A ratio with several names
The golden ratio is the number (1 plus the square root of 5) divided by 2, which is about 1.618. It is also called the golden section, the golden mean and the divine proportion. Each name carries a different history, and a good deal of what is repeated about it online mixes mathematics with later enthusiasm. The most reliable way to sort the two is to start with what Euclid wrote and then follow what was added.
Euclid’s extreme and mean ratio
In Book VI of the Elements, around 300 BCE, Euclid defines a way of cutting a line. A straight line is said to be cut in extreme and mean ratio when the whole line is to the greater segment as the greater segment is to the less. If the whole line has length 1 plus x, with the greater part 1 and the lesser part x, the condition says that (1 plus x) divided by 1 equals 1 divided by x. That leads to the golden ratio for the longer part compared with the shorter.
Earlier, in Book II, Proposition 11, Euclid shows how to construct such a division using only compass and straightedge. He does not call the result golden, and he makes no claim about beauty. For him it was a problem in geometry with a useful application: the construction is needed in Book XIII to build the regular pentagon and to relate the sides of the dodecahedron and icosahedron, two of the Platonic solids. The ratio appears because it is built into the geometry of the pentagon, whose diagonal and side stand in this proportion.
Some historians suggest the problem was studied before Euclid, possibly in the circle of Plato or earlier, but the surviving evidence is slim and the attribution is uncertain.
A number that is not a fraction
The golden ratio is irrational, so no fraction equals it exactly. Ratios of successive terms of the Fibonacci sequence, such as 8 to 5 (1.6), 13 to 8 (1.625), and 21 to 13 (about 1.615), settle closer and closer to it. The relationship was noticed in the early seventeenth century; Johannes Kepler is often cited as describing it. It is a true mathematical property but does not by itself say that Fibonacci numbers are everywhere in nature.
Medieval and Arabic transmission
The Elements reached medieval Europe through Arabic and Latin translations, and the extreme and mean ratio remained a technical topic in geometry. Campanus of Novara’s thirteenth-century Latin edition of Euclid, for example, carried the material on the pentagon and the division of a line to later readers. At this stage no one was calling it divine or golden, and it was not an aesthetic principle.
Luca Pacioli and the divine proportion
The change in tone comes with the Franciscan friar and mathematician Luca Pacioli. His book De divina proportione was written in the 1490s and printed in Venice in 1509. It discusses the ratio in the language of theology, comparing its properties to attributes of God, which is how it earned the name divine proportion. Pacioli listed several of these properties, which he linked to features such as unity and irrationality, and these were symbolic comparisons, not scientific claims.
The book is also known for its illustrations of polyhedra, which were drawn by Leonardo da Vinci, who was Pacioli’s friend and collaborator at the Milanese court. Those drawings are real and well documented. What is less secure is the often-repeated claim that Leonardo deliberately applied the golden ratio in paintings such as the Mona Lisa or the Last Supper. No surviving writing by Leonardo says so, and measurements of paintings can be made to fit many ratios.
The name “golden”
The term “golden section” in German, “goldener Schnitt”, appears in print in the nineteenth century, and its spread is associated with writers of that era, including Martin Ohm and later Adolf Zeising, who argued that the proportion underlies beauty in nature and art. These claims became popular but were not based on systematic evidence. In the twentieth century, the architect Le Corbusier built his Modulor system partly around the ratio, which shows that the ratio can be used as a design tool by choice, not that it is inherent to good design.
Checking the popular claims
Many statements about the golden ratio deserve caution:
- The Parthenon and the Great Pyramid are often said to embody it. Such claims typically depend on choosing which dimensions to measure, and the builders left no record of an intention to do so.
- Human body proportions are sometimes said to follow it, but real bodies vary and the supposed matches are loose.
- Plant spirals often show consecutive Fibonacci numbers, and some plants do grow in ways related to the golden angle, but many plants do not.
- Psychological experiments on whether people prefer rectangles of this proportion have given mixed and uncertain results.
None of this makes the ratio less interesting as mathematics. It shows only that a short list of well-documented facts, namely Euclid’s construction, the pentagon, the link to Fibonacci numbers and Pacioli’s book, supports much less than the legend of a universal law of beauty.
Reading the history carefully
A fair account therefore has three layers. Euclid supplied a geometric definition and construction. Renaissance writers such as Pacioli supplied a symbolic vocabulary. The nineteenth and twentieth centuries added the aesthetic claims. Keeping the layers apart makes it easier to appreciate the first without accepting the third.