“All truth passes through three stages. First, it is ridiculed. Second, it is violently opposed. Third, it is accepted as being self-evident.” - Arthur Schopenhauer
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PART ONE
Path of Science
THE SEARCH FOR PATTERNS
Why do we recognize patterns? …Maybe it’s because we are part of the same pattern and we see things that are familiar to us. The pattern and the equation that describes it is already in our mind even before we see the pattern with our eyes.
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.
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. As our understanding grows, so is the need to come up with new and more powerful equations to describe the universe, e.g. from Newtonian Mechanics to General Relativity.
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.
Also, patterns will only be accepted as science once we become familiar with the physics of the patterns. Kepler’s Third Law of Planetary Motion p^2 = k(a^3) is a pattern, until Newton comes along and develops the physics and introduced the concept of gravity.
"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.
Newton's First Law of Motion
In an inertial frame of reference, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a force. The force that is mentioned above is an external force. But what IF the force that creates the motion is part of the system, an internal or Intrinsic force?
If we use the concept of GRAVITY as a force (torque in this case) and this force will cause a planet to rotate, then it is expected that the spin of the planets will continue to increase since the force is applied continuously. This contradicts our observation since planets are spinning at an almost constant speed.
To address this, a retarding force (resistance) must be present such that it balances the force of gravity to maintain an almost constant speed. This retarding force is still unknown, but a good candidate for this is SPACE itself. That SPACE counteracts the effect of GRAVITY.
General Relativity: The Kerr metric solution of General Relativity is a good candidate for describing rotating massive objects. There are two (2) parameters for this metric; mass (M) and angular momentum (J). The GAV equation can be used to derive the angular momentum (J) of a rotating body which is a function of “mass and density” and this can be inserted to the metric.
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.
Planetary Dynamics
Planetary dynamics is a very a complicated subject due to the different parameters involved in the analysis. Mass, density, rotation period, orbital period, orbital velocity, eccentricity, obliquity to orbit, distance, etc., all play a crucial role in this “Endless Dance of Cause and Effect”.
I believe everything is connected to everything else so changing one parameter will affect the other. The magnitude of the perturbation depends on the type of parameter being associated to the change (e.g. mass to orbit, orbit to rotation, rotation to orbital eccentricity, orbital eccentricity to obliquity of orbit, etc.).
The approach is to study the dynamics of the system as a whole and search for symmetry. The relationship between the different parameters can then be established by performing simultaneous calculation. Refer to the "METHOD OF CALCULATION" from the first image.
One practical application of the graphs and equations is to include it on planetary simulation programs to improve the capability of predicting planetary motions via analytical solutions and also to simulate how a change in one parameter affects the other parameters.
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.
Natural Exponential Function: Values of Rd can be determined using Natural Exponential Functions.
Below are exponential functions for Rd_j (Jupiter), Rd_s (Saturn), Rd_n (Neptune), Rd_u (Uranus), Rd_v (Venus), Rd_m (Mars), Rd_{me} (Mercury) and Rd_{pl} (Pluto).
Rd_j = e^{(1/Rd_e)}
Rd_m = e^{(-1/3)(1/Rd_j)}
Rd_n = e^{(1/3)(Rd_m)}
Rd_s = e^{(1/Rd_n)}
Rd_u = e^{(1/20)(Rd_j)}
Venus rotation is very unusual. Not only is it rotating very slow,
it also spins in the opposite direction compared to the other Terrestrial planets.
Rd values on the left side of Rd_e should have a factor of (1/10)
To satisfy the requirement, Rd_v will be multiplied by 1000
Rd_m/Rd_u = Rd_s/Rd_j = e^{(Rd_v)(1000)}
Rd values on the right side of Rd_e should have a factor of 1.
To satisfy the requirement, Rd_v will be multiplied by 10,000
Rd_u/Rd_j = Rd_m/Rd_s = e^{(1/3)(Rd_v)(10,000)}
Mercury & Pluto - The same left side requirement for Rd also applies to the two planets.
To satisfy the requirement, Rd_{me} will be multiplied by 100
Rd_m/4 = e^{(-1/3)(1/(Rd_{me})(100))}
To satisfy the requirement, Rd_{pl} will be multiplied by 10
Rd_s/Rd_n = e^{(Rd_{pl})(10)}
Graph 3B - Exponential Function
Graph 3C - Gnomonic Projection
Graph 3D - P1/P2/P3
Graph 3E - Add another axis (Z) perpendicular to the Rd, Mp/Me plane. Rotate the coordinate (Z) so it will be
parallel to the line from M3 to M6 while keeping the Rd, Mp/Me plane perpendicular to the rotated axis (Z).
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.
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.
CONCLUSION:
The Fibonacci Spiral and the Golden Ratio can be used to calculate 117 phenomena within the solar system.
Planet Mass → 9 planets
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
Orbital Eccentricity → 9 planets
Obliquity to Orbit → 9 planets
Prograde/Retrograde Motion → 9 planets
Observed Precession of Perihelion → 9 planets
Precession of Perihelion (General Relativity) → 9 planets
sin(432) / cos(234) = -φ
(sin(234) / cos(432)) + 1 = -φ
[(432)(234)]^(1/2) ≈ M(Jupiter) / M(Earth) ≈ 1000 / π
432 + 234 = ???
(Not part of this blog. I will leave the interpretation to the readers and will not go down the rabbit hole.)
YouTube Version
"There is no need to search for patterns in nature...find the "TRUTH" in the physical equation and the pattern will eventually reveal itself." (Randy Evangelista)
Imagine that the Universe is a very large painting and let’s call the complete image the “TRUTH”. We are a part of this “TRUTH”, like a speck of paint on a canvas. We cannot detach ourselves from the painting and expect to see a complete image. We cannot see the “TRUTH” from the outside looking in.
Next question, can we see the “TRUTH” from “consciousness”?
We are all made of stardust, so how is it even possible that something which comes from the same element that powers the furnace of the stars evolve to have consciousness? But then again, we are here, we experience it, so it is possible. However, to disassociate consciousness from the physical realm and make it a new entity is a challenge to our way of reasoning. To think that there is an entity out there that roams in our mind and visits us in our dreams to show us the equations of the universe is a captivating thought but at the same time very hard to embrace. Unless we call in another guest, one that is “DIVINE” in nature.
Another possibility is that we may never see the “TRUTH” in our present form since our senses are not enough to truly understand the universe. Our senses basically sets the limit on what we can comprehend. It’s almost like our DNA is genetically engineered to limit our capacity to acquire new knowledge. Just enough knowledge for us to survive but not enough to do something crazy and destroy ourselves or maybe we still need to evolve to a higher level of consciousness.
To give an example, the concept of TIME only exist because our brain can store memories which we use to differentiate between past, present and predict future events. Without this ability, the concept of TIME will cease to exist since there is no way to observe and measure it. Imagine a brain that only process present events with no memory of the past and cannot process future actions.
Our sense of “sight” combined with our ability to measure TIME gives us the concept DISTANCE. Without the two, the concept of DISTANCE will cease to exist since there is also no way to observe and measure it. Try this experiment, go to a dark room and at time = 0 (instantaneous) try to measure the dimensions of the room.
The very foundation of Newtonian Mechanics is based on these 2 concepts. Remove it and the whole Newtonian physics will collapse.