Loads of websites, articles and research papers are written about PHI and the golden ratio, and as many photo’s showing it being expressed in flowers, pine cones, shells, galaxy’s and even the human body, its relation to the Fibonacci Series Algorithm and it being part of ancient knowledge. But no-where do you find anything on HOW it comes about, we are given the impression that science knows because it can calculate its value, but far from it, they have no clue as to how it comes about in nature.
So let’s look at what they do know first.
Phi is the basis for the Golden Ratio, Section or Mean. The ratio, or proportion, determined by Phi (1.618 …) was known to the Greeks as the “dividing a line in the extreme and mean ratio” and to Renaissance artists as the “Divine Proportion”. It is also called the Golden Section, Golden Ratio and the Golden Mean.
In 1202 AD, Leonardo Fibonacci wrote in his book “Liber Abaci” of a simple numerical sequence that is the foundation for an incredible mathematical relationship behind Phi. This sequence was known as early as the 6th century AD by Indian mathematicians, but it was Fibonacci who introduced it to the west after his travels throughout the Mediterranean world and North Africa. He is also known as Leonardo Bonacci, as his name is derived in Italian from words meaning “son of (the) Bonacci”.
Starting with 0 and 1, each new number in the sequence is simply the sum of the two before it.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377 . . .
This sequence is shown in the right margin of a page in Liber Abaci, where a copy of the book is held by the Biblioteca Nazionale di Firenze.
Phi for “Neo-Phi-tes”:
Phi ( Φ = 1.618033988749895… ), most often pronounced fi like “fly”, is simply an irrational number like pi ( p = 3.14159265358979… ), but one with many unusual mathematical properties. Unlike pi, which is a transcendental number, phi is the solution to a quadratic equation.
A circle with a diameter of 1 and circumference of pi, 3.14 Phi, like Pi, is a ratio defined by a geometric construction.
Just as pi (p) is the ratio of the circumference of a circle to its diameter, phi ( ) is simply the ratio of the line segments that result when a line is divided in one very special and unique way.
Divide a line so that:
Sectioning a line to form the golden section or golden ratio based on phi,
the ratio of the length of the entire line (A)
to the length of larger line segment (B)
is the same as
the ratio of the length of the larger line segment (B)
to the length of the smaller line segment (C).
This happens only at the point where:
A is 1.618 … times B and B is 1.618 … times C.
Alternatively, C is 0.618… of B and B is 0.618… of A.
Phi with an upper case “P” is 1.618 0339 887 …, while phi with a lower case “p” is 0.6180339887, the reciprocal of Phi and also Phi minus 1.
What makes phi even more unusual is that it can be derived in many ways and shows up in relationships throughout the universe.
Phi can be derived through:
A numerical series discovered by Leonardo Fibonacci
Phi appears in:
The proportions of the human body
The proportions of many other animals
The solar system
Art and architecture
The stock market
The Bible and in theology
In the Bible:
The Ark of the Covenant uses Fibonacci numbers, approximating a Golden Rectangle.
In Exodus 25:10, God commands Moses to build the Ark of the Covenant, in which to hold His Covenant with the Israelites, the Ten Commandments, saying:
“Have them make a chest of acacia wood- two and a half cubits long, a cubit and a half wide, and a cubit and a half high.”
The ratio of 2.5 to 1.5 is 1.666…, which is as close to phi (1.618 …) as you can come with such simple numbers and is certainly not visibly different to the eye. The Ark of the Covenant is thus constructed using the Golden Section, or Divine Proportion. This ratio is also the same as 5 to 3, numbers from the Fibonacci series.
In Exodus 27:1-2, we find that the altar God commands Moses to build is based on a variation of the same 5 by 3 theme:
“Build an altar of acacia wood, three cubits high; it is to be square, five cubits long and five cubits wide.”
Note: A cubit is the measure of the forearm below the elbow.
Noah’s Ark uses Fibonacci Numbers in its Dimensions.
In Genesis 6:15, God commands Noah to build an ark saying;
“And this is the fashion which thou shalt make it of: The length of the ark shall be three hundred cubits, the breadth of it fifty cubits, and the height of it thirty cubits.”
Thus the end of the ark, at 50 by 30 cubits, is also in the ratio of 5 to 3, or 1.666…, again a close approximation of phi not visibly different to the naked eye. Noah’s ark was built in the same proportion as ten arks of the covenant placed side by side.
The Number 666 is related to Phi.
Revelation 13:18 says the following:
“This calls for wisdom. If anyone has insight, let him calculate the number of the beast, for it is a man’s number. His number is 666.”
This beast, regarded by some as the Anti-Christ described by John, is thus related to the number 666, one of the greatest mysteries of the Bible. Curiously enough, if you take the sine of 666º, you get -0.80901699, which is one-half of negative phi, or perhaps what one might call the “anti-phi”. You can also get -0.80901699 by taking the cosine of 216º, and 216 is 6 x 6 x 6.
The trigonometric relationship of sine 666º to phi is based on an isosceles triangle with a base of phi and sides of 1. When this triangle is enclosed in a circle with a radius of 1, we see that the lower line, which has an angle of 306º on the first rotation and 666º on the second rotation, has a sine equal to one-half negative phi.
The relationship of phi, the golden ratio, and 666.
In this we see the unity of phi divided into positive and negative, analogous perhaps to light and darkness or good and evil. Could this “sine” be a “sign” as well?
In addition, 666 degrees is 54 degrees short of the complete second circle and when dividing the 360 degrees of a circle by 54 degrees you get 6.66… The other side of a 54 degree angle in a right angle is 36 degrees and 36 divided by 54 is .666.
Phi appears throughout creation, and in every physical proportion of the human body. In that sense it is the number of mankind, as the mysterious passage of Revelation perhaps reveals.
The colours of the Tabernacle are based on a phi relationship.
The Phi Bar program produces the colours that the Bible says God gave to Moses for the construction of the Tabernacle.
As it says in Exodus 26:1, “Make the tabernacle with ten curtains of finely twisted linen and blue, purple and scarlet yarn, with cherubim worked into them by a skilled craftsman.”
Set the primary colour of the Phi Bar program to blue, the secondary colour of the Phi Bar to purple and it reveals the Phi colour to be scarlet.
This reference to the combination blue, purple and scarlet in the construction of the tabernacle appears 24 times in Exodus 25 through 39, describing the colours to be used in the curtains, waistbands, breast pieces, sashes and garments.
Phi as an insight to deeper spiritual connection and oneness.
Phi can be calculated in an iterative process, such as those shown in the equations below:
Φ = 1 + (1/Xn),e.g.,
The act of “part of the whole adding the whole onto itself” can be thought of as a “mathematical way of describing conscious recognition of being part of the whole”. In application, this would suggest that “when someone sees himself/herself as part of the world without personal attachment to it, and stays firmly in that perspective, then he/she will also become into a harmonic relationship with the world. One cannot help but to become in Phi relationship with the nature. This could suggest even a healing process”.
The Golden Section as a universal constant of design.
The teachings of most religions express the thought that part of God is within each of us and that we are created in His image. The pervasive appearance of phi throughout life and the universe is believed by some to be the signature of God, a universal constant of design used to assure the beauty and unity of His creation.
Changes in stock prices largely reflect human opinions, valuations and expectations. A study by mathematical psychologist Vladimir Lefebvre demonstrated that humans exhibit positive and negative evaluations of the opinions they hold in a ratio that approaches phi, with 61.8% positive and 38.2% negative.
In biology, once an egg is fertilized, it divides and multiplies in count until it reaches a point at which the ratio of the succeeding number of cells to the previous number of cells is phi (1.618 …).
Musical scales are related to Fibonacci numbers.
The Fibonacci series appears in the foundation of aspects of art, beauty and life. Even music has a foundation in the series, as:
- There are 13 notes in the span of any note through its octave.
- A scale is composed of 8 notes, of which the
- 5th and 3rd notes create the basic foundation of all chords, and are based on a tone which are combination of 2 steps and 1 step from the root tone, that is the 1st note of the scale.
Note too how the piano keyboard scale of C to C above of 13 keys has 8 white keys and 5 black keys, split into groups of 3 and 2. While some might “note” that there are only 12 “notes” in the scale, if you don’t have a root and octave, a start and an end, you have no means of calculating the gradations in between, so this 13th note as the octave is essential to computing the frequencies of the other notes. The word “octave” comes from the Latin word for 8, referring to the eight tones of the complete musical scale, which in the key of C are C-D-E-F-G-A-B-C.
In a scale, the dominant note is the 5th note of the major scale, which is also the 8th note of all 13 notes that comprise the octave. This provides an added instance of Fibonacci numbers in key musical relationships. Interestingly, 8/13 is .61538, which approximates phi. What’s more, the typical three chord song in the key of A is made up of A, its Fibonacci & phi partner E, and D, to which A bears the same relationship as E does to A. This is analogous to the “A is to B as B is to C” basis for the golden section, or in this case “D is to A as A is to E”.
Here’s another view of the Fibonacci relationship presented by Gerben Schwab in his YouTube video. First, number the 8 notes of the octave scale. Next, number the 13 notes of the chromatic scale. The Fibonacci numbers, in red on both scales, fall on the same keys in both methods (C, D, E, G and C). This creates the Fibonacci ratios of 1:1, 2:3, 3:5, 5:8 and 8:13:
8 notes of the octave scale
13 notes of the chromatic scale
Musical frequencies are based on Fibonacci ratios.
Notes in the scale of western music are based on natural harmonics that are created by ratios of frequencies. Ratios found in the first seven numbers of the Fibonacci series ( 0, 1, 1, 2, 3, 5, 8 ) are related to key frequencies of musical notes.
The calculated frequency above starts with A440 and applies the Fibonacci relationships. In practice, pianos are tuned to a “tempered” frequency, a man-made adaptation devised to provide improved tonality when playing in various keys. Pluck a string on a guitar, however, and search for the harmonics by lightly touching the string without making it touch the frets and you will find pure Fibonacci relationships.
* A440 is an arbitrary standard. The American Federation of Musicians accepted the A440 as standard pitch in 1917. It was then accepted by the U.S. government its standard in 1920 and it was not until 1939 that this pitch was accepted internationally. Before recent times a variety of tunings were used. It has been suggested by James Furia and others that A432 be the standard. A432 was often used by classical composers and results in a tuning of the whole number frequencies that are connected to numbers used in the construction of a variety of ancient works and sacred sites, such as the Great Pyramid of Egypt. The controversy over tuning still rages, with proponents of A432 or C256 as being more natural tunings than the current standard. And those of you who studied the articles will agree that 432 should be the standard.
The dimensions of the Earth and Moon are in Phi relationship, forming a Triangle based on 1.618.
The illustration shows the relative sizes of the Earth and the Moon to scale.
- Draw a radius of the Earth (1).
- Draw a line from the centre point of the Earth to the centre point of the Moon (square root of Phi).
- Draw a line to connect the two lines to form a Golden Triangle (Phi).
Using dimensions from Wikipedia and geometry’s classic Pythagorean Theorem, this is expressed mathematically as follows:
|Radius of Earth||6,378.10||1.000||A|
|Radius of Moon||1,735.97||0.272|
|Earth + Moon||8,114.07||1.272||B|
(Earth Radius +
Another way of looking at the relationship is to take 10320.77² / 8114.07², which is 106,518,293.39 / 65,838,131.96, which is 1.618.
This triangle is known as the Kepler triangle This geometric construction is the same as that which appears to have been used in the construction of the Great pyramid of Egypt.
Certain solar system orbital periods are closely related to phi.
Certain planets of our solar system seem to exhibit a relationship to phi, as shown by the following table of the time it takes to orbit around the Sun:
|Power of Phi||-3||-1||0||5||7|
Saturn reveals a golden ratio phi relationship in several of its dimensions.
The diameter of Saturn is very close to a phi relationship with the diameter of its rings, as illustrated by the green lines. The inner ring division is in a relationship that is very close to phi with the diameter of the rings outside the sphere of the planet, as illustrated by the blue lines. The Cassini division in the rings of Saturn falls at the Golden Section of the width of the lighter outside section of the rings.
A closer look at Saturn’s rings reveals a darker inner ring which exhibits the same golden section proportion as the brighter outer ring.
Venus and Earth reveal a golden ratio phi relationship.
Venus and the Earth are linked in an unusual relationship involving phi. Start by letting Mercury represent the basic unit of orbital distance and period in the solar system:
in km (000)
Curiously enough we find:
Ö Period of Venus * Phi = Distance of the Earth
Ö 2.5490 * 1.6180339 = 1.5966 * 1.6180339 = 2.5833
In addition, Venus orbits the Sun in 224.695 days while Earth orbits the Sun in 365.242 days, creating a ratio of 8/13 (both Fibonacci numbers) or 0.615 (roughly phi). Thus 5 conjunctions of Earth and Venus occur every 8 orbits of the Earth around the Sun and every 13 orbits of Venus.
Mercury, on the other hand, orbits the Sun in 87.968 Earth days, creating a conjunction with the Earth every 115.88 days. Thus there are 365.24/115.88 conjunctions in a year, or 22 conjunctions in 7 years, which is very close to Pi!
Relative planetary distances average to Phi.
The average of the mean orbital distances of each successive planet in relation to the one before it approximates phi:
|Degree of variance||(0.00043)|
Note: We sometimes forget about the asteroids when thinking of the planets in our solar system. Ceres, the largest asteroid, is nearly spherical, comprises over one-third the total mass of all the asteroids and is thus the best of these minor planets to represent the asteroid belt.
The shape of the Universe itself is a dodecahedron based on Phi.
New findings in 2003 based on the study of data from NASA’s Wilkinson Microwave Anisotropy Probe (WMAP) on cosmic background radiation reveal that the universe is finite and shaped like a dodecahedron, a geometric shape based on pentagons, which are based on phi. You should also look at http://www.scholarpedia.org/article/Cosmic_Topology
The DNA molecule, the program for all life, is based on the golden section. It measures 34 angstroms long by 21 angstroms wide for each full cycle of its double helix spiral.
34 and 21, of course, are numbers in the Fibonacci series and their ratio, 1.6190476 closely approximates phi, 1.6180339.
B-DNA has spirals in phi proportions.
DNA in the cell appears as a double-stranded helix referred to as B-DNA. This form of DNA has a two groove in its spirals, with a ratio of phi in the proportion of the major groove to the minor groove, or roughly 21 angstroms to 13 angstroms.
The DNA cross-section is based on Phi.
It has been reported but not yet confirmed that a cross-sectional view from the top of the DNA double helix forms a decagon:
A decagon is in essence two pentagons, with one rotated by 36 degrees from the other, so each spiral of the double helix must trace out the shape of a pentagon.
The ratio of the diagonal of a pentagon to its side is Phi to 1. So, no matter which way you look at it, even in its smallest element, DNA, and life, is constructed using phi and the golden section!
Plants illustrate the Fibonacci series in the numbers and arrangements of petals, leaves, sections and seeds. Plants that are formed in spirals, such as pinecones, pineapples and sunflowers, illustrate Fibonacci numbers. Many plants produce new branches in quantities that are based on Fibonacci numbers.
Fibonacci numbers in plant branching.
Here a sunflower seed illustrates this principal as the number of clockwise spirals is 55 (marked in Red, with every tenth one in white) and the number of counter clockwise spirals is 89 (marked in Green, with every tenth one in white).
Here a plant illustrates that each successive level of branches is often based on a progression through the Fibonacci series.
Fibonacci numbers in plant sections.
You might expect symmetry in plants, but if you cut a fruit or vegetable you will often find that the number of sections is a Fibonacci number:
Bananas have 3
Apples have 5
Fibonacci numbers in flower petals.
Many flowers have petals that total a number in, or very close to, the Fibonacci series:
5 Buttercups, Roses
21 Black-eyed susans
And in math.
Phi, Φ, 1.618…, has two properties that make it unique among all numbers.
If you square Phi, you get a number exactly 1 greater than itself: 2.618…, or
Φ² = Φ + 1.
If you divide Phi into 1 to get its reciprocal, you get a number exactly 1 less than itself: 0.618…, or
1 / Φ = Φ – 1.
These relationships are derived from the dividing a line at its golden section point, the point at which the ratio of the line (A) to the larger section (B) is the same as the ratio of the larger section (B) to the smaller section (C).
This relationship is expressed mathematically as:
A = B + C, and
A / B = B / C.
Solving for A, which on both sides give us this:
B + C = B²/C
Let’s say that C is 1 so we can determine the relative dimensions of the line segments. Now we simply have this:
B + 1 = B²
This can be rearranged as:
B² – B – 1 = 0
Note the various ways that this equation can be rearranged to express the relationship of the line segments, and also Phi’s unique properties:
B2 = B + 1
1 / B = B – 1
B2 – B1 – B0 = 0
Note: Bx means n raised to the x power. Some browsers may not display exponents as superscripts or raised characters.
Now we have a formula that can be solved using the Quadratic formula. This formula allows you to solve a quadratic equation for an unknown x, with a, b, and c as constants. A quadratic equation has this form:
ax² + bx + c = 0
The solution to this is found with the quadratic formula:
So our formula for the golden ratio above (B2 – B1 – B0 = 0) can be expressed as this:
1a2 – 1b1 – 1c = 0
The solution to this equation using the quadratic formula is (1 plus or minus the square root of 5) divided by 2:
( 1 + √5 ) / 2 = 1.6180339… = Φ
( 1 – √5 ) / 2 = -0.6180339… = -Φ
The reciprocal of Phi (denoted with an upper case P), is known often as by phi (spelled with a lower case p).
Phi, curiously, can also be expressed all in fives as:
5 ^ .5 * .5 + .5 = Φ
I could fill page after page but this should be more than enough for you to see that this ratio ( and others mentioned in the articles) so let’s move to the cause of it, the how it comes to be! For this you need to look at the cell division again. Where the one (circle) becomes two.
As you can see the single circle in the square is placed for half its size over the other circle so that the size if the diameter is two cubit now makes a total of 1,5 in diameter. If the radius of the circle and square is one cubit then the total height is 3 cubit. The problem here is that you cannot fit one part of the circle (say it is a solid ) in the other circle. Now imagine this space is full and I mean FULL of energy, a force, then one part must be forced out. For this reason I divided the part of the Vesica Pisces in two, or half a cubit, as you know the cubit is 5,236 therefore half a cubit is equal to,2,618 yes indeed Phi.
Now where would this energy or force escape from as there is no space for it? Right there where the circles join, and at the very points where the hexagram and pentagram join with its 60 and 36 degrees making 96.
The first drawing was a 2D presentation, above you have the 3D version of it.
But let’s remember that the eight rotates within the bigger circle and now look at how this force would look like, taking that in consideration.
Although this drawing isn’t correct you will get the idea and see it reflected in the galaxy’s and how the Jing yang sign represents these two opposing forces. and originate from the same knowledge.
While the math is a bit more complicated than is presented here, it is sufficient for the purpose of this article.
Over the years more and more researchers have come to see that this structure plays a vital role in the creation of matter and the universe, while they all like to give it their own twist and name, very few are willing to look at a way to join forces, as each branch of science would find their questions answered in it. Which brings us to the real important issue here, and that is, we are made in his image therefore looking within makes us see what is out there. And I like to say, I think therefore I am NOT(hing), but a passing of a shadow on the wall of Plato’s cave.
If we really want to make this big evolutionary jump we will need to embrace this inner science with all our might.
Moshiya van den Broek