Morya’s Comments on the Golden Mean

The Golden Mean brings grief and joy into balance.

Again we come to the need for balance, called by Our Teacher the golden mean, which may also be seen as the fullness of understanding.

In everything one can study the golden mean, the very path of justice.

A proper relationship between the impulsiveness of the individuality and the infallibility of natural law is the golden mean, which gleams in the depth of each expanded consciousness.

Please point out to the newcomers that every Teaching advises the Golden Mean.

The Royal Path or the Path of Balance. This Middle Way, or Golden Mean, was also advocated by all the great Teachers of humanity.

The Golden Mean, or Path of Great Equilibrium, has been decreed by all the Great Teachers.

It is said in the Teaching that a man who does not realize what is co-measurement cannot be considered spiritual. Co-measurement is the Golden Mean

Buddha pointed out that the golden mean, or the middle path, should be understood as the realization of harmony.

Therefore, all the Teachings have always stressed the Golden Mean, or Balance. All extremes are harmful. In ancient Teachings the "Golden Mean," or Equilibrium, was indicated. And those who wanted to approach the great knowledge were expected not to go to any extremes. Nothing is so much distorted as this concept of Equilibrium.

The Thinker used to say, "Let the Golden Mean indicate the right measure of needed strength."

The body is man's only instrument for the accumulating of the new spiritual possibilities. In the ancient Teachings the foundation of all achievements was wisely linked with the great Golden Path, or the Golden Mean.

People sometimes attempt to advance by leaps, prompted by fear or prejudice or by their passions, but it is impossible to advance by leaps. A steady, systematic motion is needed in everything, and only through the Golden Mean can one progress.

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Golden Mean:

About 300 B.C., Euclid calls dividing a line at the .6180399.. point dividing a line in the extreme and mean ratio. This later gave rise to the name Golden Mean. mes.surrey.ac.uk

Luca Pacioli wrote a book call The Divine Proportion in 1509. It contains drawings made by Leonardo da Vinci. It was probably da Vinci who first called it the “golden section.” mes.surrey.ac.uk

The history of western art keeps using mysterious phrases like “Golden Mean,” “Golden Proportion,” “Golden Rectangle,” and “Golden Section.” Then came Fibonacci Numbers and their relationship to the Golden Section and how the Golden Proportion is within us and around us everywhere. For centuries, artists, architects and engineers have studied and used the Divine Equation. spyrock.com

A Golden Rectangle is a rectangle in which the ratio of the length to the width is the Golden Ratio. If one side of a Golden Rectangle is 2 feet long, the other side will be approximately equal to 2 times (1.62) = 3.24. If you have a Golden Rectangle and you cut a square off it so that what remains is a rectangle, that remaining rectangle will also be a Golden Rectangle. You can keep cutting these squares off and getting smaller and smaller Golden Rectangles. mathform.org

The Golden Mean goes on forever and ever. The whole universe is based on that mathematical proportion … your body, the fish, the trees, the galaxy, tornadoes and the flow of wind and water. Any rectangle constructed to Golden Mean proportion holds what is called the golden spiral coiled within it. Unlike a regular spiral, the distance between the golden spiral’s coils keeps increasing, growing wider as it moves away from the center. This is the spiral of constant expansion and growth. It is found in all of Nature, from galaxies to ram’s horns. The golden spiral is the template of growth, the mathematical formula for evolution. archdome.com

Phi: 1.618033988749894848204586834365638117720…

The American mathematician Mark Barr used the Greek letter phi to represent the golden ratio. The latest calculation of Phi took three hours of computation to find 1.5 billion places in May 2000. Neither the decimal form of Phi, nor the binary one nor any other base have any ultimate repeating pattern in their digits. Phi has the value (sq.rt. of 5 + 1)/2 and phi is (sq.rt. of 5 - 1)/2. Later we will show why Phi and phi cannot be written as exact fractions. The value of phi is the same as Phi but begins with 0.6.. instead of 1.6.. Such values are called irrational since they cannot be represented as a ratio of two whole numbers (i.e. a fraction). A simple consequence of this is that their decimal fraction expansions go on for ever and never repeat at any stage. mes.surrey.ac.uk

Arithmetic proportion exists when a quantity is changed by adding some amount. Geometric proportion exists whan a quantity is changed by multiplying by some amount. Phi possesses both qualities. The resulting proportion is both arithmetic and geometric. It is thus perfect proportion; you could think of it as the place on some imaginary graph where the curved line of multiplication crosses the straight line of addition. So it is not surprising that phi turns out to be an ideal rate of growth for things which grow by adding some quantity - the Nautilus shell grows larger on each spiral by phi. The sunflower has 55 clockwise spirals overlaid on either 34 or 89 counterclockwise spirals, a phi proportion. vashti.net

One can derive Phi mathematically by solving the equation of 1 plus the square root of 5 divided by 2. If you divide phi into 1, you get a number exactly 1 less than phi: 0.61804… evolutionoftruth.com

Phi and Fibonacci Numbers:

The Ratio of neighboring Fibonacci Numbers tends to Phi - and soon settles down to a particular vaalue near 1.6. In fact, the exact value of Phi, and the larger Fibonacci numbers, the closer their ratio is to Phi. mes.surrey.ac.uk

The Fibonacci Numbers are: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946 etc. Each number is the sum of the previous two numbers. Except for the number 1, any two adjacent numbers in the series, when seen as a ratio, are very close to the Golden Mean: 89/144 = 0.61805555556; when seen as fractions, they make it easy to remember how to create or identify the Golden Section: 2/3, 5/8, 8/13, etc. spyrock.com

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765 … The ratio between any two Fibonacci Number approaches a limit as the numbers get larger, and that limit is the Golden Ratio. Thus, 6765/4181 (the 20^th and 19^th Fibonaccis) is 1.618033963, which only differs from the Golden Ratio by 0.000000025. friesian.com

Where

A recent confirmation of the vitality of the Golden Mean as an organizing principle of this emerging art movement can be found in a group of crop circle patterns that appeared in southwest England during the summer of 1996. Many of these huge formations made prominent use of sacred geometry. heall.com

The family trees of rabbits, cows and bees, sea shell shapes, branching plants, flower petals and seeds, leaves and petal arrangements on pineapples and in apples, pine cones and leaf arrangements - all involve the Fibonacci numberss. Botanists know that cells become new branchs, flowers, petals and stamens. The amazing thing is that a single fixed angle can produce the optimal design no matter how big the plant grows. What is the fixed angle of turn - it is Phi cells per turn or phi turns peer new cell. The arrangements of leaves is the same as for seeds and petals. All are placed at 0.618034.. leaves, seeds, petals per turn. In terms of degrees this is 0.618034 of 360^o which is 222. 492…^o. However we tend to see the smaller angle which is 137.50776..^o. If there are Phi leaves per turn or equivalently, phi = 0.618.. per leaf, then we have the best packing so that each leaf gets the maximum exposure to light, casting the least shadow on the others. This also gives the best possible exposure to falling rain so that rain is back along the leaf and down the stem to the roots. For flowers or petals, it gives the best possible exposure to insects to attract them for pollination. The whole of the plant seems to be based upon the golden number. The Golden Section has been used in many designs, from the ancient Parthenon in Athens (400 B.C.) to Stradivari’s violins. Stradivari was aware of the golden section and used it to place the f-holes in his famous violins. It was known to artists such as Leonardo da Vinci and musicans and composers. Analysis of many of Mozart’s sonatas are divided into two parts exactly at the golden section points an almost all cases. Beethoven’s Fifth Symphony uses the Golden Mean point 0.618 of the way through the symphony (bar 372) and also at the start of the recapitulation which is phi or 0.382 of the way through the piece. It also appears that Bartok, Debussy, Schubert, Bach and Satie used this design. Virgil consciously used Fibonacci numbers to structure his poetry, especially in his Aeneid. Other Roman poets of the time used these numbers also. In films, the Russian director Sergie Eisenstein directed his movies and divided the film up using golden section points to start important scenes in the film, measuring by the length on the film. mcs.surrey.ac.uk

Musical scales are based on Fibonacci numbers. 13 notes separate each octave of 8 notes in a scale, of which the 5^th and 3^rd notes create the basic foundation of all chords, and are based on whole tone which is 2 steps from the root tone, this is the 1^st note of the scale. Note too how the piano keyboard scale of 13 keys had 8 white keys and 5 black keys, split into groups of 3 and 2. Notes in the scale of western music have a foundation in the Fibonacci Series. The climax of songs is often found at roughly the phi point (61.8%) of the song. In a 32 bar song, this would occur in the 20th bar. Musical instruments are often based on phi.

The average of the mean orbital distances of each successive planet in relation to the one before it approximates phi - averaging 1.61874.

Phi, the Golden Mean and Fibonacci numbers have been used with great success to analyze and predict stock market moves. Fibonacci numbers define the movements of stocks. In Elliot Wave Theory theory, all major market moves are described by a five-wave series. The classic Elliot Wave series consists of an initial wave up, a second wave down (often retracing 61.8% of the initial move up), then the third wave (usually the largest) up again, then another retracement, and finally the fifth wave, which would exhaust the movement. In addition, each of the major waves (1, 3, 5) could themselves be separated into subwaves, and so on, and exhibit other Fibonacci relationships.

A peaceful heartbeat is said to beat in a Phi rhythm. A normal human heat beats in a phi rhythm, with the T point of normal electrocardiogram (ECG or EKG) falling at the phi point of the heart’s rhythmic cycle. While the heartbeats vary, some believe that a heartbeat that reflects this perfect phi relationship represents a state of being that is one of health, peace and harmony. Human expectations occur in a ratio that approaches Phi. Changes in stock prices largely reflect human opinions, valuations and expectations. 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. evolutionoftruth.com

Part 2 continued next month

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