Some contemporary literature that explores this subject is a book published in 2002 by astrophysicist and art aficionado Mario Livio called "The Golden Ratio". Livio became interested in the ratio while preparing a lecture on the "aesthetics of physics" and spent many years researching the origin of understanding and written history of this ratio, as well as the existence of it in our universe. Through two hundred and fifty plus pages, Livio shares the fascinating story of phi. With intellect and insight, he shares true, solid facts about phi. He painstakingly and scientifically dispels the myths, legends and hyperbole surrounding this number ( it is more than highly unlikely phi had anything to do with the building of the Great Pyramids, etc.). It's a fun ride through euclidean and fractal geometry, and touches on more recent discoveries such as the equally "peculiar" irrational number 1.13198824.
If you're a believer in PHI and you prefer your comfortable ideas undisturbed, you might not want to read the book. Livio connects the dots through an assemblage of names from Piero della Francesca, Dürer,
Luca Pacioli and Leonardo da Vinci to Seurat, Picasso, Rivera, Gris, Le Corbusier and Mondrian; Kepler Liebnitz, Bach, Mozart and Debussy make appearances. If you are a painter and you haven't yet mastered the use of phi for your compositions, I wouldn't worry much about it.. While it's great fun to study and to use to create visual art, Livio's exploration reaches this ultimate conclusion:
"All the attempts to disclose the (real or false)Golden Ratio in various works of art, pieces of music, or poetry rely on the assumption that a canon for ideal beauty exists and can be turned to practical account. History has shown however, that the artists who produced works of lasting value are precisely those who have broken away from such academic precepts. In spite of the Golden Ratio's importance for many areas of mathematics, the sciences, and natural phenomena, we should, in my humble opinion, give up it's application as a fixed standard for aesthetics, either in the human form or as a touchstone for the fine arts."
Albrecht Dürer, Piero della Francesca and Leonardo da Vinci, were gifted mathematicians as well as artists, the most active in mathematics being Piero. Pieros book "On Perspective" became the most important handbook of it's time for artists regarding the subject of perspective in painting. He also wrote two other books, one about the five Platonic solids, and one about the abacus.
Luca Pacioli(the ‘father’ of modern accounting) translated much of Piero’s work from Latin into Italian. It was Luca Pacioli who wrote an important book about the Golden Ratio called "La Davina Proportione", The Divine Proportion (1509). Historians to this day debate for whether Luca plagiarized much of Pieros work as his own. Pacioli was a great "borrower" and his famous book "Summa" which captured all of the important mathematical knowledge of his time, reported heavily about "double entry accounting". Pacioli didn't invent that either, but he documented this system already in use by Venetian merchants.
Luca Pacioli was passionate about the arts and he was sought to establish a mathematical ideal that could be used in the arts that would reveal to artists the “secret” of harmonic forms. Leonardo da Vinci was in Venice at the time Pacioli was writing his three volume treatise on the "Divine Proportion". Luca hired Leonardo as his illustrator. Da Vinci contributed many drawings of solids and polyhedra for the work. Pacioli's second volume of “Divina” attempted to apply this ‘sacred geometry’ to the human body and architecture and was largely based on the work of Marcus Vitruvius Pollio(70-25 BCE). Based on Vitruvius writings Renaissance scholars created further linkage between the organic and geometrical basis of beauty which led to the concept of the Vitruvian man, so famously drawn by Da Vinci. Contrary to popular misconception however, while the book describes “every sort of proportionality” that can be found in the human body, “Pacioli does not insist on the Golden Ratio as determining the ratio of all works of art. Rather, he specifically advocates the use of the Vitruvian system instead, which is based on simple (rational) ratios. That’s right; the Vitruvian man is NOT an example of the ‘Golden Ratio”. It is not based on the irrational number phi, it is based on RATIONAL numbers and proportions. While the Golden Ratio can be found, and is found in many things including humans and animals, there is a large variation in the real measures of these elements in specific individuals, and the proportion measured is often greatly different from the golden ratio,” ie it is NOT the building block of “everything”, only one of many.
Albrecht Dürer knew Pacioli. It has been speculated that the ‘pupil’ in the famous Barbari portrait of Pacioli is indeed Dürer himself. This is very possible as Albrecht Dürer knew Barbari as well. He published writing on mathematics and geometry, and some of these ideas did infuse some of his work. However, beyond Dürer, Da Vinci and Pacioli’s mathematical and artistic investigations into such things, not much of these ideas influenced visual art itself on any widespread basis in the 16th century. Pop open any great collection of Raphael’s works and you will see many examples of centered and circular composition.
Among non-artists, Da Vinci seems to be linked the most to the idea of the Golden Ratio being used ‘secretly’. For years I heard that his painting “Madonna of the Rocks” was created using these “divine “measurements. However, while the focal points come close to phi, they come much closer to a RATIONAL number.
It either IS phi, or it is NOT. Close doesn’t count. You don’t have ‘close’ in nature. Imagine a world of logarithmic spirals out of whack, of crooked seashells and pine-cones and roses and apple cores, all lopsided. Doesn’t happen regularly; either it IS phi, or it ISN’T! It also turns out that the first version of this painting he created was some years prior to his association and subsequent work with Luca Pacioli.
The book goes on to explore much, much more regarding its appearance, or not, in music, architecture, etc. and the reappearance of the exploration of this idea in Academic art circles, and even in the 20th century. Dali and Seurat both painted works using it. If you look at those works, and compare them to a centrally composed painting by Raphael, ask yourself if you think they are really stronger or more powerful, or more beautiful. Personally, I prefer some of the Raphael’s if I am considering beauty alone.
I would close with this request; show me a master work that for certain was composed with phi; show me one of YOUR works composed with phi. Show me one with the actual focal points plotted at phi; not a work that might be ‘close’ to phi. Close means it might be simply a rational number, or some other irrational number. I love the idea of phi. It is fascinating. It is infused in our universe and appears many places. However, it is not alone. As science and mathematics progress, they are discovering other peculiar irrational numbers. Livio’s discussion of fractal geometry itself is worth the price of the book. In sum, it is worth remembering that beautiful, harmonious, powerful or arresting composition can come from many simple ratios, balance and shapes; it can come in many forms.
Mario Livio’s “The Golden Ratio: The Story of PHI. The Worlds Most Astonishing Number” is a wonderful education for any artist seeking further understanding of this subject.