III
THE DIVINE PROPORTION
“Without mathematics there is no art.” 1
—Luca Pacioli
“Where the spirit does not work with the hand there is no art.”
—Leonardo da Vinci
Now we will examine a variety of applications of the golden ratio in Renaissance art and beyond. In doing so, we step from the world of the absolute precision and verifiable proofs of mathematics and geometry into the more subjective world of beauty and aesthetics. Thus venturing into a domain where our heart tells us what logic cannot, we’ll also be stepping into a world of controversy, full of conflicting and polarizing claims that lead to much misinformation and confusion about the golden ratio. This is where you play the role of detective, judge, and jury. Did the Renaissance masters truly and intentionally incorporate the golden proportion into some of their most revered works? I’ll present the best evidence available, and your task will be to examine the evidence and come to your own conclusions.
French artist Horace Vernet’s 1827 painting shows Pope Julius II ordering architect Donato Bramante and Renaissance masters Michelangelo and Raphael to build the largest church in the world, St. Peter’s Basilica.
Up to this point in this book we’ve seen that there is beauty in mathematics, but as the Italian friar Luca Pacioli astutely observed, there is also mathematics in beauty. Euclid’s Elements was reintroduced to Europe via a Latin translation around the year 1120, and it became one of the most widely circulated books after the invention of the printing press in the 1450s. Although no other written reference to the golden ratio appeared until the late 1490s, there is clear and compelling evidence that some of the greatest artists of this period were applying it in the compositions of their paintings as early as the 1440s. The application of the golden ratio to the arts was later revealed to be a “secret science,” and, as we’ll see next, it seems that many of the great Renaissance masters were in on the secret, including the likes of Piero della Francesca, Leonardo da Vinci, Botticelli, Raphael, and Michelangelo. However, it was Pacioli who produced the first comprehensive study of this special number, which he dubbed “the divine proportion.”
This painting from 1495 shows Pacioli in his Franciscan habit drawing a mathematical diagram with his left hand upon an open book. In the right corner of the table is a dodecahedron. The young man behind him is probably a student—possibly the German artist and polymath Albrecht Dürer, who was in his early twenties and visiting Italy when the painting was made.
DE DIVINA PROPORTIONE
Luca Pacioli, who lived from 1447 to 1517, was a man of varied interests and talents. He was a Franciscan friar, mathematician, and friend of Leonardo da Vinci, with whom he collaborated. Known as the “Father of Accounting and Bookkeeping,” he was also the first author in Europe to publish a detailed work on the double-entry system of accounting. Soon after the publication of his six-hundred-page Summa de arithmetica (Summary of Arithmetic) in 1494, he was invited by the Duke of Milan, Ludovico Sforza, to take up residence. This led to his fateful meeting with da Vinci, who became his pupil in mathematics as Pacioli worked on De Divina Proportione. Written between 1496 and 1498, and published in 1509, this book connected mathematics to art and architecture, exploring the presence and uses of phi throughout history. His illustrator was none other than da Vinci himself, who lived with Pacioli during the late 1490s.
The title pages from Pacioli’s Summa de Arithmetica and De Divina Proportione, with Ludovico Sforza, who presided over the final and most productive stage of the Milanese Renaissance. Famed as a patron of Leonardo da Vinci and other artists, he commissioned The Last Supper around 1495 and brought Pacioli and da Vinci together.
In his monumental three-volume treatise, Pacioli captured the breadth and depth of this topic in the opening words of his introduction and statement of intent:
“A work necessary for all the clear-sighted and inquiring human minds, in which everyone who loves to study philosophy, perspective, painting, sculpture, architecture, music and other mathematical disciplines will find a very delicate, subtle and admirable teaching and will delight in diverse questions touching on a very secret science.” 2
By discussing mathematical proportion—especially the mathematics of the golden ratio—and its application in art and architecture, Pacioli hoped to enlighten the general public about the secret of harmonic forms. As we’ve seen already, some geometric solids, such as dodecahedrons and icosahedrons, have inherent golden ratios within their dimensions and in the spatial positions of their intersecting lines. However, he revealed other examples of golden ratio proportions in the dimensions of Greco-Roman structures and Renaissance paintings. We even find the golden ratio in the letter G of his beautiful architectural script letters!
Until Pacioli’s time, phi was known as the “extreme and mean ratio” described by Euclid. Although long recognized for its uniqueness and beauty, it was Pacioli who first dubbed 1.618 as “divine.” The theological implications, coupled with da Vinci’s precise renderings of three-dimensional skeletonic solids, popularized the study of phi and geometry among artists, philosophers, and more.
Italian artist Raffaello Sanzio Morghen created this engraving of a middle-aged Leonardo da Vinci in 1817.
This woodcut of the beautiful gate of Solomon’s Temple in Jerusalem, which appears in the 1509 edition of De Divina Proportione, contains golden proportions.
Da Vinci drew all of Paciolil’s original polyhedrons in his book, including the dodecahedron (above) and the Archimedean truncated icosahedron (below).
Pacioli’s G displays clear golden ratio proportions.
PIERO DELLA FRANCESCA
The third volume of Pacioli’s De Divina Proportione was an Italian translation of Piero della Francesca’s Short Book on (the) Five Regular Solids, which was written in Latin. While known in his own time mostly as a mathematician and geometer, Piero della Francesca (1415–1492) is now primarily recognized for his works as an artist.
Piero wrote De Prospectiva Pingendi (On Perspective for Painting) later in his career, but his understanding and appreciation of perspective and proportion is evident in his earlier works. In the very first of his extant paintings, The Baptism of Christ (c. 1448-1450), we see that Piero has Christ perfectly positioned between the two golden ratios formed by the sides of the canvas, and also between the two trees.
The Flagellation of Christ (see here) was probably painted between 1455 and 1460, and it is recognized for its complex composition on a panel of only 23 by 32 inches (58 by 81 cm). British art historian Kenneth Clark called it “the greatest small painting in the world.” 3 Using my PhiMatrix software, it’s easy to see that Piero carefully applied the golden ratio in the room to the left. There we find Christ at the golden ratio of the width of the room, whether measured where the floor tiles change or from the columns at its entry. The architectural features of the buildings also show alignment with the golden ratio gridlines (green).
Another painting that displays golden proportions is Polyptych of the Misericordia, (see here) completed between the years of 1445 and 1462. Here we see the crowned Madonna standing with arms outstretched. At the golden ratio of her height, there is a sash tied around her waist. The width of the sash at her waist is in golden ratio proportion to the length between her outstretched hands.
Examining the painting even more closely, we can see that Piero applied the golden ratio twice more—once horizontally in the off-centered knot of the belt, and again vertically in the lengths of rope hanging from the knot.
Thus, sixty years before De Divina Proportione was published by Luca Pacioli, we find evidence that Renaissance painters implemented the golden ratio as a means of creating visual harmony within paintings. Furthermore, in religious art the golden ratio may have been used by the artists to incorporate an element of the eternal or the divine into their works.
Baptism of Christ, c. 1449.
Madonna della Misericordia (Our Lady of Mercy), 1445-1562.
The Flagellation, c. 1457.
Golden proportions also abound in this painting of the burial of Christ, which appears directly below the Madonna in Piero della Francesa’s Polyptych of the Misericordia.
LEONARDO DA VINCI
A half millennium after his death in 1519, we still celebrate Leonardo da Vinci for his brilliant insights as an inventor and scientist. But this polymathic genius was a legend in his own time as well, being described by his contemporaries as a “divine” painter. As the illustrator of Pacioli’s De Divina Proportione and as a central figure in the plot of Dan Brown’s 2003 bestseller The Da Vinci Code, Leonardo has long been associated with the golden ratio. However, as we will see, da Vinci’s association with the golden ratio runs much longer and deeper than many of us realize.
While still a young man under the tutelage of the Florentine artist and sculptor known as Verrocchio (“true eye”), da Vinci painted Annunciation—a scene showing the announcement to Mary Virgin by the angel Gabriel that she would become the mother of Christ—which displays some interesting proportions. Painted around 1472–1475, it is thought to be his earliest surviving work.
As shown, the golden ratio appears to be the basis for the dimensions of the walls and entryway of the courtyard, as well other key elements of the composition. The ornamental carvings at the bottom of the table are positioned at the golden ratios of its width, and Mary’ neckline is at the golden ratio from her sash to the top of her head. Furthermore, a basic golden grid reveals that the painting can be divided into three vertical sections, with the two outer sections having a phi-based relationship to the middle one.
Annunciation by Leonardo da Vinci c. 1472–1475.
Perhaps one of the best illustrations of the use of the golden ratio is in da Vinci’s The Last Supper, which he painted between 1494 and 1498. Various design and architectural features show very precise golden ratio relationships. For example, examining the space between the table top and the ceiling, the top of Jesus’s head appears at the midpoint, while the tops of the windows are at the golden ratio. The width of the shields is the golden ratio of the width of the circular arcs, and the stripes within the center shield are at golden ratio points of its width. Some believe that even the positions of the disciples around the table were placed in divine proportions to Jesus.
The Last Supper, 1494-1498.
Another of da Vinci’s most famous works is a drawing created around 1490, the official title of which is Le Proporzioni del Corpo Umano Secondo Vitruvio (The Proportions of the Human Body According to Vitruvius). As indicated, it is based on the ideal human proportions as conceived by the ancient Roman architect and military engineer Vitruvius (c. 75–c. 15 BCE). In Book III of his treatise De Architectura, Vitruvius described the human figure as being the principal source of proportion in architecture, with the ideal body being eight heads high:
“The navel is naturally placed in the center of the human body, and, if in a man lying with his face upward, and his hands and feet extended, from his navel as the center, a circle be described, it will touch his fingers and toes. It is not alone by a circle that the human body is thus circumscribed, as may be seen by placing it within a square.
For measuring from the feet to the crown of the head, and then across the arms fully extended, we find the latter measure equal to the former; so that lines at right angles to each other, enclosing the figure, will form a square.” 4
Vitruvius measured the entire human body in integer fractions of the height of a man, as shown by the gridlines overlaying da Vinci’s illustration.
This illustration shows the height divided into quarters and fifths, while the horizontal extension is divided into eighths and tenths. As you can see, the gridlines align vertically at the collar bone, nipples, genitals, and knees. Horizontally they align with the wrists, elbows, and shoulders.
However, the Vitruvian Man also has some dimensions that suggest a golden ratio relationship. In the distance from the top of the forehead to the bottom of the foot, the following are all at golden ratio points:
Vitruvian Man, c. 1490.
• the navel (which is most often associated with the golden ratio of the total height).
• the pectoral nipples.
• the collar bone.
In the distance from the elbow to the fingertips, the base of the hand begins at the golden ratio point.
In 2011, the discovery of a lost painting by Leonardo da Vinci was announced to the world. This painting, entitled Salvator Mundi (Christ as Savior of the World), had been in the art collection of King Charles I of England in 1649. In 1763 it was sold at auction and then lost for many years. Robert Simon, an art historian and private art dealer, led the effort to recover the lost painting, which was later restored to its former glory by Dianne Dwyer Modestini. Many unique qualities of this painting led experts to confirm that it is indeed an original work of Leonardo da Vinci—one of only fifteen now in existence. In 2017, the painting was sold in a Christie’s auction for a record-shattering $450 million to Saudi Prince Bader bin Abdullah bin Mohammed bin Farhan al-Saud, for display in the then recently opened branch of the Louvre in Abu Dhabi. 5
Portraits generally have fewer distinct lines than paintings of landscapes and architecture, but there are some very interesting features in the overall composition of this painting that exhibit golden ratio proportions. The dimensions of its key elements are in golden ratio proportion to one another, for example. Starting with a golden rectangle based on the height of the head, we then find:
• the dimensions of the hand are based on a golden ratio of its width.
• the dimensions of the orb are based on a golden ratio of its height.
• the dimensions of embroidered emblem are based on the golden ratio of its height and width.
Further analysis reveals golden proportions horizontally in the outside of the eyes relative to the width of the canvas, the width of the center emblem to the width of the neckline, the width of the jewels to the emblems, and the positions of the fingers to the hand. Golden proportions appear vertically in the height of the head to the neckline (as with the Mona Lisa), the height of the jewels to the emblems, the positions of the fingers to the hand, and the positions of the reflections on the glass orb.
Salvator Mundi, c. 1500, the most expensive painting ever sold.
We cannot know with certainty where Leonardo intentionally applied the divine proportion in this painting’s composition. We just know that he had used it extensively before, that this painting of Christ was begun within a few years of his collaboration with Pacioli on De Divina Proportione. As Leonardo once said:
“There are three classes of people: those who see, those who see when they are shown, those who do not see.” 6
Although Luca Pacioli was Leonardo da Vinci’s mentor for mathematics, perhaps Pacioli’s appreciation of the unique aesthetics of the golden ratio that inspired him to write De Divina Proportione came from da Vinci and Francesca, both of whom used the golden ratio in their works many years before its writing.
These inferior versions of Salvator Mundi by Italian painter Michele Coltellini (above) and Bohemian etcher Wenceslaus Hollar (below) helped to alert art historians to the existence of Leonardo’s version.
SANDRO BOTTICELLI
The Birth of Venus, painted by Sandro Botticelli between 1482 and 1485, is one of the most famous pieces of fifteenth-century Italian art. It is based on Ovid’s Metamorphoses, a classic of Latin literature, and portrays Venus, the goddess of love, between her handmaid, the Hora of Spring, and Zephyros, whose breath creates the blowing wind.
Here, too, we find evidence of knowledge and application of the golden ratio well before the writing of De Divina Proportione. The first clue is found in the dimensions of the canvas itself, which is 67.9 × 109.6 inches (172.5 × 278.5 cm). 7 The ratio of the width to the height is thus 1.6168, a variance of only 0.08 percent from the golden ratio of 1.618. To put in perspective, for the canvas to have been an exact golden ratio, the height of the canvas would need to be reduced by less than one twentieth of an inch! The width of the painting at 109.6 inches (278 cm) seems somewhat arbitrary. That is, until one realizes that the units of measure were not standardized in this era. For example, the Spanish foot, or pie, of the Middle Ages was 10.96 modern inches (27.8 cm), which could indicate that the dimensions were not haphazard at all, but rather carefully planned to be exactly 10 “feet” wide. Either way you look at it, it’s quite reasonable to conclude that Botticelli’s intent here was to begin this great work of art with the perfection of the golden ratio.
The Birth of Venus, c. 1485.
Botticelli’s patron Lorenzo “Il Magnifico” de Medici is portrayed in this detail of The Procession of the Youngest King (1459–1461) by Benozzo Gozzoli.
Interestingly, the Birth of Venus is the first work ever painted on a canvas in Tuscany. A revolutionary work, it was created by Botticelli as a wedding present for a member of his patron family, the politically and financially powerful Medici family. Nudity was rarely portrayed in this era of Christian-inspired art, and its intended display above the marital bed added a rather shocking undertone of sensuality and desire. The painting was so controversial that it remained behind closed doors for another fifty years.
Several key elements of the painting are also precisely positioned at golden ratio points:
• The vertical golden ratio line from the left side to the right side falls exactly at the point at which Hora’s thumb and finger are touching, as though she is grasping the golden ratio proportion embodied in the painting, perhaps even reaching for something divine.
• The vertical golden ratio line from the right side to the left side falls at the point where the land on the horizon meets the sea.
• The horizontal golden ratio line from the top to the bottom crosses exactly at the top of the seashell.
• The horizontal golden ratio line from the bottom to the top crosses at the horizon line, most perfectly on the left side of the painting, and passes directly through Venus’ navel.
In addition, the subject Venus has her navel at the golden ratio point of the height of her body, whether measured from the top of her hair to the bottom of her lower foot, from her hairline at the top of her forehead to the bottom of her upper foot, or from the middle of the feet to the top of her head at the back part in her hair.
Botticelli also created a number of paintings of the Annunciation between 1485 and 1490. This event, which clearly captures the meeting of the divine with the mortal, is an excellent opportunity to apply the divine proportion. Note that the golden ratio gridlines are based simply on height and width of the canvas in all but one case, so no creative interpretation of placement is required.
Botticelli included this self-portrait in his c. 1475 painting The Adoration of the Magi.
The Cestello Annunciation, 1489.
This version of Annunciation by Botticelli is held at the Pushkin Museum of Fine Arts in Moscow, Russia.
A modern panorama of Florence, the birthplace of the Renaissance.
Annunciation, c. 1488–1490, from Botticelli’s Altarpiece of Saint Mark.
RAPHAEL
Raffaello Sanzio da Urbino, popularly known as Raphael, was an Italian painter and architect of the High Renaissance who lived from 1483 to 1520. He is recognized as one of the three great masters, alongside Michelangelo and Leonardo da Vinci, of that period. One of his most famous works is The School of Athens, a fresco in the Apostolic Palace in the Vatican. It captures the spirit of the Renaissance and is revered as his masterpiece. This work was begun in 1509, the year that Pacioli’s De Divina Proportione was published, and finished two years later.
If there’s any question whether Raphael used the golden ratio in this painting’s composition, it can be eliminated with a good degree of confidence by the golden rectangle that was placed front and center in the painting. It’s as though Raphael made a small but undeniable statement to answer the question before it was asked. This small rectangle is about 18 by 11.1 inches (46 × 28 cm) and is a rather unusual feature. Perhaps it once bore the title or some description of the painting? We may never know.
Self-portrait, by Raphael, c. 1504-1506.
No other ratio would accomplish the same result in this composition. The painting has thousands of intricate lines, so some might say that finding golden ratios within it would be a simple exercise in pattern recognition, whether they were intended or not. There are two ways to overcome such an objection:
1. Set the Line Ratio option in PhiMatrix software program to any other ratios and see if you get the same abundance and consistency of results than with the ratio set to the golden ratio.
2. Focus on the major elements of the composition alone. For example, note that simple golden ratios of the width and height of the painting define the position of the closest arch, the top of stairs, and the top of the farthest arch.
Other golden ratios define other key elements of the composition, as shown. Raphael’s intricate application of the golden ratio is obvious as well as brilliant. To appreciate the detail and depth of Raphael’s planning and application of the dimensional proportions in this painting, take a look at the image opposite:
• Each rectangle begins at the left side of the left column in the painting. This point represents the first architectural reference point of the actual school building as viewed through the arched portal of the fresco.
• Each rectangle extends to a prominent composition feature on the right side of the painting.
• Each dividing line illustrates a golden ratio formed within another prominent feature of the composition.
The School of Athens, by Raphael, 1509-1511.
MICHELANGELO
The paintings of the other great master of the High Renaissance, Michelangelo (born Michelangelo di Lodovico Buonarroti Simoni in 1475), provide yet another brilliant example of the golden ratio’s prominence in Renaissance art. Analysis of the Sistine Chapel has revealed more than two dozen examples of golden ratio dimensions in major elements of the composition.
Perhaps the most stunning example appears at the point at which Adam’s finger is touched by the finger of God in Michelangelo’s iconic painting The Creation of Adam. This is found at the golden ratio of both their horizontal and vertical dimensions.
Michelangelo by Italian artist Daniele da Volterra, c. 1544.
A view of Michelangelo’s finished Sistine Chapel ceiling, which was completed between 1508 and 1512.
The Creation of Adam.
Michelangelo repeated this theme of the characters touching the golden ratio point in other paintings of the Sistine Chapel. The gridlines in the photo opposite show the golden ratio of the height and/or width of each painting. In some cases, the hands are positioned as if grasping this golden proportion, which can be viewed as a visual metaphor of the human desire to grasp the Divine.
The Fall and Expulsion from the Garden of Eden.
The Creation of Eve.
The Separation of the Earth from the Waters.
The last of the series of nine biblical narration paintings on the center ceiling of the Sistine Chapel is of Noah’s disgrace. The painting itself is within 2 percent of golden rectangle proportions. In it, the fingers of two of Noah’s sons point directly to the golden ratio lines from the painting’s sides. It’s done as if to show the viewer exactly where they are, and that Michelangelo had indeed applied the divine proportion.
The Drunkenness of Noah.
A modern view of Vatican City, home of the Roman Catholic Church, with St. Peter’s Basilica at its center.
This Sistine Chapel lunette bears the names of Salmon, Boaz, and Obed, who are mentioned in the Old Testament’s Book of Ruth. In this fresco, Ruth nurses baby Obed.
If there remains any doubt that Michelangelo used the divine proportion in his epic paintings, look to the tablets listing the ancestors of Jesus on the side walls of the Sistine Chapel. The height to width of the name plates form a golden rectangle, within a pixel or two. The average height to width ratio of all the paintings is 1.62, accurate to within 1/1000th of the golden ratio of 1.618.
Michelangelo’s magnificent collection of paintings was created between 1508 and 1512 for Pope Julius II and successor Popes of the Roman Catholic Church. Given their religious significance, it really should be no surprise that Michelangelo used the divine proportion extensively to bring both mathematical and visual harmony to the biblical accounts of scripture. In retrospect, it would be much more of a surprise if he and the other masters of the Renaissance had not.