived from the Golden Ratio.

GAV[intinsic property] = f(mass, density)

The GAV Equation describes the Equatorial Rotation Velocity as a function of mass and density for both Jovian and Terrestrial planets. Inserting the (Rd) values on the equation and the Equatorial Rotation Velocity of the planets can be calculated.

Graph 1 - Rotating Golden Spiral (Polar Coordinates)
The red spiral is rotated until it intersects a planet or group of planets shown in different colors. Note the angle of the red line to the horizontal line is very close to the value of the Golden Angle.

Graph 2 - Semi-Log Plot
Below is an application of the Golden Ratio on the rotation of planets. The black dots (M1 to M9) on the Semi-Log Plot (Graph 2) are planets in the Solar System and the dot on the top far right is an Exoplanet (Beta Pictoris b). The position of the dots (Rd) on Graph 2 matches that of (Graph 1).

Graph 3A - Projection of the Golden Spiral on the Semi-Log Plot
The graph below is a rotating logarithmic spiral superimposed on a semi-logarithmic plot.

Graph 3B

Graph 4 - Exoplanet
Wikipedia: Beta Pictoris b (abbreviated as β Pic b) is an exoplanet orbiting the young debris disk A-type main sequence star Beta Pictoris located approximately 63 light-years (19.4 parsecs, or nearly 5.986214×10^14 km) away from Earth in the constellation of Pictor.

Calculated GAV = 22,534 m/s (based on calculated density of 5142 kg/m^3)

Calculated Rotation Period = 8.084 hrs

Second Test of the Golden Ratio - Planetary Arrangement (Optimal Arrangement, e.g. sunflower seed pattern)

Notice the color group of Graph 1 and compare it to the actual planet arrangement. M5 is moved to the closest group, M8. P1/P2/P3 are excluded from the color group. Also, the center is located on the Asteroid Belt (or a planet, Titius-Bode Law) between Mars and Jupiter.

The Golden Ratio can describe both planetary rotation and planetary arrangement.

Planetary Rotation < —— Golden Ratio —— > Planetary Arrangement

Additional Test of the Golden Ratio (Hypothetical Planets)

  1. From Graph 1, Pluto doesn’t have a companion planet. If such a planet exists (Vulcan), it will be close to the Sun. Also, the mass of this planet is small enough not to cause a measurable perturbation on the orbit of Mercury.
  2. From Graph 3A, draw a vertical line starting from M8 until it intersects the Brown Spiral and continue downwards tangent to the Cyan Spiral. The Hypothetical Planet at the intersection of the Brown Spiral and the vertical line will have a mass equal to ten times (10x) the mass of Earth.
Of course, there is also the possibility that this ratio is just a kind of mental construct we use to describe the physical world and there is no physical meaning to it. This is best described by one of the greatest minds of the 20th century.

“Physical concepts are free creations of the human mind, and are not, however it may seem, uniquely determined by the external world.” — Albert Einstein
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