Math and the Mona Lisa (Excerpts from Bulent Atalay's classic)

(used with author's permission)


If an abiding message is to be gleaned from an examination of Leonardo’s scientific and artistic legacy, it is the insatiable curiosity, the persistent questioning that defined his life – from resolving every day problems to exploring grand-scale issues pertaining to the workings of nature. He observed and pondered: the commonplace became wondrous, the wondrous commonplace. “Nature is the best teacher,” he wrote. “Learn from nature, not from each other.” His curiosity encompassed diverse intellectual worlds: technical and nontechnical, scientific and artistic; and indeed it is by the conjoining of these intellectual worlds that he was able to produce works of dazzling quality and diversity.
The typical liberal arts curriculum offered in the American undergraduate education system gives students exposure to different intellectual cultures. But then the need to develop specialized skills in one field or another tends to discourage further forays across the cultural divide. The higher education systems in most other countries call for this specialization to begin even earlier. Beyond formal education, normal maturation or aging itself is sadly accompanied by the monotonic dimming of one’s curiosity. Once the specialization process begins, those individuals who are more technically oriented have greater facility communicating with others who are technical, and those who are artistic or less technical similarly feel comfortable in the company of individuals with skills and interests akin to their own. In short, birds of a feather most clearly flock together, communicating with each other in their own languages. …
Under the rubric “Leonardo’s model” this book presented the mathematical and scientific basis underlying both science and art, and the tenets of the laws of nature, albeit all of it in an essentially qualitative manner. The interdisciplinary approach, it was seen, invited a discussion of the human side of doing science and art. Among the most creative artists and scientists – especially those who launched revolutionary transformations in the prevailing order in their fields – there abound tortured souls, baffling personalities very different from those in everyday life. There are also the historical and social conditions that made certain times ripe for such unusually creative individuals to spring up and leave their marks.
Beethoven on his deathbed reportedly raised his fist in defiance at a thunderstorm raging outside and cried, “I know I am an artist!” In the last hours of his death, Leonardo with an air of forlorn resignation, remarked to an assistant, “Tell me, did anything get done?” For us, these are puzzling and poignant pleas for reassurance by two of the greatest creative individuals in history. In Leonardo’s case there also resonates the tone of frustration at unfulfilled ideas and unfinished work. One cannot deny that the reasoning out of solutions, especially in art, and the creating of mental inventions, especially theories in science and technology, were more important for Leonardo than bringing them to actuality. Thus his chronic problem of darting from one beckoning source of curiosity to the next was real. And as his reputation for failing to complete projects and commissions was well-deserved. Through it all, however, he was already breaking significant new ground. From the threshold of the twenty-first century we can view the dizzying array and quality of his accomplishments, and we can wonder: “Could there have existed a dozen different and supremely gifted individuals all operating with the same name – Leonardo da Vinci? …
Early in the book I had expressed, with some trepidation, the faith that presenting science through art and art through science would lead to an understanding of Leonardo’s mind. Who could be presumptuous enough to claim to fathom the mind of someone as complex as Leonardo? Complex he certainly was, but certain components of his psyche have indeed emerged that are surprisingly simple.
In the surviving paintings and notebooks are strewn a jumble of clues. Leonardo’s life can be viewed as a sort of kaleidoscope – creating art of sublime beauty and awesome power, abruptly darting to thoughts on turbulent flow in white water, the geometric patterns of polyhedra, the anatomical studies of man and beast metamorphosing into each other, human flight, spring-driven vehicles, then returning to create one more miraculous work of art. But amidst it all, there are the scattered unfinished projects, endlessly frustrating the most loyal patron.
Leonardo’s unusual style, dizzying in pace and diversity, defined his true modus operandi. The depiction of turbulent water in the daily notebooks is reflected in the golden curls of his subjects. The gently twisted helix, a product of mathematical musings, is suspiciously evocative of the poses struck by the subjects of his portraits. Regarding painting, he wrote, “It embraces all the forms that are and are not found in nature.” The Platonic solids informed his architectural drawings, were seen as rough sketches in the codices, and are formalized in De divina proportione. There is the golden pyramid, which he appears to have discovered independently of past civilizations, organizing the grouping of the characters in the Virgin of the Rocks (and most likely also in the unfinished lost mural, The Battle of Anghiari). In studying the flight of birds he is at once a naturalist, an aerodynamic engineer, and an artist capturing the motion of wings in serial animated drawings; but his thoughts ultimately speculated on human flight. The helical spiral used earlier to give dynamism to the subjects in his paintings returned as the aerial screw, giving lift to the Leonardo helicopter. Totally preoccupied with manned flight, he examined free-fall and gravitation, issues of fundamental significance for physics. The experiments he performed were magnificently framed, leading to conclusions of constant acceleration, independent of the weight of the falling body. In Leonardo’s entire process there was total symbiosis, total synergy. His personal growth can be seen in all areas of his creativity, but also seen is the evidence of the deliberate and systematic effort, the mental exercises he performed to attain this growth. An “unlettered man,” he left behind lists of daily words to be mastered in the manner of the autodidact striving for self-improvement….
Through the ages both the arts (including the visual arts and literature) and the sciences (the natural sciences and mathematics) have generally evolved gradually, incrementally. But there have also taken place, however infrequently, fundamental changes in the prevailing order – changes of kind rather than degree, akin to a phase transition between vapor and liquid, or liquid and solid. Initiating these “revolutions” have been a finite number of artistic and intellectual revolutionaries – better characterized as transformative geniuses or “magicians.” Their reasoning processes do not appear to be constrained by the familiar topography of logic, a topography that includes the slopes and valleys. Rather, reaching conclusions for them is reminiscent of leaping from mountaintop to mountaintop. But their works represent no less than the periodic syntheses of the accumulated knowledge in a field; they succeed by defining human limits and elevating the human spirit. Leonardo, Michelangelo, Shakespeare, Newton, Beethoven, and no more than three or four others – all magicians, all towering intellects, all consumed by a fire raging within – also appear to share one common quality: they rarely allow any profound penetration into their psyche; their creative processes are forever steeped in mystery. One thing is clear: the more time one devotes to understanding the mind of the magician, the deeper he recedes into mist. This is a curious phenomenon that distinguishes the magician from others, including the ordinary genius….
Leonardo Da Vinci, paragon artist-scientist-engineer, a cynosure of the Italian Renaissance, more so than any other magician manifests that quality. Does it really matter that Leonardo will always remain this mysterious and mythical creature, who as a part-time artist produced the epitome in art? His works number so few that even the expression “part-time artist” overstates the case. But it is certainly not the quantity that counts. Goethe’s throwaway line, “In art, the best is good enough,” encapsulates the circumstance. Leonardo also developed scientific methodology a century before Galileo, and anticipated future technologies centuries before they were realized. What really matters is that he demonstrated a model of supreme efficacy, a lesson for the ages: seeking connections, conjoining different intellectual worlds, is indeed the source of profound understanding and appreciation in any of these worlds or in all of them.
What exactly can account for a creature of such transcendent gifts will most likely never be known. It is too easy to say that he was a happy accident of nature and nurture – the concatenation of his mother’s and father’s genes, the effects of the prevailing social, political, and intellectual winds sweeping Renaissance Florence. In Leonardo’s case, there is so much self-nurture. We simply do not know much about his upbringing. He repeatedly admonished others – artists and scientists alike – to “learn from nature, not from each other.” In a timeless irony, we must first learn from him, then observe and ponder. Avoid taking anything for granted, test it before accepting it. Don’t ever give up the aspiration of personal growth no matter what stage of your life you are in, read incessantly, read critically, even look up words you don’t know with a view toward increasing your vocabulary. Carry a small pad with you and draw sketches (even if you have convinced yourself that you cannot draw). Creating sketches will make you more observant. Observe in the manner of the scientist, savor in the manner of the artist. Record your observations. Experiment, knowing full well that some experiments will fail. But that is how to attain a deeper understanding. It is important to be curious, and important to explore intellectual worlds, but it is essential to seek their connections. The model that worked magnificently for him will never make any of us another Leonardo, a man called by so many scholars “the greatest genius who ever lived.” But it cannot fail to make us each far more creative and more effective practitioners in the intellectual world that we inhabit.