The History of Phi & The Golden Ratio
The Golden Ratio – Phi – is known by many other names such as; The Golden Section, Golden Number, Golden Mean, Golden Proportion, Golden Rectangle, Golden Triangle, Golden Spiral, Golden Cut, The Divine Proportion, Fibonacci Sequence, and Tau (τ).
Who first discovered The Golden Ratio?
No one really knows for sure as to who originally discovered the The Golden Ratio – 1.618 – (a.k.a. the Phi ratio) in Human history, but some suggestions point to the ancient Egyptians and even as far back as the ancient Sumerian civilization. However, the ratio was also used by many artists, mathematicians, scientists and philosophers throughout history. The most renowned include the Greeks, such as:
Pythagoras (570–495 BC) was an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism. He is best known for the Pythagorean theorem which bears his name to this day. Although, because legend and obfuscation cloud his work even more than with the other pre-Socratic philosophers, one can say little with confidence about his teachings, and some have questioned whether he contributed much to mathematics and natural philosophy. Many of the accomplishments credited to Pythagoras may actually have been accomplishments of his colleagues and successors.
Pythagoras thought that absolute happiness lay in the contemplation of the harmony of the rhythms of the Universe, (“tes teleiotetos arithmon”- the perfection of numbers, the number being both rhythm and proportion). In other words, Pythagoras was looking for a numerical pattern that would explain it all. A goal of Pythagoras’ disciples was to develop a ratio theory. The ratio, a comparison between two sizes represented in mathematics a basic operation of judgment: the perception of the relation between ideas.
Hippocrates of Chios (470–410 BC) was an ancient Greek mathematician, (geometer), and astronomer. He is reported to have been kicked out of a secretive group for having divulged the construction of the pentagram. The pentagram and pentagon are constructed using Phi ratios (as well as deriving Phi from these geometric shapes; as I also discovered for myself back in 2004 and is mentioned under the article “What Followed After My Discovery of OneEye“). Supposedly, the construction of the pentagram or pentagon was one of the secrets of the Medieval Mason’s guilds. The Pentagram was used as a sign of salutation by the Pythagoreans, its construction supposed to have been a jealously guarded secret.
Phidias (490–430 BC) was a Greek sculptor, painter and architect, who lived in the 5th century BC and is said to exhibit Phi ratios in his body of artistic works without a mention of using mathematics. He made the Parthenon statues that seem to embody the Golden Ratio.
Plato (428–347 BC) was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world.
He wrote a dialogue named Timaeus (Greek: Τίμαιος, Timaios) one of Plato’s dialogues, mostly in the form of a long monologue given by the title character, written circa 360 BC. The work puts forward speculation on the nature of the physical world and human beings. [source]
“For whenever in any three numbers, whether cube or square, there is a mean, which is to the last term what the first term is to it; and again, when the mean is to the first term as the last term is to the mean—then the mean becoming first and last, and the first and last both becoming means, they will all of them of necessity come to be the same, and having become the same with one another will be all one”; thereby he implies the aesthetically perfect proportion known as Golden ratio or Golden mean. (31c – 32a).” [source]
Euclid (323–283 BC) is often referred to as the “father of geometry”, where he wrote the collection of 13 books named “The Elements” and was written in Greek around 300 BC. In “The Elements”, he gave the first recorded definition of the golden ratio, which he called, as translated into English, “extreme and mean ratio”.
Many historians do not believe that Euclid’s Elements represents his original work so there is the question of who studied the golden ratio before Euclid. Proclus, a Greek mathematician who lived several centuries after Euclid, wrote in his commentary of the Elements:
“Euclid, who put together the Elements, collecting many of Eudoxus’ theorems, perfecting many of Theaetetus’, and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors”.
Some historians believe that Book II of The Elements covers material originally studied by Theodorus of Cyrene while others attribute the material to Pythagoras, or at least to the Pythagoreans. Proclus, writing in the fifth century AD, claims:-
Eudoxus … multiplied the number of propositions concerning the section which had their origin in Plato, employing the method of analysis for their solution.
Many believe that by ‘section’ Proclus means ‘golden ratio’. Eudoxus certainly attended lectures by Plato so it is entirely reasonable that he might work on topics suggested during these lectures. Heath writes in his edition of Euclid’s Elements:-
This idea that Plato began the study of [the golden ratio] as a subject in itself is not in the least inconsistent with the supposition that the problem of Euclid II, 11 was solved by the Pythagoreans. [source]
The Golden Ratio (Phi) was known and used widely during the European Renaissance of the 14th–17th centuries, particularly by the renowned inventor/artist Leonardo Da Vinci. It is believed that Leonardo, as a mathematician, purposefully made his paintings line up within Golden Ratios in order to further the incorporation of mathematics into art. Such works were the “Mona Lisa”, “The Last Supper” and “The Vetruvian Man”. Further evidences will be provided in another chapter here at EyePhi.com (“Phi in Art, Music, Beauty and Design”).
Going back even further than the Greeks, we see that the Egyptians built their Pyramids and designed them within Phi ratios. It is also a well known fact that the dimensions of the Great Pyramid of Khufu at Giza also exhibits the other irrational ratio – Pi (3.1416). Evidence of this will be shown on another chapter here at EyePhi.com (“Phi in Theology & Ancient Monuments”).
More recently in history…
Leonardo of Pisa (a.k.a. Fibonacci) (1170–1250) mentioned the numerical series and is now famously named after him in his Liber Abaci; the ratio of sequential elements of the Fibonacci sequence approaches the golden ratio asymptotically.
Johannes Kepler (1571–1630) proves that the golden ratio is the limit of the ratio of consecutive Fibonacci numbers, and describes the golden ratio as a “precious jewel”: “Geometry has two great treasures: one is the Theorem of Pythagoras, and 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.” These two treasures are combined in the Kepler triangle.
Charles Bonnet (1720–1793) points out that in the spiral phyllotaxis of plants going clockwise and counter-clockwise were frequently two successive Fibonacci series.
Edouard Lucas (1842–1891) gives the numerical sequence now known as the Fibonacci sequence its present name.
Mark Barr (20th century) suggests the Greek letter Phi (Φ), the initial letter of the Greek sculptor Phidias, as a symbol for the Golden Ratio (1.618…). Mark Barr was an American mathematician who, according to Theodore Andrea Cook (a British art critic and writer), in about 1909, gave the Golden Ratio its name – “Phi“.
If one was to take the first letter of Phidias, one would naturally think of using the uppercase Greek letter Φ. Unfortunately, the lowercase representation φ of the Phi ratio (1.618…) has become a more recent phenomenon amongst some sites including Wikipedia, but I will stick to the traditional representation of Phi as the uppercase Greek letter Φ (1.618…) and the inverse of this to be represented by the lowercase Greek letter φ (0.618…); in line with the more renowned academic establishments and scholarly figures such as Mario Livio and Roger Penrose.
Roger Penrose (b.1931) discovered a symmetrical pattern that uses the golden ratio in the field of aperiodic tilings, which led to new discoveries about quasicrystals.
Where did the terms “Golden Ratio”, “Divine Proportion”, and “Phi” originate?
The term “Phi” and its symbol Φ was first coined by Mark Barr at the beginning of the 20th century in commemoration of the Greek sculptor Phidias.
The term “The Divine Proportion” is said to be first mentioned by Luca Pacioli (1445–1517) after defining the Golden Ratio as the “Divine Proportion” in Divina Proportione. It contains drawings made by Leonardo Da Vinci of the 5 Platonic solids. It is reasonable to say that Leonardo (da Vinci) was the first to call it the “sectio aurea” (Latin for the Golden Section).
If you search on Google for the phrase “the history of the golden ratio” you will also find a great resource of information with a time line included from 3000 BC to now.
Although not mentioned, I believe that other ancient races also knew of this ratio. One such race I believe were the Mayans. The Mayans were masters in Astronomy and of calculating the exact movements and positions of cosmic bodies within our Galaxy. As such, they created a calendar that is considered cyclic – The Great Cycle of the Mayan Long Count. You may ask what this has to do with “Phi”, but rest assured I will elaborate on this under “Phi in Theology & Ancient Monuments”.
<< Back to an intro on Phi Φ – The Golden Ratio
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After much thought, I have decided to stick with traditional representations by using the uppercase Greek letter Φ to represent the Phi ratio (1.618…), and the lowercase Greek letter φ to represent the inverse of the Phi ratio (0.618…), despite a more recent phenomenon amongst some sites including “Wikipedia” which includes works from people who have done things the other way around.