• Randy Evangelista


Updated: May 2

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"For nothing is hidden, except to be revealed; nor has anything been secret, but that it would come to light."

Different views / interpretation of the Fibonacci Spiral and 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 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.

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).

First Test - Planetary Rotation

Fibonacci Spiral & Golden Angle

Wikipedia: Fibonacci Spiral - an approximation of the Golden Spiral created by drawing circular arcs connecting the opposite corners of squares in the Fibonacci tiling.

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 and Fibonacci Spiral. The values of “Rd” (x-axis) can be derived from the Golden Ratio and Fibonacci Spiral.

The Gravitational Angular Velocity (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.

NOTE: Column 4 are the values of equatorial rotational velocities from Wikipedia.

Refer to my blog for information on P1, P2 and P3.

Below is an application of the Fibonacci Spiral and the Golden Ratio on the rotation of planets. The dots (M1 to M9) 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.

PRINCIPLE OF LEAST ENERGY (Minimum Energy Configuration)

Graph 1 - Rotating Fibonacci 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.

Earth-Moon System

These objects are interlocked and with a large moon/planet ratio. Instead of describing the rotation of the Earth or Moon separately, it is more meaningful to describe it as a whole, the “Earth-Moon System” and derive the angular velocity of Earth from this point of view.

Pluto-Charon System

These objects are also interlocked and with a large moon/planet ratio. Instead of describing the rotation of Pluto or Charon separately, it is more meaningful to describe it as a whole, the “Pluto-Charon System” and derive the angular velocity of Pluto from this point of view.

Graph 2 - Semi-Log Plot

X axis = Rd

Y axis = Mass(planet) / Mass(earth)

Graph 3A - Projection of the Fibonacci Spiral on the Semi-Log Plot

The graph below is a rotating Fibonacci Spiral superimposed on a semi-logarithmic plot.

Graph 3B - Gnomonic Projection

Graph - P1/P2/P3

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

Rotation Period (Wikipedia) = 8.1 hrs

Second Test - Planetary Evolution and Optimal Arrangement (e.g. sunflower seed pattern, phyllotaxis, nautilus shell, galaxies, etc.)

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 of the spiral is located in the Asteroid Belt (or a planet) between Mars and Jupiter, “Titius-Bode Law”.

Graph 5

Third Test - Planet Distance from the Sun (link to Newton’s Law of Universal Gravitation)

Use the same values of "Rd" and the angular separation of planets to compute the planet distance from the Sun.

Aphelion and Perihelion of Mars (M3)

Perihelion (Mars) / Aphelion (Mars) ≈ Rd (Mars) x |cos (Golden Angle / 10)|

Θ ≈ 137.5° (golden angle)

1.41 / 1.64 ≈ Rd (Mars) x |cos (Θ / 10)|

1.41 / 1.64 ≈ 0.885 x |cos (137.5° / 10)|

Mars (M3) average distance from the Sun

Θ ≈ 137.5° (golden angle)

1.52 ≈ 1 / Rd(Mars) x |cos (Θ)|

1.52 ≈ 1 / 0.885 x |cos (137.5°)|

Information from NASA website:

Graph 6 - Projection of the mass of Terrestrial Planets on the Rd axis

Graph 7 - Planet Group

Multiplying the mass of all the planets per group will result in a value approximately equal to the combined mass of the Earth and Moon.

Table 3

Fourth Test - Planet Type

Trace the Spirals in a counterclockwise direction;

  • if the direction is "up", it is a Jovian (gas planet)

  • if the direction is "close to horizontal", it is a Jovian (ice planet)

  • if the direction is "down", it is a Terrestrial planet

Fifth Test - Hypothetical Planet

Pluto doesn’t have a companion planet. If such a planet exist, it will be further from the Sun. The planet type, "Terrestrial / Jovian" will depend on the planet location on the Brown Spiral (Fourth Test). This hypothetical planet will have a mass of about 4.2 times the mass of Earth as shown on the Third Test.

NASA Science: Caltech researchers have found mathematical evidence suggesting there may be a "Planet X" deep in the solar system. This hypothetical Neptune-sized planet orbits our Sun in a highly elongated orbit far beyond Pluto. The object, which the researchers have nicknamed "Planet Nine," could have a mass about 10 times that of Earth and orbit about 20 times farther from the Sun on average than Neptune. It may take between 10,000 and 20,000 Earth years to make one full orbit around the Sun.


The Fibonacci Spiral and the Golden Ratio can be used to calculate 108 phenomena within the solar system.

  • Planet distance from the Sun → 9 planets (link to Newton’s Law of Universal Gravitation)

  • Planet Rotation (spin) → 9 planets

  • Planet Orbit (counterclockwise) → 9 planets

  • Planet Arrangement → 9 planets

  • Planet Type (Terrestrial / Jovian) → 9 planets

  • Orbital Period → 9 planets

  • Orbital Velocity → 9 planets

  • Prograde/Retrograde Motion → 9 planets

  • Obliquity to Orbit → 9 planets

  • Orbital Eccentricity → 9 planets

  • Precession of Perihelion (General Relativity) → 9 planets

  • Observed Precession of Perihelion → 9 planets

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