Golden Ratio Quotes

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. ~ Alexey Stakhov

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

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. ~ R. A. Schwaller De Lubicz

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. ~ Jonathan Black

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. ~ Marja De Vries

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. ~ Sharon Weil

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

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

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. ~ Alison Charlotte Primrose

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

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. ~ Drunvalo Melchizedek

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. ~ Roshani Chokshi

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. ~ John Michell

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. ~ Ibrahim Ibrahim

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. ~ Einar Thorsteinn

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 ~ Samuel Colman

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

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. ~ Darrin Crescenzi

This relationship, often called the Golden mean, has been discovered and rediscovered at various times in history as a unique proportion believed to have both aesthetic and mystic significance. That the Egyptians knew of it and used it seems certain. ~ John Pile

The two solutions of the equation for the Golden Ratio are:

x1 = (1+ Sqr5) / 2

x2 = (1 - Sqr5) / 2 ~ Mario Livio

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. ~ Mario Livio

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. ~ Mario Livio

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. ~ Steven L. Griffing

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. ~ Richard A. Dunlap

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. ~ Anjan Chatterjee

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. ~ Euclid

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. ~ Schwaller De Lubicz

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. ~ Peter Tompkins

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 int ~ Mario Livio

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. ~ John Anthony West

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' ... ~ Willem Witteveen

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. ~ H.E. Huntley

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 .... ~ Carl Johan Calleman

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. ~ Bonnie Gaunt

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

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. ~ Robert K. G. Temple

It is arrogance to believe you can know all things. Perhaps you are right. But perhaps you are not. ~ Christie Golden

The character as well as the fortunes of the gospel is committed to the preacher. He makes or mars the message from God to man. The preacher is the golden pipe through which the divine oil flows. ~ Edward McKendree Bounds

[The eyes] are the most expressive part of a woman's body. ~ Arthur Golden

The golden rule would be to write a great, authentic song that is well produced and it will find its home. The audience can feel whether or not the artist is being genuine in their music. It's up to the artist to have the courage to reveal their truth through their songs. ~ Wendy Starland

I woke up remember the good times and the bad. I know one can't really exist without the other, so I always remind myself that life isn't about hanging onto memories, it's about experience. It's about learning from experience and understanding how to use it. It's about being intelligent enough to turn memories into golden moments, no matter what they may be. ~ Nicole DSettemi

I have never tried anything psychedelic, but I've always wanted to. Would I be wrapped up in colors, attacked by sounds, filled with insights about worlds I don't even know exist? ~ Rita Golden Gelman

Neighbours watching neighbours, twisting ropes to bind us all. ~ Jessie Burton

When you grow older you miss that eagerness; life may be happy, you may have health and wealth and love and success, but the odds are that you never look forward as you once did to a single golden day. You never count the hours to it, you never see some moment ahead beckoning like a goddess across a fourth dimension. ~ James Hilton

Percy was waiting for them. He looked mad.
He stood at the edge of the glacier, leaning on the staff with the golden eagle, gazing down at the wreckage he'd caused: several hundred acres of newly open water dotted with icebergs and flotsam from the ruined camp.
The only remains on the glacier were the main gates, which listed sideways, and a tattered blue banner lying over a pile of now-bricks.
When they ran up to him, Percy said, "Hey," like they were just meeting for lunch or something.
"You're alive!" Frank marveled.
Percy frowned. "The fall? That was nothing. I fell twice that far from the St. Louis Arch."
"You did what?" Hazel asked.
"Never mind. The important thing was I didn't drown. ~ Rick Riordan

There is no eleventh hour with Cosmic Ordering, only the golden hour. ~ Stephen Richards

I think I'm no different to my friends who are doctors or businessmen or architects - we all started watching films of the golden age together. But whether I'm making films or writing poetry or doing photography, it's very much rooted in my sense of unease. And that's really where everything goes back to. ~ Abbas Kiarostami

Why do they call it the Catholic Church? Because if they called it the Pedophiles Club, nobody would join. ~ Mike Golden

When I was little, we had a Golden Book that had all these Disney characters in one portrait on the first page. My dad used to read from it every night. We'd play this game of find Pluto or find Donald Duck. He'd read us stories and do all the voices. Those are great memories. ~ Danica McKellar

John Kerry went hunting today. He said he killed a goose. He didn't bring Teresa along because he was a little rusty and he was afraid he might kill the goose that laid the golden egg. ~ Jay Leno

(Golden Globe acceptance speech in the style of Jane Austen's letters):

"Four A.M. Having just returned from an evening at the Golden Spheres, which despite the inconveniences of heat, noise and overcrowding, was not without its pleasures. Thankfully, there were no dogs and no children. The gowns were middling. There was a good deal of shouting and behavior verging on the profligate, however, people were very free with their compliments and I made several new acquaintances. Miss Lindsay Doran, of Mirage, wherever that might be, who is largely responsible for my presence here, an enchanting companion about whom too much good cannot be said. Mr. Ang Lee, of foreign extraction, who most unexpectedly apppeared to understand me better than I undersand myself. Mr. James Schamus, a copiously erudite gentleman, and Miss Kate Winslet, beautiful in both countenance and spirit. Mr. Pat Doyle, a composer and a Scot, who displayed the kind of wild behavior one has lernt to expect from that race. Mr. Mark Canton, an energetic person with a ready smile who, as I understand it, owes me a vast deal of money. Miss Lisa Henson -- a lovely girl, and Mr. Gareth Wigan -- a lovely boy. I attempted to converse with Mr. Sydney Pollack, but his charms and wisdom are so generally pleasing that it proved impossible to get within ten feet of him. The room was full of interesting activitiy until eleven P.M. when it emptied rather suddenly. The lateness of the hour is due therefore not to the da ~ Emma Thompson

You'd be teary, too, yes you would be," he said, "if a girl and her bed had crashed into your head. ~ Renata Bowers

A tablecloth restaurant is still one of the great rewards of civilization. ~ Harry Golden

Under the sad end-of-days spell of the smoky dusk and the waning year, of the moon and its ostentatious superiority to the trashy, petty claptrap of his sublunar existence, why does he even hesitate? The Kamizakis are your enemies whether you do or not, so you might as well do it. Yes, yes, if you can still do something, you must do it - that is the golden rule of sublunar existence, whether you are a worm cut in two or a man with a prostate like a billiard ball. If you can still do something, then you must do it! Anything living can figure that out. ~ Philip Roth

Leah's baby was nestled protectively in the crook of Devon's arm.
In the golden firelight, his expression appeared grim. The neck of his shirt was open, the sleeves rolled up. With a soft gasp of mother love, Leah reached for her child. But Devon's deep, silky voice stopped her.
"Who is the baby's father? And do you love him?"
She froze. Her mouth went dry. Her heart pounded in her chest.
He waited. ~ Cathy Maxwell

Sleep pressed upon him with an urgency that became resistless. And April Bell was calling to him. Her voice came clearly to him, above all the subdued murmur of traffic noises. It was a ringing golden chime, more penetrating than the occasional beep of a driver's horn or the far clamor of a streetcar. It shimmered out of the dark, in waves of pure light as green as her malachite eyes. Then he thought he could see her, somehow, far across the slumberous town. Only she wasn't a woman. ~ Jack Williamson

The wind had blown the fog away, and the sky was like a monstrous peacock's tail, starred with myriads of golden eyes. ~ Oscar Wilde

I had to wonder if men were so blinded by beauty that they would feel privileged to live their lives with an actual demon, so long as it was a beautiful demon. ~ Arthur Golden

A poor girl may have an illusion that a prince will come and fetch her home. It is possible, some such cases have occurred. That the Messiah will come and found a golden age is much less probable. ~ Sigmund Freud

The goodness of the true pun is in the direct ratio of its intolerability. ~ Edgar Allan Poe

Usually, the murmur that rises up from Paris by day is the city talking; in the night it is the city breathing; but here it is the city singing. Listen, then, to this chorus of bell-towers - diffuse over the whole the murmur of half a million people - the eternal lament of the river - the endless sighing of the wind - the grave and distant quartet of the four forests placed upon the hills, in the distance, like immense organpipes - extinguish to a half light all in the central chime that would otherwise be too harsh or too shrill; and then say whetehr you know of anything in the world more rich, more joyous, more golden, more dazzling, than this tumult of bells and chimes - this furnace of music - these thousands of brazen voices, all singing together in flutes of stone three hundred feet high, than this city which is but one orchestra - this symphony which roars like a tempest. ~ Victor Hugo

The river - with the sunlight flashing from its dancing wavelets, gilding gold the grey-green beech-trunks, glinting through the dark, cool wood paths, chasing shadows o'er the shallows, flinging diamonds from the mill-wheels, throwing kisses to the lilies, wantoning with the weirs' white waters, silvering moss-grown walls and bridges, brightening every tiny townlet, making sweet each lane and meadow, lying tangled in the rushes, peeping, laughing, from each inlet, gleaming gay on many a far sail, making soft the air with glory - is a golden fairy stream. ~ Jerome K. Jerome

Now, writing every day, and being paid for it and encouraged to do it, it was as if, in the midst of the clich?d dark and stormy night, I found the magical inn, its windows golden lit, and Summer was due to start tomorrow. I can only work at one thing well. Deprive me of that, and my "back-up plan," even now, will be the empty, stormy, darkened heath
where, incidentally, even unpublished, somehow I'll still be writing. ~ Tanith Lee

Indian summer comes gently, folds over the hills and valleys as softly as the fall of a leaf on a windless day. It is always unexpected. After a sharp cold spell, we wake one morning and look out and the very air is golden. The sky has a delicate dreamy color, and the yet unfallen leaves on the bravest trees have a secure look, as if they would never, never fall. ~ Gladys Taber

So, I said, when does the enchantment start? We were sitting side by side, facing the mountains. "It started when the earth was born." Her eyes were closed. Her face was golden in the setting sun. "It never stops. It is, always. It's just here. ~ Jerry Spinelli

After all, as Edward Tufte once said, "Overload, clutter, and confusion are not attributes of information, they are failures of design. ~ Golden Krishna