A quote from Joseph Knecht, the Magister Ludi, in "The Glass Bead Game";
ah....Herman Hesse's Nobel Prize winning 1942 novel, I've
got Steppenwolf and Sidhartha by Hesse........ when Mike
came to visit I had Siddhartha out and he commented about
that book.
in Shakyumani Budhha's (the awakened one)recounting of his
own story to his close disciple Ananda, he had
spent many lives he was a bodhisattva, one who is on the
path of awakening. when he came back the last time as
being enlighted, he was born and on the 5th day was
named Siddhartha which means, "Acomplishment of
the Goal this is from "The Buddha and an introduction
to his teachings"
It's interesting that you start talking about Hesse because
he was such an explorer of our world, our consensus
reality and the different systems of thought.
I was talking with Barbara Rockefeller last month at
the TAG 22 meeting in Dallas, she worked at Citi
as well we got to talking about Andy Kreiger,
the BT FX trader who made the 300 million (later lowered
downward a fair bit) back in 1987-88. He was a front
page story in the WSJ and it was interesting to
note that he had studied Sanskrit extensively, and had
probably seen some of these mathematical concepts right
from the Hindu texts.
Krieger took his BT bonus(a large part of the WSJ coverstory
was that no one on Wall Street had a very good formula
calculate his bonus unless it was a 30-40 million dollar
one) and so Krieger took some of the money and
started a Foundation named "Karma" again a hindu concept.
Since the Arabs learned their numbering systems and so much
math from the Hindu's and then the Europeans learned
their math from the Arab's. this and the knowledge of
PI and PHI is chronicled in a neat little article below
the line in a couple of paragraphs.
The 20th century was one of realization that a cross-
disciplinary approach of combining scientific disciplines,
like Astronomy, biology, chemistry, Mathematics, Music
Theory and many others could produce some
inherent common ground.
Fractals, could appear in the lay out of the universe,
the way a tree grows and also in area's of mass psychology
such as the financial markets. Fractals unfold in
logrithmic spiral proportions which adhere to the
Greek symbol PHI PHI = 1.618
Thus all the folks going Fibonacci retracement and
projection levels in the markets. Many of Gann's ideas
where dealing with some of these types of mathematical
relationships.
Here is some back ground on PHI in nature......
and let me tell you and anyone else reading.
this stuff made absolutely no sense to me when I started reading it 17-20 years ago, or only a shadow of a
bit of sense.
It was like a 1000 piece jigsaw puzzle with 25 pieces
in place around the perimeter. Today I have many more
pieces in place, where I can kind of see what so many
people are talking. It takes Time to assimilate
many of the things we see in life
John
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Phi in Nature. The mathematical properties of Phi have caused the value to appear repeatedly in nature. Rotation of plant leaves on a stalk follow the ratio phi which insures that the same position is not used twice, and thus affording optimal exposure of underlying leaves to sunlight. Spiral seashells (typified by the ancient organism the nautilus) expand at the ratio of phi, which insures the partitions never align, providing optimal structural strength. (The massive blocks used in building the lower levels of the Great Pyramid were overlapped by the ratio phi to optimize stability.) The number of pedals of many flowers, such as the daisy and aster, are related to the diameter of the pod by the number phi. Spiral formations such as in the sunflower pod, pine cones, and lettuce heads correspond with radius vectors generated by nested golden section rectangles. These examples illustrate how the force of ORDER operates upon this ratio, and the mathematical reason for this attraction is detailed in the section below on the powers of Phi.
Today the pleasing proportion of a snapshot is 3x5, an approximation of the Golden Section. People recognize the ratio phi without knowing it, responding to an innate sense of balance. In a photo, objects trailing off the edges look best when intersecting an edge at the phi ratio. Perceptions of beauty are based on the number phi in the features of a face. Symmetry considerations in perceptions of beauty combine with golden section rectangle intersection zones in defining attractive proportions in the features.
Plato wrote in Timaeus that this formula was the key to understanding the physics of the cosmos and the operation of natural processes.
Leonardo Fibonacci, a mathematician of the 11th century, was introduced to the Arab version of the Hindu system of decimal numerals, and in turn introduced it into Europe where the clumsy Roman numerals were still in use. Fibonacci was exposed in Egypt to the formulas (phi + 1 = phi2) and (1 + 1/phi = phi). These formula are both specific cases of the general formula phin = phin-1 + phin-2.
In the first formula of phi + 1 = phi2 if you divide by phi you obtain phi/phi + 1/phi = phi2/phi, which reduces to 1 + 1/phi = phi, the second formula. The first formula is the specific case of the general formula governing the powers of phi, (that phin = phin-1 + phin-2) using the powers phi2 = phi1 + phi0, where phi0 is shown as 1. The second formula in effect describes the case where the length of the long segment is 1 (in other words, setting phi=1), and defines the short segment as the reciprocal of the entire line. So the short segment is 1/1.618 = .618. The appearance of the decimal .618 in both phi and its reciprocal reflects the symmetry of phi, the "natural balance" number.
. The Fibonacci Series. The first formula (phi + 1 = phi2) is very useful when building with phi as a basic unit. It means there is a series of numbers which relate to each other by the value phi. When you build with these numbers for relevant dimensions, the ratio phi will exist between them. This series of numbers is known as the Fibonacci Series which, beginning with the numbers 1 and 2, progresses to a number which is the sum of the two prior numbers in the series. 1,2 ; 3,5,8,13,21,34,55,89,144,233,377, ...
Each new number has a ratio to the previous number which gets ever closer to the value of phi (to seven decimals 1.6180339 as shown below).
1, 2 ; 3, (1.66) 5, (1.6) 8, (1.625) 13, (1.615) 21, (1.619) 34, (1.6176) 55, (1.61818...) 89, (1.61798) 144, (1.61805) 233, (1.61803) 377, (1.618037) 610, (1.618033) 987, (1.6180344) 1597, (1.6180338) 2584, (1.6180340) 4181, (1.6180339) 6765, (1.6180339) 10946 (1.6180339) 17711 ......
Notice that in the formula 1 + 1 / (1 + 1 / (1 + 1 / (1 + ... ...))) = phi, working out this number from a finite portion of the fraction, we have:
1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / 1))))) =
1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / (1 + 1 / 2)))) =
1 + 1 / (1 + 1 / (1 + 1 / (1 + 2 / 3))) =
1 + 1 / (1 + 1 / (1 + 3 / 5)) =
1 + 1 / (1 + 5 / 8) =
(1 + 8/13) = 1.615
You can see the Fibonacci number pairs in the final fractions of each step of the reduction. In effect, the reduction of the infinite fraction produces the Fibonacci series in reverse.
. Phi in the Great Pyramid. The paired numbers beginning with 3 and 5 can be used with increasing precision to build highly stable structures utilizing the ratio phi. The convenient 3-4-5 right triangle (32 + 42 = 52 or 9 + 16 = 25) was used by ancient Egyptian builders extensively. Of these paired numbers it is the repeating value of 1.618181818... from 55 and 89 which was used in building the Great Pyramid.
A geographical degree is 1/360 the circumference of the Earth at the equator. Using geographical degrees as a measure, the base of the pyramid is exactly 1/480 of one degree. The apothem (line from peak to midpoint of base) is close to 1/600 of a geographic degree, but not exactly. There are 100 geographic feet in a geographic second, so the base is exactly 750 geographic feet (3600 seconds in a degree, so 3600/480= 7.5 seconds of 100 feet each, yielding 750 feet). The apparent reconstructed apothem of the capstone-stripped pyramid is about 606.8 geographic feet. In the triangle formed by the apothem and the center vertical line (height), the height is the square root of the sum of the square of the apothem and the square of half the base; (apothem2 = height2 + base-line-from-center2); height2 = (606.82 - 3752); height = 477.
The ancient Egyptian measure of the cubit reveals directly how the Fibonacci Series was utilized. Applying the Egyptian measure of 440 cubits for the base, the apothem is 606.818 x 440/750 = 355.999 cubits. Using 356 cubits, the height (height2 = 3562 - 2202) works out to 280 cubits, and the proportions are (half-base = 220/220 =1), (height = 280/220 = 1.272727 = root of phi), and (apothem =356/220 = 1.6181818 = phi), or (1, square-root of phi, and phi). This reveals that the builders used 4 x 55 = 220 for the half base and 4 x 89 = 356 for the apothem, the two sequence numbers (55 and 89) on either side of the ratio value 1.61818 in the Fibonacci series.
As a result of these dimensions the face of the pyramid is (base/2 x height) [where height of face is pyramid's apothem] = 220 x 356 = 78320. This converts to (220/220 x 356/220) = (1 x phi) = phi, as the value of the area of the face of the pyramid. Thus the face of the pyramid has an area equal to the square of the height of the pyramid (since the height is the square-root of phi). These formulas reveal the balance of the Great Pyramid rests on the golden section applied to its dimensions. But they also lead to the number pi, which is the transcendental fraction relating the square to the circle. Taking the square of the pyramid base and a circle with the radius of the height arrives at a nearly equal circumferences for both the square and the circle. The circumference of the square is 440 x 4 = 1760, and knowing the pyramid squares the circle you can calculate the approximate value of pi as a ratio of the diameter from the formula (circumference = diameter x pi). In fact the circumference divided by the diameter is 1760/560 = 22/7 = 3.1428, just slightly over the value of pi. The convenience number of 22/7 for pi was extensively utilized in Egyptian construction.
Oddly enough one can calculate the circumference of a circle from the diameter using the Fibonacci series, and the formula (pi = phi2 x 6/5). For example, if the diameter is 34 then since each number in the series increases by phi, two numbers in the series increase by phi2, and by substitution the circumference is 89 x 6/5 = 106.8 and pi has a value of 106.8/34 = 3.1412. Because of this, building with Fibonacci series dimensions, such as with the Great Pyramid using the dimensions of (4 times) 55 and 89, provides a built-in conversion that squares the circle. |
