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> Phi, TESSELATION.
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post Mar 09, 2005, 08:20 PM
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TESSELATION

Phi

Phi = 1.61803399887

The Golden Ratio is a geometric proportion discovered in antiquity that turns up in sculpture, botany/leaves/seashells, and galaxies/planetary orbits, and in the thermodynamics of black holes.

Nature appears to have chosen the logarithmic spiral as one of its favourite shapes.

The golden ratio is embedded in such geometric forms as the cube, dodecahedron (twelve faces), icosahedrons (twenty faces), octahedron (eight faces), and tetrahedron (four faces).

Euclid stumbled upon its significance in antiquity.

Make a square inside a rectangle and the draw a square outside it. The new rectangles are also golden rectangles, and the ratio between the squares is the golden ratio.

Leonardo Da Vinci, an Italian 15th century artist, inventor, and sculptor, rediscovered the balanced perfection of the golden rectangle and pencilled it into his masterpieces. Connecting a curve through the concentric golden rectangles, you generate the mythical golden spiral.
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post Mar 18, 2005, 06:03 PM
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http://www.michaelbach.de/ot/mot_reverse-phi/index.html
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post Mar 24, 2005, 07:44 PM
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The Golden Ratio

The golden ratio (a.k.a. phi ratio a.k.a. sacred cut a.k.a. golden mean a.k.a. divine proportion) is another fundamental measure that seems to crop up almost everywhere, including crops.

(The golden ratio is about 1.618033988749894848204586834365638117720309180...)

The golden ratio is the unique ratio such that the ratio of the whole to the larger portion is the same as the ratio of the larger portion to the smaller portion. As such, it symbolically links each new generation to its ancestors, preserving the continuity of relationship as the means for retracing its lineage.

The golden ratio has some unique properties and makes some interesting appearances:

phi = phi^2 - 1; therefore 1 + phi = phi^2; phi + phi^2 = phi^3; phi^2 + phi^3= phi^4; ad infinitum.

phi = (1 + square root(5)) / 2 from quadratic formula, 1 + phi = phi^2

phi = 1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1/...)))))...

phi = (sec 72)/2 =(csc 18)/2 = 1/(2 cos 72) = 1/(2 sin 18) = 2 sin 54 = 2 cos 36 = 2/(csc 54) = 2/ (sec 36) for all you trigonometry enthusiasts

ratio of segments in 5-pointed star (pentagram) considered sacred to Plato & Pythagoras in their mystery schools. Note that each larger (or smaller) section is related by the phi ratio, so that a power series of the golden ratio raised to successively higher (or lower) powers is automatically generated: phi, phi^2, phi^3, phi^4, phi^5, etc.

phi = apothem to bisected base ratio in the Great Pyramid of Giza

phi = ratio of adjacent terms of the famous Fibonacci Series evaluated at infinity; the Fibonacci Series is a rather ubiquitous set of numbers that begins with one and one and each term thereafter is the sum of the prior two terms, thus: 1,1,2,3,5,8,13,21,34,55,89,144...

(interesting that the 12th term is 12 "raised to a higher power", which appears prominently in a vast collection of metaphysical literature)

The mathematician credited with the discovery of this series is Leonardo Pisano Fibonacci and there is a publication devoted to disseminating information about its unique mathematical properties, The Fibonacci Quarterly

Fibonacci ratios appear in the ratio of the number of spiral arms in daisies, in the chronology of rabbit populations, in the sequence of leaf patterns as they twist around a branch, and a myriad of places in nature where self-generating patterns are in effect.

The sequence is the rational progression towards the irrational number embodied in the quintessential golden ratio. This most aesthetically pleasing proportion, phi, has been utilized by numerous artists since (and probably before!) the construction of the Great Pyramid.

As scholars and artists of eras gone by discovered (such as Leonardo da Vinci, Plato , & Pythagoras), the intentional use of these natural proportions in art of various forms expands our sense of beauty, balance & harmony to optimal effect. Leonardo da Vinci used the Golden Ratio in his painting of The Last Supper in both the overall composition (three vertical Golden Rectangles, and a decagon (which contains the golden ratio) for alignment of the central figure of Jesus. The outline of the Parthenon at the Acropolis near Athens, Greece is enclosed by a Golden Rectangle by design.
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post Mar 24, 2005, 07:45 PM
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Phi or Φ, was described by Johannes Kepler as one of the "two great treasures of geometry." (The other is the Theorem of Pythagoras.)

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trojan_libido
post Sep 19, 2006, 11:35 PM
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Pi = female
Phi = male

Pi + Phi = Party on dude!

I love the Golden Ratio and its strangeness, its clearly a part of the blueprint of life.
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post Sep 20, 2006, 01:24 PM
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I have a math handbook that lists hexadecimal! Westergren, "mathematics". 1.9E377 9B97F.
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post Jun 01, 2007, 02:39 PM
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QUOTE(Trip like I do @ Mar 24, 2005, 07:45 PM) *

Phi or Φ, was described by Johannes Kepler as one of the "two great treasures of geometry." (The other is the Theorem of Pythagoras.)


"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 [1571-1630]

Everywhere on the Internet there are loads of sayings like that by Kepler with no reference to the source.

Kepler in his work "Mysterium Cosmographicum" of 1962 expressed such a thought on p.42, p.47 and p.50-51.
However he used completely different words to express it.

So, from which work of Kepler was the saying above quoted?
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post Sep 09, 2008, 06:52 AM
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the golden ratio is what first introduced me to the fun of mathematics, in 9th grade. I was given a book called The Divine Proportion, and this short book held my fascination for quite some time. Its interesting the broad and varied scope that we find the golden ratio involved with, from geometric shapes to living creature and logarithmic spirals. one of the most interesting aspects to me is its involvement in the Fibonacci Sequence, and the growth of trees and flowers.

The way i see it, the golden ratio is an optimization constant. At least for nature, on the planet earth, it functions in such a way as to provide the ideal situation for plants (and humans and animals) to grow and flourish. Nature is not one to waste resources and I believe it is the golden ratio that makes this possible.

Examples: spacing of flower petals around a stem are rotated by ~.618, so that each petal receives maximum sunlight exposure. Branches on a tree grow upward and outward following a fibonacci sequence of growth, so that as you move up the tree, the number of branches approximates the next fibonacci number. Our human limbs are very close to the golden ratio, making sure our arms and legs are not awkward, and our bodies can move with relative grace. There are many other examples with the shapes and sizes of fish, other animals, plants, and living things. (If only I could find its presence at the cellular level!!)

We must take care not to attribute some arbitrary things that are "close" to the golden ratio. Fanatics and mystics like to go around measuring everything and claiming the golden ratio's presence. with accurate measurement and and a skeptic mind, we can see some amazing things!
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Phi
post Sep 14, 2008, 10:31 PM
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Where are you pi?
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trojan_libido
post Sep 14, 2008, 11:18 PM
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I think the points of interest of Phi are enough to include with mysticism and spirituality without having to find additional examples.

I just look at the relationship and shape of the Universe-to-Galaxy-to-Solar System-to-Stars-to-Orbits, and all I see is everything spinning in relationship to each other. If this spinning isn't part of our spirituality, we need to update it!
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Phi
post Sep 14, 2008, 11:22 PM
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QUOTE(trojan_libido @ Sep 15, 2008, 12:18 AM) *

I think the points of interest of Phi are enough to include with mysticism and spirituality without having to find additional examples.

I just look at the relationship and shape of the Universe-to-Galaxy-to-Solar System-to-Stars-to-Orbits, and all I see is everything spinning in relationship to each other. If this spinning isn't part of our spirituality, we need to update it!


Excellent point. I believe we share interests in the movement toward unity of the scientific mind with the spiritual mind. Phi is an excellent example, as it portrays both sides in that unity.
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post Sep 15, 2008, 04:11 AM
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The Yin-Yang is another example in which ancient philosophy seems spot on.



Animals polarise into sexes, and yet this obviously important aspect of biology is almost never touched upon. The Yin-Yang symbol is the embodiment of that, and it also shows the spinning motion described above. Another interesting aspect is in the shape of DNA, something the ancients could not have known about, and yet if that symbol is shown rotating and moving forward then the path would show the double helix. The proteins are even in pairs, which is how DNA can replicate itself, CG/GC AT/TA. Then the same symbol shows up in the ancient medical Caduceus:


The symbol itself is a shamen symbol. The pole is the world axis, the wings are the enlightened state of spiritual union.

I'm not saying Aliens came and told the ancients about these things. What I am saying is that the nature and path of our biology/conciousness keeps these images at the forefront of our spirituality.

Sorry if I've diverged.
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post Nov 14, 2009, 05:44 PM
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Phi = (sqrt 5 + 1) / 2 = 1.618033989...

Phi to 27 decimal places = 1.618033988749894848204586834...
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Phi
post Nov 14, 2009, 06:18 PM
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thank you for the tribute
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post Nov 15, 2009, 05:47 AM
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QUOTE(Phi @ Sep 14, 2008, 10:31 PM) *

Where are you pi?

Eer! In your hand?
QUOTE(Phi @ Sep 14, 2008, 11:22 PM) *

QUOTE(trojan_libido @ Sep 15, 2008, 12:18 AM) *

I think the points of interest of Phi are enough to include with mysticism and spirituality without having to find additional examples.

I just look at the relationship and shape of the Universe-to-Galaxy-to-Solar System-to-Stars-to-Orbits, and all I see is everything spinning in relationship to each other. If this spinning isn't part of our spirituality, we need to update it!


Excellent point. I believe we share interests in the movement toward unity of the scientific mind with the spiritual mind. Phi is an excellent example, as it portrays both sides in that unity.

I wish you all elaborated on this! Because I don't see the point!
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post Nov 15, 2009, 07:30 AM
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QUOTE(trojan_libido @ Sep 15, 2008, 07:18 AM) *
I just look at the relationship and shape of the Universe-to-Galaxy-to-Solar System-to-Stars-to-Orbits, and all I see is everything spinning in relationship to each other. If this spinning isn't part of our spirituality, we need to update it!
They won't spin forever!
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post Nov 15, 2009, 08:21 AM
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QUOTE(Hey Hey @ Nov 15, 2009, 07:30 AM) *

QUOTE(trojan_libido @ Sep 15, 2008, 07:18 AM) *
I just look at the relationship and shape of the Universe-to-Galaxy-to-Solar System-to-Stars-to-Orbits, and all I see is everything spinning in relationship to each other. If this spinning isn't part of our spirituality, we need to update it!
They won't spin forever!

Then "escape" to the moon or Mars is futile. Why not spend the money to make as many people here on earth now more comfortable? Or do you know that the slowing spin of galaxies is in sequence and we're the last one to stop spinning? In that case where would we escape to, to get away from our slowing galaxy then. If we're not the last slowing one then how do we know if we're not escaping to another slowing galaxy? Why not be satisfied with living right here on earth now? If we chase the light of a present galaxy would it even still be there by the time that we got to it? If not then what?
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post Nov 16, 2009, 08:05 AM
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wink.gif
QUOTE(PJS @ Nov 15, 2009, 08:21 AM) *

Then "escape" to the moon or Mars is futile. Why not spend the money to make as many people here on earth now more comfortable?

I believe tha Mayans have something to say about the timing for your plan. wink.gif
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post Nov 17, 2009, 10:08 AM
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QUOTE(code buttons @ Nov 16, 2009, 08:05 AM) *

wink.gif
QUOTE(PJS @ Nov 15, 2009, 08:21 AM) *

Then "escape" to the moon or Mars is futile. Why not spend the money to make as many people here on earth now more comfortable?

I believe tha Mayans have something to say about the timing for your plan. wink.gif

The Mayans calculated that the time for the sun to eclipse the center of the Milky Way Galaxy would occur in December 2012. This event would effect the moon and Mars as well wouldn't it?
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post Nov 18, 2009, 05:53 PM
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1.618033988749894848204586834




just stare at that sequence of numbers.... do they sooth or somehow feel appeasing or appealing to the eyes and the mind! Could the visual sequence of numbers that represent the golden ratio somehow act on an unconscious level?
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post Nov 18, 2009, 05:58 PM
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http://en.wikipedia.org/wiki/Golden_ratio
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post Nov 18, 2009, 06:03 PM
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post Nov 19, 2009, 12:34 PM
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QUOTE(Trip like I do @ Nov 18, 2009, 05:53 PM) *
... Could the visual sequence of numbers that represent the golden ratio somehow act on an unconscious level?

No. Not any more than any sequence of pseudo-random digits.
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post Nov 19, 2009, 01:20 PM
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QUOTE(Rick @ Nov 19, 2009, 03:34 PM) *

QUOTE(Trip like I do @ Nov 18, 2009, 05:53 PM) *
... Could the visual sequence of numbers that represent the golden ratio somehow act on an unconscious level?

No. Not any more than any sequence of pseudo-random digits.

so you believe that there is nothing cognitively soothing about this sequence of numbers that represents the quantifiable existence of something so innately concrete to the fundamental fixture of both the macro cosmic and micro cosmic universes!
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post Nov 19, 2009, 07:39 PM
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post Nov 19, 2009, 07:40 PM
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post Nov 19, 2009, 07:42 PM
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post Nov 19, 2009, 07:44 PM
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