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.
Full Version: Phi
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.
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.
Phi or Φ, was described by Johannes Kepler as one of the "two great treasures of geometry." (The other is the Theorem of Pythagoras.)
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.
Phi = male
Pi + Phi = Party on dude!
I love the Golden Ratio and its strangeness, its clearly a part of the blueprint of life.
I have a math handbook that lists hexadecimal! Westergren, "mathematics". 1.9E377 9B97F.
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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