Leonardo's Divine Proportions

Chapter 5— Leonardo's Divine Proportions

Leonardo da Vinci
Vitruvian Man (1487)
Gallerie dell'Accademia,
Venice, Italy

Leonardo da Vinci (1452-1519) is one of my all-time heroes.
I read his Notebooks for philosophical insights. While attending the 1st Sickle Cell Conference in Washington DC (1967), I left some of the boring lectures and went to the National Gallery of Art to see Leonardo's Ginevra de' Benci, bought from Prince of Liechtenstein for a record price of $5 million. His artistic genius brought me to Milan to study his Last Supper on my first trip to Europe (August 1972). Visited Louvre in Paris (July 22, 1979) and waited till the crowd had gone in the Gallery. Spent an hour in communion with Mona Lisa's smile. Then it happened— Mona Lisa graced me with her smile and inspired the poem "Mona Lisa". My epiphany came when I realized that Mona Lisa's smile was the same as Mahakasyapa's during Buddha's silent Flower Sermon. Went to the Legion of Honor to view his Lady with an Ermine in San Francisco (May 13, 2003). Wowed by the exhibit "Leonardo: 500 Years into the Future" at San Jose's Tech Museum (November 1, 2008). Gathered 9 of my Leonardo postage stamps and exhibited them on a web page.
Vitruvian Man shows his fascination with Divine Proportions.

Postage Stamps Showing Leonardo's Vitruvian Man

UN Geneva #24, 0.80 franc
World Health Day
(issued 4-7-1972)
Japan #1355, 50 yen
Western Medicine
(issued 4-7-1979)
Italy #1911, 3200 lira
Columbus Discovery 1492
(issued 9-18-1992)
Monaco #2175, 0.99 Euro
Year of Mathematics
(issued 9-4-2000)

Cuba #1526, 30 centavos
Telecommunication Day
(issued 5-17-1970)
Argentina #1592, 0.30 Austral
Health Care: Fight Drug Abuse
(issued 8-15-1987)
San Marino #1684, 0.62 Euro
Vitruvian Man as Jigsaw Puzzle
(issued 6-19-2006)

Leonardo's Studies with Luca Pacioli

Luca Pacioli was an Italian mathematician and Franciscan friar. In 1494, his first
book, Summa de arithmetica, geometria, Proportioni et proportionalita, was
published in Venice. In 1497, he accepted an invitation from Duke Ludovico
Sforza to work in Milan. There he met, taught mathematics to, collaborated,
and lived with Leonardo da Vinci. Their De divina proportione was written in
Milan, 1496-1498, and published in Venice, 1509. The subject was mathematical and artistic proportion, especially the mathematics of the golden ratio and its
application in architecture. Leonardo da Vinci drew the illustrations of the five Platonic solids in De divina proportione. Leonardo's drawings are probably the first illustrations of skeletonic solids which allowed an easy distinction between front & back. He was first to draw a truncated icosahedron with hollow faces.
Luca Pacioli (1447-1517)
wrote De divina proportione (1498)

Stanley Morison's Pacioli's
Classic Roman Alphabet Luca Pacioli's Type Used as Logo
for Metropolitan Museum of Art
Took "Typography with the Computer" class
at Foothill College (1994). Knew the British
typographer Stanley Morison invented the font Times New Roman (1931), one of the most popular typefaces in history. When seeing his book Pacioli's Classic Roman Alphabet (1994), bought a copy (January 1995) for study. Amazed how Pacioli designed each letter painstakingly with lines, circles, and square. The iconic Met "M" was from the Divina proportione woodcut by Pacioli, designed after Leonardo da Vinci.
Luca Pacioli's "Letter M"
used as Logo for MMA (1971)

The design, which overlays the letter M on top of a circle and a square, with smaller circles resting on each serif, recalls Leonardo's famous Vitruvian Man drawing with its proportional geometry. Pacioli's commentary on M: "This letter is made from circle & its square. The thin limbs must be half thickness of thick ones, like left limb
of A. The outer limbs must be slightly inside the square; middle limbs between them & intersection of diameter, their width, both thick & thin, like those of A" (p. 51). The logo was adapted in 1971, and is quite beautiful in
its simplicity. After 46 years, it was replaced by a new logo on March 1, 2016 that has become unpopular.

Golden Ratio Found in Nature

Nautilus Shell
Pineapple Scales
Dall Sheep Horns
Sunflower Spirals

Stephen Skinner
Sacred Geometry (2006) The golden ratio φ = (1 + $\sqrt{5}$ )/2 = 1.6180339887... is an irrational number, also called golden section and divine proportion. In his book Sacred Geometry, Skinner has a chapter on "Living Spirals", citing Nautilus shell & Dall sheep horns as examples of logarithmic spiral found in nature. More examples of golden ratio φ include sunflower spirals and pineapple scales. From the microscopic DNA double-helix molecule to the macroscopic Spiral Galaxy, we find the presence of the golden ratio. In Skinner's "Chapter 6: Sacred Geometry in Archictecture" (pp. 116-139), the golden ratio φ was not found in the Giza Pyramids nor at the Parthenon, despite earlier claims. British astronomer Piazza Smyth claimed that the pyramid inch was a God-given measure handed down from Shem (Noah's Son). However, Paul Brunton noted in Search in Secret Egypt (p. 41): "a disappointed follower of Smyth's, whom he found trying to file down the granite boss in the ante-room to the King's Chamber to the size required for Smyth's theory!" Skinner found no φ in Milan Cathedral, Chartres, and St. Paul's Cathedral. More on "Golden ratio in architecture".

Books on Divine Proportions & Golden Ratio

H.E. Huntley
Divine Proportion
(published 1970)
Robert Lawlor
Sacred Geometry
(published 1979)
György Doczi
The Power of Limits
(published 2005)
Priya Hemenway
Divine Proportion
(published 2005)
Lieselotte Kugler (Ed.)
Divine Golden Ingenious
(published 2017)

H.E. Huntley: The Divine Proportion (1970)
Professor Huntley explores the fascinating relationship between geometry and aesthetics. Poetry, patterns like Pascal's triangle, philosophy, psychology, music, and dozens of simple mathematical figures are enlisted to show that the "divine proportion" or "golden ratio" is a feature of geometry and analysis which awakes answering echoes in the human psyche. When we judge a work of art aesthetically satisfying, according to his formulation, we are making it conform to a pattern whose outline is laid down in simple geometrical figures; and it is the analysis of these figures which forms the core of Professor Huntley's book.
Robert Lawlor: Sacred Geometry (1979)
Thinkers of ancient Egypt, Greece and India recognized that numbers governed much of what they saw in their world and hence provided an approach to its divine creator. Robert Lawlor sets out the system that determines the dimension and the form of both man-made and natural structures, from Gothic cathedrals to flowers, from music to the human body. By also involving the reader in practical experiments, he leads with ease from simple principles to a grasp of the logarithmic spiral, the Golden Proportion, the squaring of the circle and other ubiquitous ratios and proportions.
György Doczi: The Power of Limits (2005)
One of the delights of life is the discovery and rediscovery of patterns of order and beauty in nature— designs revealed by slicing through a head of cabbage or an orange, the forms of shells and butterfly wings. These images are awesome not just for their beauty alone, but because they suggest an order underlying their growth,
a harmony existing in nature. What does it mean that such an order exists; how far does it extend? The Power
of Limits was inspired by those simple discoveries of harmony. The author went on to investigate and measure hundreds of patterns— ancient and modern, minute and vast. His discovery, vividly illustrated here, is that certain proportions occur over and over again in all these forms. Patterns are also repeated in how things grow and are made— by the dynamic union of opposites— as demonstrated by the spirals that move in opposite directions in the growth of a plant.
Priya Hemenway: Divine Proportion (2005))
The number Phi, φ, simply defined, is one plus the square root of five, all divided by two. But its myriad occurrences in art, nature, and science have been a source of speculation and wonder for thousands of years. Divine Proportion draws upon both religion & science to tell the story of Phi & to explore its manifestations in such diverse places as structure of the inner ear, spiral of a hurricane, majesty of the Parthenon, and the elusive perfection of the . A universal key to harmony, regeneration, and balance, Phi is at the heart of a tantalizing
story begun on clay tablets in ancient Babylon, and which will continue to be written for centuries to come.
Lieselotte Kugler (Ed.): Divine Golden Ingenious: The Golden Ratio as a Theory of Everything? (2017)
What do shells of nautilus, pineapples, Marilyn Monroe's face, and an Aston Martin all have in common?
Is it the same mystic, divine formula that lies behind everything that is beautiful? For centuries this formula, often referred to as the golden mean, has been a subject of endless fascination. The ratio of proportion that formulates the golden mean can be found in nature, artistic design processes, and in how we perceive our surroundings. But how much of this universal formula is true, and how much of it is myth? Divine Golden Ingenious is a collection of essays from many fields— architecture, mathematics, science, art, and design—
who explore history & applications of the golden mean. Search for the mean dates back to Euclid in the 3rd century BC, but it was only in the 19th century that it reached fame as a universal constant of beauty. From
this point onwards the "golden section" was described in flora and fauna, established as foundation for the Fibonacci number, and optimized by Le Corbusier in architecture. Sorting fact from fiction, the essays
analyze the golden mean with regard to function and relevance while exploring its use in current examples
from art & design. Together, they reveal the extent to which this art-historical phenomenon plays a role in
organization and presentation of the world around us.

2011 Poetry Workshop at Stanford

Stephen Dobyns' first Stanford poetry class (January 18, 2011) was to write a sonnet. I wrote "Platonic Lambda Sonnet" with Notes and Cornford's diagram of Plato's "Soul of the Universe". While writing this sonnet, I realized that Plato's "World Soul" is not something abstract and invisible, but quite tangible when we're walking and while breathing, since its shape is our nose in the center of our face!

Plato's Soul of the Universe is placed as the nose in the center of our face

Leonardo da Vinci
(1452-1519) The Platonic Lambda Λ which Plato calls "Soul of the Universe" (Timaeus 35b) appeared abstract to me until I realized its concrete example in Giacometti's Walking Man that is present in every step we take. Likewise, if God created man by breathing into his nostrils a living soul (Genesis 2.7), the nose is the prime conduit of air in keeping us alive. So the Soul is not hidden but right in the center of our face. Leonardo's Vitruvian Man (1487) shows a man inscribed in a circle and square. The drawing and text are called the Canon of Proportions, showing the correlations of ideal human proportions with geometry described by the ancient Roman architect Vitruvius. [Images: Leonardo da Vinci,
Self Portrait (1515) & Proportion of the Face (1490)]
Leonardo da Vinci
Proportions of the Face

— Peter Y. Chou
Mountain View, 11-7-2018

Leonardo's Studies with Luca Pacioli
Luca Pacioli was an Italian mathematician and Franciscan friar. In 1494, his first book, Summa de arithmetica, geometria, Proportioni et proportionalita, was published in Venice. In 1497, he accepted an invitation from Duke Ludovico Sforza to work in Milan. There he met, taught mathematics to, collaborated, and lived with Leonardo da Vinci. Their De divina proportione was written in Milan, 1496-1498, and published in Venice, 1509. The subject was mathematical and artistic proportion, especially the mathematics of the golden ratio and its application in architecture. Leonardo da Vinci drew the illustrations of the five Platonic solids in De divina proportione. Leonardo's drawings are probably the first illustrations of skeletonic solids which allowed an easy distinction between front & back. He was first to draw a truncated icosahedron with hollow faces.	Luca Pacioli (1447-1517) wrote De divina proportione (1498)

Stanley Morison's Pacioli's Classic Roman Alphabet	Luca Pacioli's Type Used as Logo for Metropolitan Museum of Art Took "Typography with the Computer" class at Foothill College (1994). Knew the British typographer Stanley Morison invented the font Times New Roman (1931), one of the most popular typefaces in history. When seeing his book Pacioli's Classic Roman Alphabet (1994), bought a copy (January 1995) for study. Amazed how Pacioli designed each letter painstakingly with lines, circles, and square. The iconic Met "M" was from the Divina proportione woodcut by Pacioli, designed after Leonardo da Vinci.	Luca Pacioli's "Letter M" used as Logo for MMA (1971)
The design, which overlays the letter M on top of a circle and a square, with smaller circles resting on each serif, recalls Leonardo's famous Vitruvian Man drawing with its proportional geometry. Pacioli's commentary on M: "This letter is made from circle & its square. The thin limbs must be half thickness of thick ones, like left limb of A. The outer limbs must be slightly inside the square; middle limbs between them & intersection of diameter, their width, both thick & thin, like those of A" (p. 51). The logo was adapted in 1971, and is quite beautiful in its simplicity. After 46 years, it was replaced by a new logo on March 1, 2016 that has become unpopular.

H.E. Huntley Divine Proportion (published 1970)	Robert Lawlor Sacred Geometry (published 1979)	György Doczi The Power of Limits (published 2005)	Priya Hemenway Divine Proportion (published 2005)	Lieselotte Kugler (Ed.) Divine Golden Ingenious (published 2017)
*H.E. Huntley: The Divine Proportion* (1970) Professor Huntley explores the fascinating relationship between geometry and aesthetics. Poetry, patterns like Pascal's triangle, philosophy, psychology, music, and dozens of simple mathematical figures are enlisted to show that the "divine proportion" or "golden ratio" is a feature of geometry and analysis which awakes answering echoes in the human psyche. When we judge a work of art aesthetically satisfying, according to his formulation, we are making it conform to a pattern whose outline is laid down in simple geometrical figures; and it is the analysis of these figures which forms the core of Professor Huntley's book. Robert Lawlor: Sacred Geometry (1979) Thinkers of ancient Egypt, Greece and India recognized that numbers governed much of what they saw in their world and hence provided an approach to its divine creator. Robert Lawlor sets out the system that determines the dimension and the form of both man-made and natural structures, from Gothic cathedrals to flowers, from music to the human body. By also involving the reader in practical experiments, he leads with ease from simple principles to a grasp of the logarithmic spiral, the Golden Proportion, the squaring of the circle and other ubiquitous ratios and proportions. György Doczi: The Power of Limits (2005) One of the delights of life is the discovery and rediscovery of patterns of order and beauty in nature— designs revealed by slicing through a head of cabbage or an orange, the forms of shells and butterfly wings. These images are awesome not just for their beauty alone, but because they suggest an order underlying their growth, a harmony existing in nature. What does it mean that such an order exists; how far does it extend? The Power of Limits was inspired by those simple discoveries of harmony. The author went on to investigate and measure hundreds of patterns— ancient and modern, minute and vast. His discovery, vividly illustrated here, is that certain proportions occur over and over again in all these forms. Patterns are also repeated in how things grow and are made— by the dynamic union of opposites— as demonstrated by the spirals that move in opposite directions in the growth of a plant. Priya Hemenway: Divine Proportion (2005)) The number Phi, φ, simply defined, is one plus the square root of five, all divided by two. But its myriad occurrences in art, nature, and science have been a source of speculation and wonder for thousands of years. Divine Proportion draws upon both religion & science to tell the story of Phi & to explore its manifestations in such diverse places as structure of the inner ear, spiral of a hurricane, majesty of the Parthenon, and the elusive perfection of the . A universal key to harmony, regeneration, and balance, Phi is at the heart of a tantalizing story begun on clay tablets in ancient Babylon, and which will continue to be written for centuries to come. Lieselotte Kugler (Ed.): Divine Golden Ingenious: The Golden Ratio as a Theory of Everything? (2017)** What do shells of nautilus, pineapples, Marilyn Monroe's face, and an Aston Martin all have in common? Is it the same mystic, divine formula that lies behind everything that is beautiful? For centuries this formula, often referred to as the golden mean, has been a subject of endless fascination. The ratio of proportion that formulates the golden mean can be found in nature, artistic design processes, and in how we perceive our surroundings. But how much of this universal formula is true, and how much of it is myth? Divine Golden Ingenious is a collection of essays from many fields— architecture, mathematics, science, art, and design— who explore history & applications of the golden mean. Search for the mean dates back to Euclid in the 3rd century BC, but it was only in the 19th century that it reached fame as a universal constant of beauty. From this point onwards the "golden section" was described in flora and fauna, established as foundation for the Fibonacci number, and optimized by Le Corbusier in architecture. Sorting fact from fiction, the essays analyze the golden mean with regard to function and relevance while exploring its use in current examples from art & design. Together, they reveal the extent to which this art-historical phenomenon plays a role in organization and presentation of the world around us.