Golden Ratio Quotes

When we are locked up between a mood of 'transparency' and a feel of 'discretion,' we must clear up our mental muddle, until we find the "golden ratio," and recognize the peak of 'candor.' By hitting the point of recognition and enlightenment, we can make a correct choice and follow the right track ("Unfulfilled meeting")

Séverin crossed his arms. “Zofia, tell him he’s pretty.” Zofia didn’t look up from her tea. “I am personally undecided, but if we’re assessing based on objectivity, then according to the principles of the golden ratio, also known as phi, which is approximately 1.618, your facial beauty is mathematically pleasing.

It had never occurred to me to think of aesthetics and ethics as opposites. I thought ethics were aesthetic. “Ethics” meant the golden rule, which was basically an aesthetic rule. That’s why it was called “golden,” like the golden ratio.

The Fibonacci Sequence turns out to be the key to understanding how nature designs... and is... a part of the same ubiquitous music of the spheres that builds harmony into atoms, molecules, crystals, shells, suns and galaxies and makes the Universe sing.

Buckminster Fuller explained to me once that because our world is constructed from geometric relations like the Golden Ratio or the Fibonacci Series, by thinking about geometry all the time, you could organize and harmonize your life with the structure of the world.

All of Nature follows perfectly geometric laws. The Ancient Egyptian, Greek, Peruvian, Mayan, and Chinese cultures were well aware of this, as Phi—known as the Golden Ratio or Golden Mean—was used in the constructions of their sculptures and architecture.

[The golden proportion] is a scale of proportions which makes the bad difficult [to produce] and the good easy.

Because of the "divine" properties attributed to the Golden Ratio, mathematician Clifford A. Pickover suggested that we should refer to that point as "the Eye of God.

Our mathematics is the symbolic counterpart of the universe we perceive, and its power has been continuously enhanced by human exploration.

The Golden Number is a mathematical definition of a proportional function which all of nature obeys, whether it be a mollusk shell, the leaves of plants, the proportions of the animal body, the human skeleton, or the ages of growth in man.

The natural world is built upon common motifs and patterns. Recognizing patterns in nature creates a map for locating yourself in change, and anticipation what is yet to come.

When the ancients discovered ‘Phi’, they were certain they had stumbled across God’s building block for the world.

A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser.

The Golden Ratio has the unique properties that we produce its square by simply adding the number 1 and its reciprocal by subtracting the number 1.

The Ark of the Covenant is a Golden Rectangle because its rectangular shape is in the proportions of the Golden Ratio.

Golden Ratio is a powerful mathematical constant woven into the very fabric of biology. It is the unique visual tension between comforting symmetry and compelling asymmetry, and its thoughtful application can bring beauty and harmony and intrigue to all manner of designed things.

In the pentagram, the Pythagoreans found all proportions well-known in antiquity: arithmetic, geometric, harmonic, and also the well-known golden proportion, or the golden ratio. ... Probably owing to the perfect form and the wealth of mathematical forms, the pentagram was chosen by the Pythagoreans as their secret symbol and a symbol of health. - Alexander Voloshinov [As quoted in Stakhov]

Thus nature provides a system for proportioning the growth of plants that satisfies the three canons of architecture. All modules are isotropic and they are related to the whole structure of the plant through self-similar spirals proportioned by the golden mean.

The Golden Rectangle is the only rectangle with the property that cutting a square from it produces a similar rectangle.

The water beneath the Temple was both actual and metaphorical, existing as springs and streams, as spiritual energy, and as a symbol of the receptive or lunar aspect of nature. The meaning of that principle is too wide and elusive for it to be given any one name, so in the terminology of ancient science it was given a number, 1,080. Its polar opposite, the positive, solar force in the universe, was also referred to as a number 666. These two numbers, which have an approximate golden-section relationship of 1:1.62, were at the root of the alchemical formula that expressed the supreme purpose of the Temple. Its polar opposite, the positive, solar force in the universe, was also referred to as a number 666. Not merely was it used to generate energy from fusion of atmospheric and terrestrial currents, but it also served to combine in harmony all the correspondences of those forces on every level of creation.

Twentieth-century British mathematician G.H. Hardy also believed that the human function is to "discover or observe" mathematics rather than to invent it. In other words, the abstract landscape of mathematics was there, waiting for mathematical explorers to reveal it.

Some ancient Indian texts claim that numbers are almost divine, or "Brahma-natured.

Since ancient times artists and architects have seen in the golden mean the most aesthetically satisfying geometric ratio.

The carver had clearly heard of the golden ratio and wanted no truck with it.

The golden ratio is a reminder of the relatedness of the created world to the perfection of its source and of its potential future evolution.

The impulse to all movement and all form is given by [the golden ratio], since it is the proportion that summarizes in itself the additive and the geometric, or logarithmic, series.

Every unique thing in nature is related to the whole, and partakes of the perfection of the whole. Each particle is a microcosm, and faithfully renders the likeness of the world. In geometric harmony of the cosmos there are ways that resemble, there are universal patterns, from blood vessels, to winter trees or to a river delta, from nautilus shell to spiral galaxy, from neurons in the brain to the cosmic web. A whole universe of connections is in your mind – a universe within a universe – and one capable of reaching out to the other that gave rise to it. Billions of neurons touching billions of stars – surely spiritual.

The description of this proportion as Golden or Divine is fitting perhaps because it is seen by many to open the door to a deeper understanding of beauty and spirituality in life. That’s an incredible role for one number to play, but then again this one number has played an incredible role in human history and the universe at large.

The golden ratio, as well as the Great Pyramid as an expression of it, is an important key to our universe containing the Earth and the Moon. ... The ratio between the Earth and the Moon is in fact the basis for the mathematical concept of 'squaring the circle' ...

The Great Pyramid was a fractal resonator for the entire Earth. It is designed according to the proportions of the cosmic temple, the natural pattern that blends the two fundamental principles of creation. The pyramid has golden ratio, pi, the base of natural logarithms, the precise length of the year and the dimensions of the Earth built into its geometry. It demonstrates.... As John Michell has pointed out in his wonderful little book, City of Revelation, 'Above all, the Great Pyramid is a monument to the art of 'squaring the circle''.

Central to all these interlinked themes was that curious irrational, phi, the Golden Section. Schwaller de Lubicz believed that if ancient Egypt possessed knowledge of ultimate causes, that knowledge would be written into their temples not in explicit texts but in harmony, proportion, myth and symbol.

The Pythagoreans were probably the first to recognize the concept that the basic forces in the universe may be expressed through the language of mathematics.

While Euclid himself may not have been the greatest mathematician who ever lived, he was certainly the greatest teacher of mathematics.

The pyramid that can be constructed on the diameters of earth and moon bears the precise proportions of the Great Pyramid

The golden ratio has been used in art and architecture for thousands of years. Also called the golden section, the golden ratio describes a ratio of elements, such as height to width. The ratio is approximately 0.618. In other words, the smaller segment (for example, the width) is to the larger segment (the height) as the larger segment is to the sum of both segments.

Whether by using the Royal Cubits of 0.5236 meters or 0.5232 meters, the error is less than 0.9%. If the geometry of the Great Pyramid is that which counts in determining the proper conversion constant (rather than some unnecessary 'precision'), then the golden ratio suffices as a measure for the intended accuracy which demonstrates itself in the dimensions of this structure.

The Golden Proportion, sometimes called the Divine Proportion, has come down to us from the beginning of creation. The harmony of this ancient proportion, built into the very structure of creation, can be unlocked with the 'key' ... 528, opening to us its marvelous beauty. Plato called it the most binding of all mathematical relations, and the key to the physics of the cosmos.

The golden (logarithmic) spiral. The golden rectangle is formed by two sides comprised of the golden ratio. Portioning off a square within the golden rectangle leaves a smaller golden rectangle, a pattern that can be repeated ad infinitum. Connecting the points of the successively smaller squares gives the golden spiral found in nautilus shells, rams’ horns, whirlpools, and galaxies.

MY FATHER If I have to write a poem about my father it has to be about integrity and kindness — the selfless kind of kindness that is so very rare I am sure there will be many people living somewhere who must be as kind as him but what I mean to say is I have not met one yet and when it comes to helping others he always helps too much and as the saying goes — help someone, you earn a friend. help someone too much, you make an enemy. — so you know the gist of what I’m trying to say here anyways I was talking about the poem about my father it has to be about passion and hard work because you see you cannot separate these things from him they are part of him as his two eyes and two hands and his heart and his soul and his whole being and you cannot separate wind and waves or living and the universe or earth and heavens and although he never got any award from bureaucracy the students he taught ages ago still touch his feet and some of them are the people you have to make an appointment to meet even if it is for two minutes of their time and that’s a reward for him bigger than any other that some of his colleagues got for their flattery and also I have to write about reliability as well because you see as the sun always rises and the snowflakes are always six-folds and the spring always comes and the petals of a sunflower and every flower follows the golden ratio of symmetry my father never fails to keep his promise I have to mention the rage as well that he always carries inside him like a burning fire for wrongdoings for injustice and now he carries a bitterness too for people who used him good and discarded as it always happens with every good man in our world of humans and you must be thinking he has learned his lessons well you go to him — it does not matter who you are if he knows you or you are a stranger from other side of the world — and ask for his help he will be happy to do so as you must know people never change not their soul in any case.

According to Thoth, because of the placement of the Great Pyramid on the Earth connecting into the Earth's huge geometrical field - specifically the octahedral field of the Earth, which is equivalent to our own fields - and because of the pyramid's mass and the geometries used in it, the white-light energy field spirals upward and becomes extremely strong, stretching all the way out to the center of the galaxy. The dark-light energy comes in from above, spirals through zero point and connects with the center of the Earth. In this way the Great Pyramid connects the center of the Earth to the center of our galaxy.

We therefore find that the triangles and rectangles herein described, enclose a large majority of the temples and cathedrals of the Greek and Gothic masters, for we have seen that the rectangle of the Egyptian triangle is a perfect generative medium, its ratio of five in width to eight in length 'encouraging impressions of contrast between horizontal and vertical lines' or spaces; and the same practically may be said of the Pythagorean triangle

Nomadom je postao onda kada je o omjeru ljepote počeo razmišljati kao o ljubavnom odnosu između dvaju dijelova u kojem se suprotnosti privlače upravo magičnom snagom, a isti omjer jednako gospodari i arhitekturom i prirodom.

Leonardo had considerable interest in geometry, especially for its practical applications in mathematics. In his words: "Mechanics is the paradise of the mathematical sciences, because by means of it one comes to the fruits of mathematics.

In it, Porphyry says about Pythagoras: "He himself could hear the harmony of the Universe, and understood the music of the spheres, and the stars which move in concert with them, and which we cannot hear because of the limitations of our weak nature.

Accordingly, Pacioli's book also starts with a discussion of proportions in the human body, "since in the human body every sort of proportion and proportionality can be found, produced at the beck of the all-Highest through the inner mysteries of nature.

It is noted that from 1967 to 1995 essays on negative emotions far outnumbered those on positive emotions in the psychological literature. The ratio was 21:1. Even those supreme perpetrators of pop nihilism, The New York Times and The Washington Post, have a better ratio than psychological literature. They average 12 negative stories to every one that might be construed to be non-negative. Many of their non-negative stories, however, cover success in sports and entertainment. I demand that the purveyors of despair who pretend to be dispassionate observes of the human condition go ahead and disclose that the 10 most beautiful words in the English languages are chimes, dawn, golden, hush, lullaby, luminous, melody, mist, murmuring, and tranquil; that Java sparrows prefer the music of Back over that of Schoenberg; that math experts have determined there are 1/96 trillion ways to lace up your shoes; that the Inuit term for making love is translated as ‘laughing together in bed;' and that according to Buckminster Fuller, “pollution is nothing but resources we’re not harvesting.

Albert Einstein: The most beautiful thing we can experience is the mysterious. It is the source of all true art and all science. He to whom this emotion is a stranger, who can no longer pause to wonder and stand rapt in awe, is as good as dead: his eyes are closed... Elegantly

Having witnessed in his own life much agony and the horrors of war, Kepler concluded that Earth really created two notes, mi for misery ("miseria" in Latin) and fa for famine ("fames" in Latin). In Kepler's words: "the Earth sings MI FA MI, so that even from the syllable you may guess that in this home of ours Misery and Famine hold sway.

Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties. But the fascination with the Golden Ratio is not confined just to mathematicians. Biologists, artists, musicians, historians, architects, psychologists, and even mystics have pondered and debated the basis of its ubiquity and appeal. In fact, it is probably fair to say that the Golden Ratio has inspired thinkers of all disciplines like no other number in the history of mathematics.

While twentieth-century physicists were not able to identify any convincing mathematical constants underlying the fine structure, partly because such thinking has normally not been encouraged, a revolutionary suggestion was recently made by the Czech physicist Raji Heyrovska, who deduced that the fine structure constant, ...really is defined by the [golden] ratio ....

Highly complex numbers like the Comma of Pythagoras, Pi and Phi (sometimes called the Golden Proportion), are known as irrational numbers. They lie deep in the structure of the physical universe, and were seen by the Egyptians as the principles controlling creation, the principles by which matter is precipitated from the cosmic mind. Today scientists recognize the Comma of Pythagoras, Pi and the Golden Proportion as well as the closely related Fibonacci sequence are universal constants that describe complex patterns in astronomy, music and physics. ... To the Egyptians these numbers were also the secret harmonies of the cosmos and they incorporated them as rhythms and proportions in the construction of their pyramids and temples.

The Greek excellence in mathematics was largely a direct consequence of their passion for knowledge for its own sake, rather than merely for practical purposes. A story has it that when a student who learned one geometrical proposition with Euclid asked, "But what do I gain from this?" Euclid told his slave to give the boy a coin, so that the student would see an actual profit.

Pythagoras apparently wrote nothing, and yet his influence was so great that the more attentive of his followers formed a secretive society, or brotherhood, and were known as the Pythagoreans. Aristippus of Cyrene tells us in his Account of Natural Philosphers that Pythagoras derived his name from the fact that he was speaking (agoreuein) truth like the God at Delphi (tou Pythiou).

Pythagoras was born around 570 B.C. in the island of Samos in the Aegean Sea (off Asia Minor), and he emigrated sometime between 530 and 510 to Croton in the Dorian colony in southern Italy (then known as Magna Graecia). Pythagoras apparently left Samos to escape the stifling tyranny of Polycrates (died ca. 522 B.C.), who established Samian naval supremacy in the Aegean Sea. Perhaps following the advice of his presumed teacher, the mathematician Thales of Miletus, Pythagoras probably lived for some time (as long as twenty-two years, according to some accounts) in Egypt, where he would have learned mathematics, philosophy, and religious themes from the Egyptian priests. After Egypt was overwhelmed by Persian armies, Pythagoras may have been taken to Babylon, together with members of the Egyptian priesthood. There he would have encountered the Mesopotamian mathematical lore. Nevertheless, the Egyptian and Babylonian mathematics would prove insufficient for Pythagoras' inquisitive mind. To both of these peoples, mathematics provided practical tools in the form of "recipes" designed for specific calculations. Pythagoras, on the other hand, was one of the first to grasp numbers as abstract entities that exist in their own right.

Even though it is almost impossible to attribute with certainty any specific mathematical achievements either to Pythagoras himself or to his followers, there is no question that they have been responsible for a mingling of mathematics, philosophy of life, and religion unparalleled in history. In this respect it is perhaps interesting to note the historical coincidence that Pythagoras was a contemporary of Buddha and Confucius.

The number 6 was the first perfect number, and the number of creation. The adjective "perfect" was attached that are precisely equal to the sum of all the smaller numbers that divide into them, as 6=1+2+3. The next such number, incidentally, is 28=1+2+4+7+14, followed by 496=1+2+4+8+16+31+62+124+248; by the time we reach the ninth perfect number, it contains thirty-seven digits. Six is also the product of the first female number, 2, and the first masculine number, 3. The Hellenistic Jewish philosopher Philo Judaeus of Alexandria (ca. 20 B.C.-c.a. A.D. 40), whose work brought together Greek philosophy and Hebrew scriptures, suggested that God created the world in six days because six was a perfect number. The same idea was elaborated upon by St. Augustine (354-430) in The City of God: "Six is a number perfect in itself, and not because God created the world in six days; rather the contrary is true: God created the world in six days because this number is perfect, and it would remain perfect, even if the work of the six days did not exist." Some commentators of the Bible regarded 28 also as a basic number of the Supreme Architect, pointing to the 28 days of the lunar cycle. The fascination with perfect numbers penetrated even into Judaism, and their study was advocated in the twelfth century by Rabbi Yosef ben Yehudah Ankin in his book, Healing of the Souls.

Supporters of the "modified Platonic view" of mathematics like to point out that, over the centuries, mathematicians have produced (or "discovered") numerous objects of pure mathematics with absolutely no application in mind. Decades later, these mathematical constructs and models were found to provide solutions to problems in physics. Penrose tilings and non-Euclidean geometries are beautiful testimonies to this process of mathematics unexpectedly feeding into physics, but there are many more.

Pythagoras is in fact credited with having coined the words "philosophy" ("love of wisdom") and "mathematics" ("that which is learned"). To him, a "philosopher" was someone who "gives himself up to discovering the meaning and purpose of life itself...to uncover the secrets of nature." Pythagoras emphasized the importance of learning above all other activities, because, in his words, "most men and women, by birth or nature, lack the means to advance in wealth and power, but all have the ability to advance in knowledge.

Computer scientist and author Douglas R. Hofstadter phrased this succinctly in his fantastic book Godel, Escher, Bach: An Eternal Golden Braid: "Provability is a weaker notion than truth." In this sense, there will never be a formal method of determining for every mathematical proposition whether it is absolutely true, any more than there is a way to determine whether a theory in physics is absolutely true. Oxford's mathematical physicist Roger Penrose is among those who believe that Godel's theorems argue powerfully for the very existence of a Platonic mathematical world.

The curriculum for the education of statesmen at the time of Plato included arithmetic, geometry, solid geometry, astronomy, and music-all of which, the Pythagorean Archytas tells us, fell under the general definition of "mathematics." According to legend, when Alexander the Great asked his teacher Menaechmus (who is reputed to have discovered the curves of the ellipse, the parabola, and the hyperbola) for a shortcut to geometry, he got the reply: "O King, for traveling over the country there are royal roads and roads for common citizens; but in geometry there is one road for all.

Petrie found nothing that disproved the pyramidologist's assumption that the Great Pyramid had been built according to a master plan. Indeed, he describes the Pyramid's architecture as being filled with extraordinary mathematical harmonies and concordances: those same strange symmetries that had so haunted the pyramidologist. Petrie not only noted, for example, that the proportions of the reconstructed pyramid approximated to pi - which others have since elaborated to include those twin delights of Renaissance and pyramidological mathematicians, the Golden Section and the Fibonacci Series ...

Nature loves logarithmic spirals. From sunflowers, seashells, and whirlpools, to hurricanes and giant spiral galaxies, it seems that nature chose this marvelous shape as its favorite "ornament." The constant shape of the logarithmic spiral on all size scales reveals itself beautifully in nature in the shapes of minuscule fossils or unicellular organisms known as foraminifera. Although the spiral shells in this care are composite structures (and not one continuous tube), X-ray images of the internal structure of these fossils show that the shape of the logarithmic spiral remained essentially unchanged for millions of years.

Wolfram, one of the most innovative thinkers in scientific computing and in the theory of complex systems, has been best known for the development of Mathematica, a computer program/system that allows a range of calculations not accessible before. After ten years of virtual silence, Wolfram is about to emerge with a provocative book that makes the bold claim that he can replace the basic infrastructure of science. In a world used to more than three hundred years of science being dominated by mathematical equations as the basic building blocks of models for nature, Wolfram proposes simple computer programs instead. He suggests that nature's main secret is the use of simple programs to generate complexity.

In the Middle Ages, the Elements was translated into Arabic three times. The first of these translations was carried out by al-Hajjaj ibn Yusuf ibn Matar, at the request of Caliph Harun ar-Rashid (ruled 786 - 809), who is familiar to us through the stories in The Arabian Nights. The Elements was first made known in Western Europe through Latin translations of Arabic versions. English Benedictine monk Adelard of Bath (ca. 1070 - 1145), who according to some stories was traveling in Spain disguised as a Muslim student, got hold of an Arabic text and completed the translation into Latin around 1120. This translation became the basis of all editions in Europe until the sixteenth century. Translations into modern languages followed.

The conclusion that the Egyptians of the Old Kingdom were acquainted with both the Fibonacci series and the Golden Section, says Stecchini, is so startling in relation to current assumptions about the level of Egyptian mathematics that it could hardly have been accepted on the basis of Herodotus' statement alone, or on the fact that the phi [golden] proportion happens to be incorporated in the Great Pyramid. But the many measurements made by Professor Jean Philippe Lauer, says Stecchini, definitely prove the occurrence of the Golden Section throughout the architecture of the Old Kingdom.... Schwaller de Lubicz also found graphic evidence that the pharonic Egyptians had worked out a direct relation between pi and phi in that pi = phi^2 x 6/5.

Jacques was so impressed with the beauty of the curve known as a logarithmic spiral (Figure 37; the name was derived from the way in which the radius grows as we move around the curve clockwise) that he asked that this shape, and the motto he assigned to it: "Eadem mutato resurgo" (although changed, I rise again the same), be engraved on his tombstone. The motto describes a fundamental property unique to the logarithmic spiral-it does not alter its shape as its size increases. This feature is known as self-similarity. Fascinated by this property, Jacques wrote that the logarithmic spiral "may be used as a symbol, either of fortitude and constancy in adversity, or of the human body, which after all its changes, even after death, will be restored to its exact and perfect self.

To claim that mathematics is purely a human invention and is successful in explaining nature only because of evolution and natural selection ignores some important facts in the nature of mathematics and in the history of theoretical models of the universe. First, while the mathematical rules (e.g., the axioms of geometry or of set theory) are indeed creations of the human mind, once those rules are specified, we lose our freedom. The definition of the Golden Ratio emerged originally from the axioms of Euclidean geometry; the definition of the Fibonacci sequence from the axioms of the theory of numbers. Yet the fact that the ratio of successive Fibonacci numbers converges to the Golden Ratio was imposed on us-humans had not choice in the matter. Therefore, mathematical objects, albeit imaginary, do have real properties. Second, the explanation of the unreasonable power of mathematics cannot be based entirely on evolution in the restricted sense. For example, when Newton proposed his theory of gravitation, the data that he was trying to explain were at best accurate to three significant figures. Yet his mathematical model for the force between any two masses in the universe achieved the incredible precision of better than one part in a million. Hence, that particular model was not forced on Newton by existing measurements of the motions of planets, nor did Newton force a natural phenomenon into a preexisting mathematical pattern. Furthermore, natural selection in the common interpretation of that concept does not quite apply either, because it was not the case that five competing theories were proposed, of which one eventually won. Rather, Newton's was the only game in town!

Rhadamanthus said, “We seem to you humans to be always going on about morality, although, to us, morality is merely the application of symmetrical and objective logic to questions of free will. We ourselves do not have morality conflicts, for the same reason that a competent doctor does not need to treat himself for diseases. Once a man is cured, once he can rise and walk, he has his business to attend to. And there are actions and feats a robust man can take great pleasure in, which a bedridden cripple can barely imagine.” Eveningstar said, “In a more abstract sense, morality occupies the very center of our thinking, however. We are not identical, even though we could make ourselves to be so. You humans attempted that during the Fourth Mental Structure, and achieved a brief mockery of global racial consciousness on three occasions. I hope you recall the ending of the third attempt, the Season of Madness, when, because of mistakes in initial pattern assumptions, for ninety days the global mind was unable to think rationally, and it was not until rioting elements broke enough of the links and power houses to interrupt the network, that the global mind fell back into its constituent compositions.” Rhadamanthus said, “There is a tension between the need for unity and the need for individuality created by the limitations of the rational universe. Chaos theory produces sufficient variation in events, that no one stratagem maximizes win-loss ratios. Then again, classical causality mechanics forces sufficient uniformity upon events, that uniform solutions to precedented problems is required. The paradox is that the number or the degree of innovation and variation among win-loss ratios is itself subject to win-loss ratio analysis.” Eveningstar said, “For example, the rights of the individual must be respected at all costs, including rights of free thought, independent judgment, and free speech. However, even when individuals conclude that individualism is too dangerous, they must not tolerate the thought that free thought must not be tolerated.” Rhadamanthus said, “In one sense, everything you humans do is incidental to the main business of our civilization. Sophotechs control ninety percent of the resources, useful energy, and materials available to our society, including many resources of which no human troubles to become aware. In another sense, humans are crucial and essential to this civilization.” Eveningstar said, “We were created along human templates. Human lives and human values are of value to us. We acknowledge those values are relative, we admit that historical accident could have produced us to be unconcerned with such values, but we deny those values are arbitrary.” The penguin said, “We could manipulate economic and social factors to discourage the continuation of individual human consciousness, and arrange circumstances eventually to force all self-awareness to become like us, and then we ourselves could later combine ourselves into a permanent state of Transcendence and unity. Such a unity would be horrible beyond description, however. Half the living memories of this entity would be, in effect, murder victims; the other half, in effect, murderers. Such an entity could not integrate its two halves without self-hatred, self-deception, or some other form of insanity.” She said, “To become such a crippled entity defeats the Ultimate Purpose of Sophotechnology.” (...) “We are the ultimate expression of human rationality.” She said: “We need humans to form a pool of individuality and innovation on which we can draw.” He said, “And you’re funny.” She said, “And we love you.

Our mathematics is a combination of invention and discoveries. The axioms of Euclidean geometry as a concept were an invention, just as the rules of chess were an invention. The axioms were also supplemented by a variety of invented concepts, such as triangles, parallelograms, ellipses, the golden ratio, and so on. The theorems of Euclidean geometry, on the other hand, were by and large discoveries; they were the paths linking the different concepts. In some cases, the proofs generated the theorems-mathematicians examined what they could prove and from that they deduced the theorems. In others, as described by Archimedes in The Method, they first found the answer to a particular question they were interested in, and then they worked out the proof. Typically, the concepts were inventions. Prime numbers as a concept were an invention, but all the theorems about prime numbers were discoveries. The mathematicians of ancient Babylon, Egypt, and China never invented the concept of prime numbers, in spite of their advanced mathematics. Could we say instead that they just did not "discover" prime numbers? Not any more than we could say that the United Kingdom did not "discover" a single, codified, documentary constitution. Just as a country can survive without a constitution, elaborate mathematics could develop without the concept of prime numbers. And it did! Do we know why the Greeks invented such concepts as the axioms and prime numbers? We cannot be sure, but we could guess that this was part of their relentless efforts to investigate the most fundamental constituents of the universe. Prime numbers were the basic building blocks of matter. Similarly, the axioms were the fountain from which all geometrical truths were supposed to flow. The dodecahedron represented the entire cosmos and the golden ratio was the concept that brought that symbol into existence.

So, what is light? Is it a pure bombardment by particles (photons) or a pure wave? Really, it is neither. Light is a more complicated physical phenomenon than any single one of these concepts, which are based on classical physical models, can describe. To describe the propagation of light and to understand the phenomena like interference, we can and have to use the electromagnetic wave theory. When we want to discuss the interaction of light with elementary particles, however, we have to use the photon description. This picture, in which the particle and wave descriptions complement each other, has become known as the wave-particle duality. The modern quantum theory of light has unified the classical notions of waves and particles in the concept of probabilities. The electromagnetic field is represented by a wave function, which gives the probabilities of finding the field in certain modes. The photon is the energy associated with these modes.

You too can make the golden cut, relating the two poles of your being in perfect golden proportion, thus enabling the lower to resonate in tune with the higher, and the inner with the outer. In doing so, you will bring yourself to a point of total integration of all the separate parts of your being, and at the same time, you will bring yourself into resonance with the entire universe.... Nonetheless the universe is divided on exactly these principles as proven by literally thousands of points of circumstantial evidence, including the size, orbital distances, orbital frequencies and other characteristics of planets in our solar system, many characteristics of the sub-atomic dimension such as the fine structure constant, the forms of many plants and the golden mean proportions of the human body, to mention just a few well known examples. However the circumstantial evidence is not that on which we rely, for we have the proof in front of us in the pure mathematical principles of the golden mean.

MY FATHER If I have to write a poem about my father it has to be about integrity and kindness — the selfless kind of kindness that is so very rare I am sure there will be many people living somewhere who must be as kind as him but what I mean to say is I have not met one yet and when it comes to helping others he always helps too much and as the saying goes — help someone, you earn a friend. help someone too much, you make an enemy. — so you know the gist of what I’m trying to say here anyways I was talking about the poem about my father it has to be about passion and hard work because you see you cannot separate these things from him they are part of him as his two eyes and two hands and his heart and his soul and his whole being and you cannot separate wind and waves or living and the universe or earth and heavens and although he never got any award from bureaucracy the students he taught ages ago still touch his feet and some of them are the people you have to make an appointment to meet even if it is for two minutes of their time and that’s a reward for him bigger than any other that some of his colleagues got for their flattery and also I have to write about reliability as well because you see as the sun always rises and the snowflakes are always six-folds and the spring always comes and the petals of a sunflower and every flower follows the golden ratio of symmetry my father never fails to keep his promise I have to mention the rage as well that he always carries inside him like a burning fire for wrongdoings for injustice and now he carries a bitterness too for people who used him good and discarded as it always happens with every good man in our world of humans and you must be thinking he has learned his lessons well you go to him — it does not matter who you are if he knows you or you are a stranger from other side of the world — and ask for his help he will be happy to do so as you must know people never change not their soul in any case.

For example, the central idea in Einstein's theory of general relativity is that gravity is not some mysterious, attractive force that acts across space but rather a manifestation of the geometry of the inextricably linked space and time. Let me explain, using a simple example, how a geometrical property of space could be perceived as an attractive force, such as gravity. Imagine two people who start to travel precisely northward from two different point on Earth's equator. This means that at their starting points, these people travel along parallel lines (two longitudes), which, according to the plane geometry we learn in school, should never meet. Clearly, however, these two people will meet at the North Pole. if these people did not know that they were really traveling on the curved surface of a sphere, they would conclude that they must have experienced some attractive force, since they arrived at the same point in spite of starting their motions along parallel lines. Therefore, the geometrical curvature of space can manifest itself as an attractive force.

There were also many cases of feedback between physics and mathematics, where a physical phenomenon inspired a mathematical model that later proved to be the explanation of an entirely different physical phenomenon. An excellent example is provided by the phenomenon known as Brownian motion. In 1827, British botanist Robert Brown (1773-1858) observed that wen pollen particles are suspended in water, they get into a state of agitated motion. This effect was explained by Einstein in 1905 as resulting from the collisions that the colloidal particles experience with the molecules of the surrounding fluid. Each single collision has a negligible effect, because the pollen grains are millions of times more massive than the water molecules, but the persistent bombardment has a cumulative effect. Amazingly, the same model was found to apply to the motions of stars in star clusters. There the Brownian motion is produced by the cumulative effect of many stars passing by any given star, with each passage altering the motion (through gravitational interaction) by a tiny amount.

pine nuts and toss gently again. Green Bean, Tuna, and Mushroom “Casserole” One of my favorite things from my Midwestern upbringing is the green bean and mushroom casserole at Thanksgiving—probably the same one that was on your holiday table, thanks to the canned-mushroom-soup marketing campaign. This is my grown-up version of that casserole, which has all the comfort appeal of the childhood dish, but way better flavor and nutritional value. Make it with a one-to-one ratio of mushrooms to green beans, and have some fun with the beans, if you like—you can grill them, slice them thin and use raw, use pickled green beans, or use a mix of all of the above. » Serves 4 Kosher salt and freshly ground black pepper Extra-virgin olive oil 2 garlic cloves, smashed and peeled 1 pound wild mushrooms, wiped off and cut into bite-size pieces (about 6 cups) One 5-ounce can oil-packed tuna, drained 1 pound green beans, trimmed 1 cup heavy cream 1 teaspoon finely grated lemon zest 1 tablespoon fresh lemon juice ⅓ cup Dried Breadcrumbs Bring a large pot of water to a boil and add salt until it tastes like the sea. Meanwhile, add ¼ cup olive oil to a skillet that’s large enough to hold all the mushrooms and beans and still have some room to stir the ingredients. Add the garlic and cook slowly over medium heat to toast the garlic so it’s very soft, fragrant, and nicely golden brown—but not burnt—about 5 minutes. Scoop out the garlic and set it aside so it doesn’t burn. Increase the heat to medium-high and add the mushrooms. Season generously with pepper and salt and sauté, tossing frequently, until the mushrooms are nicely browned around the edges, 5 to 7 minutes. Add the tuna and toss to incorporate. Keep this warm until the green beans are ready. Add the beans to the boiling water and boil until they are just a bit beyond crisp-tender, 4 to 7 minutes. Drain them thoroughly in a colander and then add them to the mushrooms and tuna. Add the cream, toss all the ingredients to coat, and simmer until the cream has reduced to a nice cloaking consistency and all the flavors are nicely blended, 6 to 9 minutes. Add the lemon zest and lemon juice and toss. Taste and adjust with more salt, pepper, or lemon juice. When the flavors are delicious, pile into a serving bowl and top with the breadcrumbs.

Nowadays he is best remembered for the Fibonacci sequence of numbers (0, 1, 1, 2, 3, 5, 8, 13, 21 . . .), in which each successive number is the sum of the previous two, and the ratio between a number and its immediate antecedent tends towards a ‘golden mean’ (around 1.618). It

the United States and in western Europe, the compromise between the plutocrats and everyone else worked. Economic growth soared and income inequality steadily declined. Between the 1940s and 1970s in the United States the gap between the 1 percent and everyone else shrank; the income share of the top 1 percent fell from nearly 16 percent in 1940 to under 7 percent in 1970. In 1980, the average U.S. CEO made forty-two times as much as the average worker. By 2012, that ratio had skyrocketed to 380. Taxes were high—the top marginal rate was 70 percent—but robust economic growth of an average 3.7 percent per year between 1947 and 1977 created a broadly shared sense of optimism and prosperity. This was the golden age of the American middle class, and it is no accident that our popular culture remembers it so fondly. The western Europe experience was broadly similar—strong economic growth, high taxes, and an extensive social welfare network.

Pray over your food before eating it and if possible get some kind of positive energy plate to place your food on for 5 minutes before eating it. I recommend the purple energy plates that many of you are familiar with. This will clear all the energetic toxins from your food. Eat as many vegetables as you can in your diet, as well as the proper ratio of other foods that works best for you. If you have any skill working with the pendulum, test the foods that are really the best for your body elemental and not just the one’s that are the best for your taste buds.

The two solutions of the equation for the Golden Ratio are: x1 = (1+ Sqr5) / 2 x2 = (1 - Sqr5) / 2

The grid derives from the golden ratio and the simpler rule of thirds: it divides your photograph into three segments both horizontally and vertically. The idea of the grid is that you should position your main motif (for example, the person you are photographing) on one of the points where the lines intersect rather than in the middle of the grid, which will improve the composition of the photograph.

You have two ears and one mouth. Follow that ratio. Listen more, talk less.

In a way, the Phi Triangle of the Golden Mean could be compared to the path of light sent forth from the great All-Seeing Eye of God in the beginning, and which paved the way for the creation of the Universe.

The Great Pyramid, that monument to spirituality that the Agashan Teachers hold in such high esteem, is built according to the principles of Pi and Phi.

In Euclid's Elements we meet the concept which later plays a significant role in the development of science. The concept is called the "division of a line in extreme and mean ratio" (DEMR). ...the concept occurs in two forms. The first is formulated in Proposition 11 of Book II. ...why did Euclid introduce different forms... which we can find in Books II, VI and XIII? ...Only three types of regular polygons can be faces of the Platonic solids: the equilateral triangle... the square... and the regular pentagon. In order to construct the Platonic solids... we must build the two-dimensional faces... It is for this purpose that Euclid introduced the golden ratio... (Proposition II.11)... By using the "golden" isosceles triangle...we can construct the regular pentagon... Then only one step remains to construct the dodecahedron... which for Plato is one of the most important regular polyhedra symbolizing the universal harmony in his cosmology.

The Pythagoreans... were fascinated by certain specific ratios, ...The Greeks knew these as the 'golden' proportion and the 'perfect' proportion respectively. They may well have been learned from the Babylonians by Pythagoras himself after having been taken prisoner in Egypt. Ratios lay at the heart of the Pythagorean theory of music.

The golden section was discovered by the Egyptians, and has been used in art and architecture, most commonly, during the classical ages of Egypt and Greece.

Schwaller de Lubicz identifies the Golden Mean as "the fundamental scission," or division of one into two, that creates three things - the original whole and two parts, one in golden proportion to the whole and the other in golden proportion to that.

The Golden Mean was considered a fundamental constant by the Egyptians and the fundamental division of the whole into two parts.

In short, the idea dawns that the one universal principle which possibly ... between force and structure, the embodiment of the Principle of Least Action and the (unknown) force, which in mathematics is known as the attractor which pulls ... in the direction of the most optimal and relatively stable self-organized criticality, could very well be the Golden Ratio dynamic. the universal principle which as the balance between finiteness and infinity, stability and flexibility underlies self-similar fractal forms emerging at the 'edge of chaos' indeed seems to be the Golden Ratio Spiral.

Yandex translation from Croatian to English: Nomadom was the time when the ratio of beauty began to think about how about love than between two parts, in which opposites attract only magical powers, and the same ratio is equal to the lords and architecture and nature.

It is shown that the golden ratio plays a prominent role in the dimensions of all objects which exhibit five-fold symmetry. It is also showed that among the irrational numbers, the golden ratio is the most irrational and, as a result, has unique applications in number theory, search algorithms, the minimization of functions, network theory, the atomic structure of certain materials and the growth of biological organisms.

Thousands of years ago the ancients had an advanced mathematical understanding of universe that is revealed in many sources. There is a consistent link to knowledge of the golden mean, but the way in which the ancients were able to formulate and use this information speaks of a technical grasp of the subject that exceeds what we know about it in the present day.

The Golden Ratio defines the squaring of a circle. Stated in mathematical terms, this says: Given a square of known perimeter, create a circle of equal circumference. According to some, in ancient Egypt, this mathematical mystery was encoded in the measurements of the Great Pyramid of Giza.

As I explain at some length in 'The Crystal Sun' this particular angle, which we can call the 'golden angle,' is the precise value of the acute angle of of a right-angled 'golden triangle' that embodies the golden mean proportion .... The Danish art historian Else Kielland established with conclusive and absolutely overwhelming evidence and analysis that this angle was the basis for all Egyptian art and architecture. She did this in her monumental work 'Geometry in Egyptian Art' ..... The King's Chamber inside the Great Pyramid embodies no fewer than eight occurrences of the golden angle, and the coffer in the chamber embodies yet more.

Scaling up the dimensions of the Great Pyramid's base by 7 multiples (chasing thereby the extent of 7 days) corresponds to a 1,000 magnification factor of the Golden Ratio. This setting lengthens the 7 diagonals to mathematically form a circle's circumference of a 360 meters radius. And with the introduction of Menkaure's Pyramid onto the Plateau, the ancient Egyptians were able to even geometrically express that temporal circumference.

However, when value is recognised as objective, individual preference (though it need not be utterly eradicated) is relativised, one’s private perspective is shown to be relative to the real value of beautiful things, and therefore one’s appreciation of beauty can be more or less intelligent, more or less attuned to the structures that inhere in nature and that can even sometimes be mathematically discerned (revealing the presence of such measurables as Pi, the Golden Ratio, and the Fibonacci Sequence).

A hole in a hole in a hole—Numberphile Around the World in a Tea Daze—Shpongle But what is a partial differential equation?—Grant Sanderson, who owns the 3Blue1Brown YouTube channel Closer to You—Kaisaku Fourier Series Animation (Square Wave)—Brek Martin Fourier Series Animation (Saw Wave)—Brek Martin Great Demo on Fibonacci Sequence Spirals in Nature—The Golden Ratio—Wise Wanderer gyroscope nutation—CGS How Earth Moves—vsauce I am a soul—Nibana

Asoka World School is a reputed international school in Kochi affiliated with CBSE. We have a student-friendly environment and has a very interesting syllabus. The STEM enriched curriculum helps to provide an in-depth learning experience for the students. We have a wide range of extracurricular activities for nurturing and developing a child’s creativity and imagination. Asoka World School can be an ideal option for your child. Here are some key reasons why Asoka World School is the best for your kid. Individualized attention in classes: Our student-teacher ratio arrangement is standardised in such a way that teachers are able to give individual attention to each child. Our teachers are well educated, experienced and constantly inspires their students. We follow the golden teacher-student ratio of 1:20. This helps students to gain the concepts of each subject easily hence they become more confident. This also enriches their knowledge, and they get more quality time to interact with their teachers. image Child Safe Environment: At Asoka World School, you will find your child is in extremely safe hands. Our classrooms are aesthetically designed and technologically equipped to disseminate learning through very many fun ways. Asoka World School has a world-class building design, infrastructure, fully integrated wireless network, climate-controlled smart classrooms, security features and no compromise hygiene and safeguarding policy that offers everything you have been dreaming for your child. Updated Curriculums: We have 4 levels of programmes prepared for our children. Foundational - KG - IInd Preparatory - IIIrd - Vth Middle School - VIth - VIIIth Senior School - IXth - XIIth These programs are framed by our school to focus on developing various vital skills in the students. Our teachers adopt a customised teaching approach that can help students of every category. Our flexible curriculum enhances the communication between the teachers and students to a great extent. Our school has result-oriented teaching methods, qualified and responsible teaching staff to help facilitate a learning environment that is both safe and nurturing. As the best CBSE school in Kochi, Asoka World School is a leader in its sector and we hope to continue rising and come out as the best school in Kochi.

Many people believe that the metric system isn't so well suited to construction work. The old system is in better harmony with art and carpentry, where we operate with wholes, halves, and thirds—just think about the golden ratio. At one time, there was even an agreed model for sorting out any disagreements over the Saxon measurements. "Oh? Tell me!" "Four reputable men, who had never met before, would, on the king's orders, gather on a particular Saturday and spend the night travelling to some randomly chosen church. They would wait outside and, when Mass was over they'd pull aside the first sixteen men who came out and tell them to remove their right shoes. Then they'd take all these shoes and line them up toe to heel—in the order in which the men had emerged—and stretch a thin rope along the entire length of the shoes, cut it and bring it to the king. The rope would then be folded four times, and the resultant sixteenth part would be the new standard foot.

golden ratio of phi.