While the proportion known as the Golden Mean has always existed in mathematics and in the physical universe, it is unknown exactly when it was first discovered and applied by mankind. It is reasonable to assume that it has perhaps been discovered and rediscovered throughout history, which explains why it goes under several names.
Uses in architecture potentially date to the ancient Egyptians and Greeks
It appears that the Egyptians may have used both pi and phi in the design of the Great Pyramids. The Greeks are thought by some to have based the design of the Parthenon on this proportion, but this is subject to some conjecture.
Phidias (500 BC – 432 BC), a Greek sculptor and mathematician, studied phi and applied it to the design of sculptures for the Parthenon.
Plato (circa 428 BC – 347 BC), in his views on natural science and cosmology presented in his “Timaeus,” considered the golden ratio to be the most binding of all mathematical relationships and the key to the physics of the cosmos. It was not known as the golden ratio in his time, but he describes it with his first reference to proportion:
“Now it is not possible for two things to be combined well on their own without a third, for some bond is required between the two to draw them together. The very best bond is that which, as much as possible, makes itself and the conjoined entities, one; and it is proportion that by nature best accomplishes this. So whenever the middle item of three numbers or volumes or powers is such that the first is to the middle as the middle is to the last, and again, that the last is to the middle as the middle is to the first, then the middle becomes first and last, and the last and first for their part both become middles. Accordingly it follows, of necessity, that they all turn out to be the same, and since they have all become the same as one another, they will all be one.” Translation © 2021 by David Horan
This is analogous to what Euclid later wrote.
Euclid (365 BC – 300 BC), in “Elements,” referred to dividing a line at the 0.6180399… point as “dividing a line in the extreme and mean ratio.” This later gave rise to the use of the term mean in the golden mean. He also linked this number to the construction of a pentagram.
The Fibonacci Sequence was written of in India in about 200-300 BC and brought to the Western world around 1200 AD
What we now as the Fibonacci sequence is named after Leonardo Pisano Bonacci (aka Bigollo) of Pisa, an Italian born in 1175 AD, who later became known as Leonardo Fibonacci. His book Liber Abaci, published in 1202, introduced this sequence to Western European mathematics in the form of a math problem on the breeding of rabbits. He learned of it though while being educated in North Africa with an Arab master, where he was exposed to the much earlier knowledge of Indian mathematicians. Liber Abaci became a pivotal influence in adoption by the Europeans of the Arabic decimal system of counting over Roman numerals. (3)
The sequence itself though had been described as early as the 2nd or 3rd century BC in the works of Acharya Pingala, an Indian mathematician who enumerated the possible patterns of Sanskrit poetry that could be formed from syllables of two lengths.
The contributions of Pingala and Fibonacci are important, but it’s not apparent that anyone even realized its connection to the Golden Ratio until the 1600’s by Johannes Kepler and others.
It was first called the “Divine Proportion” in the 1500’s
Leonardo Da Vinci provided illustrations for a dissertation published by Luca Pacioli in 1509 entitled “De Divina Proportione” (1), perhaps the earliest reference in literature to another of its names, the “Divine Proportion.” This book contains drawings made by Leonardo da Vinci of the five Platonic solids.
The Renaissance artists used the Golden Mean extensively in their paintings and sculptures to achieve balance and beauty. Leonardo Da Vinci, for instance, used it to define all the fundamental proportions of his painting of “The Last Supper,” from the dimensions of the table at which Christ and the disciples sat to the proportions of the walls and windows in the background.
Johannes Kepler (1571-1630), discoverer of the elliptical nature of the orbits of the planets around the sun, also made mention of the “Divine Proportion,” saying this about it:
“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.”
The “Golden Ratio” was coined in the 1800’s
It is believed that Martin Ohm (1792–1872) was the first person to use the term “golden” to describe the golden ratio. to use the term. In 1815, he published “Die reine Elementar-Mathematik” (The Pure Elementary Mathematics). This book is famed for containing the first known usage of the term “goldener schnitt” (golden section).
The term “Phi” was not used until the 1900’s
It wasn’t until the 1900’s that American mathematician Mark Barr used the Greek letter phi (Φ) to designate this proportion. This appeared in the “The Curves of Life” (page 420) in 1914 by Theodore Andrea Cook . By this time this ubiquitous proportion was known as the golden mean, golden section and golden ratio as well as the Divine proportion. Phi is the first letter of Phidias (1), who used the golden ratio in his sculptures, as well as the Greek equivalent to the letter “F,” the first letter of Fibonacci. Phi is also the 21st letter of the Greek alphabet, and 21 is one of numbers in the Fibonacci series. The character for phi also has some interesting theological implications.
Recent appearances of Phi in math and physics
Phi continues to open new doors in our understanding of life and the universe. It appeared in Roger Penrose’s discovery in the 1970’s of “Penrose Tiles,” which first allowed surfaces to be tiled in five-fold symmetry. It appeared again in the 1980’s in quasi-crystals, a newly discovered form of matter.
Phi as a door to understanding life
The description of this proportion as Golden and 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 a single number to play, but then again this one number has played an incredible role in human history and in the universe at large.
Source – The Divine Proportion : A Study in Mathematical Beauty by H. E. Huntley
(1) Page 25
(2) Page 157
(3) Page 158
The Golden Ratio is found in:
• Geometry
• Art
• Architecture
• Nature
• The human body
• And many other things
What do u mean by many others???
Mathematics, design, the solar system, plants, animals, stock markets, beauty and cosmetic surgery to name a few. See https://www.goldennumber.net/site-map/ for the full list on this site.
There’s more to the eye that these numbers represent. Numbers are a universal language. Their origin and placement in important designs and belief systems should be noticed. Can you connect the dots?
I actually did a project on the Golden Ratio. I’m going to States for it.
Could you perhaps tell me what exactly you focused on? I am writing a conference paper on Phi and do not know where to start.
I’m about two years too late to help on your topic, but I wrote a paper on phi as it appears in the musical ratios of many cultures, most insistently in western music.
As a legend regarding Pythagoras comes down to us, he heard the golden mean proportions as he passed a metallurgy forge, and the metalsmith’s hammer on different lengths of iron clanged out our now-beloved and long-internalized series of aurally appealing, non-repeating intervals whose spans diminish as they descend from higher to lower tones through inversely larger ratios of 21/13, 13/8, 8/5, 5/3, 3/2, 2/1. On a keyboard they sound out the octave, the fifth, the fourth, the major third, minor third, second, half-tone . . . though no smaller, like Indian and Arabic scales proceed, into yet finer and evermore discerning microtones for evermore subtle and sophisticated music.
In any case, western music, like western art, is utterly beholden on phi’s beauty from its theoretical foundations through its ascending architectural proportions—diatonic scales and the anchoring dominant and subdominant tonalities as though they were columns, arches, the apse, dome and rotunda—to the Golden Ratio and its linear iteration—the Fibonacci sequence. You will dazzle any professor with a well-constructed explication on the power of Pythagoras and Phidias over Purcell and Prokofiev.
True I have ever read about golden ratio. But what is the real connection between golden ratio and Fibonacci ?
For the connection of the golden ratio and Fibonacci sequence see https://www.goldennumber.net/fibonacci-series/.
The foundation of the Asiatic sequence isthe original man other wise known as the black man P.S. not aliens
Many thanks.
We have a divine ratio described quite divinely on the front page of Shake-Speares sonnets. What is fascinating is we also appear to have e., and it’s 1609.
I will avoid posting my analysis of this to date in here, it may get spam busted anyway,, I made a short 20 minutes video analysing the accuracy of this. If you are interested in this please contact me. mc_lester@yahoo.co.uk
nice work. it gave a lot help to my group when doing our project. Thanks for your resource. :):)
This is food for thought.
i agree man
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As a geometrical relationship of relationships (and taking a Platonic view), the Golden Ratio exists beyond time and space as it can be viewed as a logical system of information, like Godel’s incompleteness theorem. It needs no matter nor energy expression or substrate to exist independently as a Platonic archetype. This is the power of geometry – of which mathematics is just a code, like html is to this graphic webpage.
Explicit reference can be traced to Plato and Euclid but since it is often attributed to Pythagoras (and since Plato and Euclid are intellectual followers of Pythagoras), and the symbol of the Pythagoreans is the Pentagram or Five-pointed star that inspired Da Vinci’s Vitruvian man, which is Golden Ratio proportioned, it’s reasonable to assume he first popularised it.
Interestingly, he was a cross between John Cage, Spinoza, Newton and Julian Assange in fusing music, philosophy physics-mysticism and political resistance to tyranny, a true renaissance man before his time. It’s said that he went to Egypt and leant geometry from the Priest-Architects there, who as mathematicians and surveyors, used simple triangular geometry with ropes & pegs to remap the temple farm lands each year after the seasonal Nile floods. They perhaps would have shown him the Golden Ratio in the proportions of the Great Pyramid as well. So we could say it’s likely the Priest-Architects of ancient Egypt and Mesopotamia first codified the Golden Ratio in a cultural-economic and agricultural measurement response to the seasonal vagaries of the flooding Tigris, Euphrates and Nile.
The wealth generated from the exploitation of those lands allowed the construction of the magnificent temples and tombs we can still see there to this day,
All this measurement history however – is statics, it’s spatial.
For the temporal manifestation of the Golden Ratio and it’s role as the signature of the new Constructal design law of nature and culture, please refer to this seminar given late last year in Shanghai:
Cosmomimetic Design in Nature&Culture – Asynsis Principle-Constructal Seminar:
ShanghaiUniNantesEcoledeDesign http://wp.me/p1zCSP-1S via @ASYNSIS
http://asynsis.wordpress.com
http://constructal.org
http://www.forbes.com/sites/anthonykosner/2012/02/29/theres-a-new-law-in-physics-and-it-changes-everything/
The fingerprints of God! 🙂
I agree. This is obviously evidence of a God. The chance of this happening from a big bang (who’s the big banger) is 1 in infinity AND BEYOND (buzz lightyear joke). There must be some sort of conspiracy for this with BIG money behind it.
Fingerprints? Uhmmm… we may need a forensic scientist to investigate this so-called god you mention. I can understand calling something you don’t understand completely, god. But if God is Ignorance it is slowly being driven further back with new discoveries and understandings. Every culture and age has a new “god”, this concept will certainly change as we go along.
And on the other side, those who choose to not believe in God just explain the things they don’t understand completely with rationalizing terms such as spontaneous generation, self-organization, punctuated equilibrium and such. These have no more scientific proof behind them than God. Paraphrasing your own words, “but if pseudo-scientific rationalization is ignorance, it is slowly being driven further back with new discoveries and understandings. Every culture and age has a new “science,” this concept will certainly change as we go along. The faith of all monotheistic religions in One God as our Creator has lasted for thousands of years, longer than the science of any culture or age.
I think you are mistaking the scientific method for practice. The scientific method, unlike religious or profane belief systems, does not seek to prove anything. The premise is to disprove (as in the use of the null hypothesis), and reduce through repetition the likelihood that notions of chance and metaphysics may be the cause of observed phenomena or results. For example, your statement the monotheism has lasted for thousands of years, is contentious and false, easily disprovable in the historic analysis of modern and medieval archaeology, extant architecture and texts, oral and written. Scientific method was being practiced by many Socratic, Platonists, Daoists. Evidence from modern hunter-gatherer observations and anthropological observations of those bands no longer allowed to survive, all show the global prevalence of scientific deductive, inductive and inferential methods as a preferred means of analysing the environment to survive.
I agree. I should have been clearer in my use of the term “science” in contrast to “scientism” or any other abuse that falls short of true “science.” Science should be nothing more or less than a methodology of investigation of the evidence that produces the most accurate and truthful conclusions about the nature of things. Scientism, by contrast, is like any other form of religion or belief system. It starts with a particular viewpoint in mind and filters out evidence that contradicts it. As described by Wikipedia, “Scientism is the promotion of science as the best or only objective means by which society should determine normative and epistemological values.” If the evidence indicates logical reasons to conclude that a supreme intelligence is behind the design of life, then true science would openly explore that possibility rather than categorically dismissing it as being “unscientific” as those practicing scientism do. Humankind has made great strides forward in applying the scientific method of investigation, and lost its way when believing that scientism and science are one in the same. Here’s a video I compiled of other videos that express this: https://www.youtube.com/watch?v=AcGw21Wbcgk
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I find that one of the more interesting things about phi is the manner in which the division of adjacent pairs of the Fibonacci series converges to the limit. Pick any pair at random, and it will be either above or below the limit. The next pair, above or below, will be on the other side: the quotient of the adjacent pairs of the Fibonacci series will converge from alternate sides of the limit all the way to infinity.
I recognize the age-old significance of the golden ratio and the worship of the female form.
The Sanskrit the golden sequence is based on is written in units of “light” and “gravity”, kind of like the universe itself.
The guy who wrote it down is Bonacci, but we call him Fibonacci because “fi” means filial, and he shares his last name with his father.
Phidias is reason the symbol “phi” is used for the golden ratio. Literally, his name means “faith in sky father” (from “fides” and “dias”/Zeus).
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That aliens will speak math when they come?
If aliens do come by any means involving moving matter across space, the technological knowledge that got them here will certainly have had a foundation in mathematics. As Galileo said, “Mathematics is the language with which God has written the universe.” Should make for interesting conversation at our first intra-galactic dinner event.
I’m leaving open the possibility that aliens could transport or appear to us as in stories of angels. That might involve a different form of energy than we now know, and not require all the same mathematical calculations to get here.
im doing a project on how the golden ratio and fibonacci numbers are used in the greek culture, how did they use them?
No record of evidence that I know of for the Greek’s use of the Fibonacci series, but for their application of the golden ratio see https://www.goldennumber.net/parthenon-golden-ratio-design/.