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Tuesday, August 16, 2011

Phi, The Golden Ratio, and the Fibonacci Sequence


                              
Here is a list of what phi is:
  1. The 21st letter of the Greek Alphabet
  2. The Golden Ratio (in Mathematics, Art, and Architecture)
  3. Electric Potential (in Physics)
  4. The Work Function (in Electronics)
  5. A Representation of Groups of Atoms in Molecules (in Organic Chemistry)
  6. The Fugacity Coefficient (in Thermodynamics)
  7. Navigation and Latitude (in Cartography/Map Making)
  8. The Equivalence Ratio (in Combustion Engineering)
  9. Representation of a Sentence (in Symbolic Logic)
  10. Porosity (in Geology and Hydrology)
  11. Strength (in Structural Engineering)
  12. The Roll Angle (in Flight Dynamics)
  13. A Generic Act (in Philosophy)
  14. Magnetic and Electric Flux (in Physics)
  15. The Cumulative Distribution Function (in Mathematics and Statistics)
  16. The Number of Phases in a Power System (in Electrical Engineering)
So, as you can see, we have many uses for Phi.

THE GOLDEN RATIO

QUOTED FROM WIKIPEDIA:

"In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant that is approximately 1.61803398874989 (but goes on infinitely never repeating itself). Other names frequently used for the golden ratio are the golden section (Latin: sectio aurea) and golden mean.Other terms encountered include extreme and mean ratio,[5] medial sectiondivine proportiondivine section (Latin: sectio divina), golden proportiongolden cut,[6] golden number, and mean of Phidias.In this article the golden ratio is denoted by the Greek lowercase letter phi (\varphi \,), while its reciprocal, 1/\varphi \, or \varphi^{-1} \,, is denoted by the uppercase variant Phi (\Phi\,).

The figure below illustrates the geometric relationship that defines this constant. Expressed algebraically:
 \frac{a+b}{a} = \frac{a}{b} \equiv \varphi\,.
This equation has one positive solution in the set of algebraic irrational numbers:
\varphi = \frac{1+\sqrt{5}}{2} = 1.61803\,39887\ldots.\, [1]
At least since the Renaissance, many artists and architects have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing. Mathematicians have studied the golden ratio because of its unique and interesting properties. The golden ratio is also used in the analysis of financial markets, in strategies such as Fibonacci retracement.


The 9 edge-transitive convex polyhedra: 
ALL OF THE FOLLOWING CONTAIN WITHIN THEM THE GOLDEN RATIO OF PHI
AT EVERY EDGE; THEY CANNOT BE FORMED CORRECTLY WITHOUT PHI
  1. Tetrahedron (Fire)
  2. Hexahedron (Earth)
  3. Octahedron (Air)
  4. Icosahedron (Water)
  5. Dodecahedron (Ether/Universe/Consciousness)
  6. Cuboctahedron (see image below)
  7. Rhombic Dodecahedron (see image below)
  8. Icosidodecahedron (see image below)
  9. Rhombic Triacontahedron (see image below)



Cuboctahedron: an Archemedian Solid; 

It is one of nine edge-transitive convex polyhedra. 
Edge-transitive means that there is only one type of edge to the object; 
given two edges, whatever happens on one edge will be translated or reflected onto every other edge, 
without changing the original region.

Formed by merging the Cube (Hexahedron) and the Octahedron.


Rhombic Dodecahedron:
This is formed by merging two Cuboctohedrons;
It is how the bee's honeycomb is formed


ICOSIDODECAHEDRON:
This is the closest we can come to the
shape of the earth geometrically.
Looking back at the Earth's Electro-Magnetic Field,
and the maps I have provided, (under the title "Earth") 
this is where those shapes (our conscious electro-magnetic field and ley lines) come from.
Icosidodecahedron

Please click here to see how the golden ratio represents the geometric relation between all ancient sites on the planet.
Rhombic Triacontahedron:
This is the duality of the Icosidodecahedron


Johannes Kepler:

"Geometry has two great treasures: one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first, we may compare to a measure of gold; the second, we may name a precious jewel" (Johannes Kepler)


Kepler's Model of the Solar System:



THE GREAT PYRAMID:

Vesica Piscis:

The Vesica Piscis from the Seed of Life & The Golden Ratio:



The Egyptians Knew This Stuff!

The Eye of Horus:



THE FIBONACCI SPIRAL:
This is the representation of PHI
This is the Fibonacci Sequence:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 
233, 377, 610, 987, 1597, 2584, 4181, 6765, 
10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, ...
The 2 is found by adding the two numbers before it (1+1);
The 3 is found by adding the two numbers before it (1+2);
And the 5 is found by adding the numbers before it (2+3);
this can go on forever in either direction (negative numbers).


Here is Pascal's triangle, which is very closely related to the Fibonacci sequence; 
you'll see this later on.


It is found in the Yin-Yang:




Here, the spiral represents FEMALE ENERGY;
and the straight lines, turning at the corners, represent MALE ENERGY.
At every 90 degree angle, the MALE and FEMALE energy meet.
MALE AND FEMALE ENERGY:

"First I would like to say something about the nature of the male and the female. These energies are two aspects of the One. They therefore are not really opposed or dualistic, they are one; they are two faces of one energy.

The male energy is the aspect that is outwardly focused. It is that part of God or Spirit that drives outward manifestation, that makes Spirit materialize and take form. The male energy therefore knows a strong creative force. It is natural to the male energy to be highly focused and goal-oriented. In this manner the male energy creates individuality. The male energy allows you to separate yourself from the One, from the Whole and to stand alone and be a specific individual.

The female energy is the energy of Home. It is the energy of the Primal Source, the flowing Light, pure Being. It is the energy that has not yet manifested, the inner aspect of things. The female energy is all encompassing and oceanic; it does not differentiate or individualize.

Now, imagine the energy of the female becoming aware of a certain movement inside of her, a slight restlessness, a desire for… reaching out, outside of her boundaries, moving outside of herself to attain experience. There is a longing for something new, for adventure! And then an energy comes to her that answers that longing. It is the male energy that wants to be of service and help her manifest in matter, in form. The male energy defines and shapes the female energy and by their cooperation the total sum of energies can take a completely new direction. A new reality can be created in which everything can be explored and experienced, in ever changing forms of manifestation.

The dance of the male and the female brings forth the fluctuating spectacle of created reality, of your creation. This is a spectacle of great beauty wherein the male and female energy worship each other and celebrate their cooperation and playful joining. And this is as it should be. The male and female energy belong together, they are two aspects of the One and together they celebrate the joyful manifestation that Creation is supposed to be.

It has been said that in the final realization of who you are, the only truth that matters is: I AM. And in this mystical mantra, precisely those two aspects merge. In the I is the male energy, in the AM is the female energy. The I is constricting, differentiating. It gives focus, it gives direction, it individuates: I, not the other, I. And then AM. AM is oceanic, all encompassing. It reflects the ocean of Home, the female energy, the inexhaustible source that knows no bounds, no differentiation. The flowing and joining aspect is the core of the female energy. In the I AM, the male and female come together and blissfully join their energies."
[FOR A CONTINUATION OF THIS QUOTE, CLICK HERE]

It can be found everywhere; from humans, to plants, to animals, to galaxies. It is the mathematical sequence of the seed of life. It is representative of a never-ending holographic universe we just so happen to live inside of. Everything is one. Everything is eternal.

^^(This is a great site.)^^


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