Book Cover design by Brian Barth, Smithsonian Books
Math and the Mona Lisa
The Essays

"da Vinci and the divine proportion" by Kiran Krishnamurthy 
(Excerpted from) Richmond Times-Dispatch Sep 9, 2004

FREDERICKSBURG — Like the man he admires, Bulent Atalay straddles the worlds of art and science.

Atalay, an artist and University of Mary Washington physics professor, suggests in his new book, "Math and the Mona Lisa: The Art and Science of Leonardo da Vinci," that the enduring appeal of da Vinci's paintings may lie in the painter-scientist-engineer's conscious use of perspective, pattern and symmetry, all of which are rooted in math.

"In the Renaissance, you don't find others doing paintings with geometric underlayings to them," Atalay said."Whether it's his unerring eye . . . or whether it was purposeful, we'll never know."

But Atalay's guess is that da Vinci built his paintings on mathematical formulations. The book, released this spring, is enjoying better-than-expected success on the heels of Dan Brown's fiction best seller, "The Da Vinci Code."

Sales for "Math and the Mona Lisa" - into its fifth printing five months after its release - have far exceeded the expectations of publisher Smithsonian Books.


While many artists imbue their works with examples of the golden ratio, Atalay suggests they do so intuitively or rather casually. A possible exception during the Renaissance was Raphael, he said, though Raphael was born 30 years after da Vinci (1452-1519.)

"In the work of any other artist, we would assume these manifestations to be coincidental," Atalay writes. "For Leonardo, who seamlessly integrated mathematics, science and art, and spent his life seeking unifying principles, perhaps not!"

"With [da Vinci], it's certainly not a cheap geometric construction. His doodling consists of complex polyhedra," Atalay said in a recent interview, referring to the term for geometric structures with multiple faces, such as pyramids.


Born in Turkey, Atalay recalls always being surrounded by art in his youth. He took art lessons and now is an accomplished artist himself. His pen-and-ink drawings are included in collections in the White House, Buckingham Palace and the Smithsonian.

Mona Lisa RTD

"I realized he was a kindred spirit, but he was the best at everything he did," Atalay, who himself has been described as a Renaissance man, said of da Vinci.

Atalay notes that da Vinci was mostly self-taught and schooled at home.

In contrast, Atalay was educated at some of the finest schools in England and America. He attended Eton College, the secondary school preferred by Britain's royal family, and St. Andrew's School, the Delaware prep school featured in the Robin Williams film "Dead Poets Society."

Atalay stumbled into physics accidentally - a Georgetown University secretary misread his application, in which he wrote his aspiration was to become a "physician." He has done post-doctoral work at Oxford, is a Princeton fellow and serves as an adjunct professor at the University of Virginia, where he has taught math and archaeology.


He said that while Beethoven is known for his music and Isaac Newton for his scientific insight, the products of da Vinci's mind spanned myriad disciplines. Da Vinci's designs for the tank, the helicopter, the submarine and the parachute predated the actual invention of each by hundreds of years. He studied anatomy, geology and, of course, produced some of the world's most renowned paintings.

"He was a transformative genius. He was inventing civilization, inventing technology from scratch," Atalay said. "Here was a man who did it all."




An Essay on Bulent Atalay's “Math and the Mona Lisa”
Barry Bressler
2004 May 10

The Book

The book is a gem, demonstrating that “good things come in small packages.” As an object, it is attractive and well made. How nice it is to be able to handle such a book, rather than stare at a computer screen. So far I have found no typographical errors. That in itself is remarkable in this age of sloppily produced stacks of computer-generated garbage.

Ellen Brown wanted to keep her autographed copy as a treasure, and would not let me read it. She also would not let me read the copies that she set aside to be used as presents. I finally got access to a copy just the other day. I have not yet had time to read it carefully, but I have skimmed it. This was enough to get my mind working. Every time I open the book and read I get a new idea. I thought perhaps the author might be interested in reading some of the ideas fostered so far by his book.

I wonder how the author might have elaborated on fractals and their relation to symmetry and art. Otherwise, I have no particular comments on the principal topic of the book. The discussion of mathematical symmetry in art and nature is presented clearly, concisely, and comprehensively. It is the thought-provoking discussions peripheral to the main theme of the book that motivated me to respond with comments.

Whether by coincidence, irony, or serendipity, the news media have announced that the “Mona Lisa” is warped and deteriorating, and needs restoration. This book may help to accomplish the needed repairs by renewing interest in the painting, and in art in general.

Incidentally, the dedication of the book - to a soldier, statesman, and father- indicates a respect for the father that studies have shown is typical of physicists - and also of businessmen - but not, for example, of social scientists.

Symmetry and Imperfection

Humans have a general bilateral symmetry, but have asymmetries as well. Leonardo (page 5) could not read the classics, but had talent for art and music, and was (page 6) left-handed or ambidextrous. This suggests a dominance of the right hemisphere of the brain over the left (page 149). Left-handedness has been associated with various other traits, such as mathematical ability, as well as autism and auto-immune disorders. Innate left-handedness is probably hereditary. On the other hand, permanent, non-situational homosexuality (distinguished by orientation as opposed to simply behavior) may most often be congenital but rarely genetic (due to hereditary or non-hereditary genetic anomalies akin to those leading to complex intersex states). Homosexuality has been tentatively linked to hormonal disruptions resulting from stressful situations at key periods during gestation. Populations of laboratory rats subjected to crowding have responded by exhibiting increases in homosexuality. (But is this merely temporary, situational behavior in individuals?) Perhaps homosexuality is a natural method of birth control in populations. Turing (note 13-1), like Leonardo and Newton, was suspected of being homosexual. In Turing's case, there is supposed to be good evidence for the suspicion.

Too much symmetry indeed can be monotonous (page 17). Regarding this, Francis Bacon, the philosopher (note 1-3), wrote that “there is no perfect beauty without some strangeness in its proportion.” It has been said that some ancient peoples built their temples imperfectly so as not to compete with the gods. Studies suggest that animals with asymmetry are less successful in attracting mates, perhaps because the asymmetry is a signal of a “defective” or discordant genetic makeup. Personally I feel that the women used as models for the Marquardt masks (page 111) were not especially beautiful. (But then, beauty is in the eye of the beholder, eh?). It is curious that nature did not create a perfect living match to the virtual Pygmalion of the mask.

Physicists make much of symmetry. Dirac (page 254) sought beauty as a test of truth in equations (page 23), and symmetry was an aspect of this beauty. The complexity of the world arises from broken symmetries (page 138), and experiments show that expected symmetries are sometimes incomplete.

The notion of imperfection is significant also with respect to mimicry. Mimicry (page 141), as a specific example of general protective camouflage, is an interesting subject. It is muddled by the problem that the perception of mimicry is subjective. The late entomologist Frank Lutz, who was not a disbeliever in mimicry, cautioned in his “Field Book of Insects” that some supposed examples of mimicry may not be so. It seems to me that, with millions of species, it is inevitable that some few would resemble each other by chance, in the way that supposed “cancer clusters” can occur. There are all degrees of apparent mimicry, from vague similarities to striking and compelling resemblances. Where does one draw the line in attempting to discern which examples really demonstrate mimicry? Almost all mimicry is imperfect. The Viceroy butterfly rather closely resembles the Monarch, but differs noticeably in having an extra dark line on the hind wing. This line should be clearly visible to a predator. If evolution could produce a resemblance that otherwise was so close, why would it leave the line? I wonder what believers in intelligent design would make of this imperfection in divine design? (For that matter, what do the believers make of mimicry?) Evolution, operating as driver of optimization through the natural selection of random mutations, would be likely to nudge species into a local minimum (or maximum?) rather than the global minimum (or maximum?) on an n-dimensional adaptation surface. Presumably intelligent design would direct evolution to the unique global optimum.

Art and Science

Art involves synthesis, science involves analysis (page 19). Mathematics and physics involve deduction, whereas classical biology involves induction. Could this be one reason why women have been more involved in biology?

Analysis leads to reductionism (page 20), whether in art (page 139) or in science, while synthesis leads to a more holistic view. Reductionism (page 20) and stratification of the sciences (page 50) are related to the issue of emergent properties. I do not see how emergent properties arise in a system if they are not inherent in the constituent particles.

Curiously, reductionism in philosophy has failed to find fundamentals, and instead has led to the reductio ad absurdum of anarchic deconstructionism that has infiltrated and subverted the humanities, but that only recently has challenged science with antiscientific claims.

Analysis tends to lead toward specialization, whereas synthesis tends to lead to a broader perspective. The sociobiologist E. O. Wilson contrasted the narrowly focused researcher with the wise scholar, and remarked about the paucity of such scholars in modern science. It is the focused researcher who is likely to make discoveries and reap the resulting recognition and rewards, but who will not have the time to pay attention to matters beyond the relevant specialty. A narrow focus is compatible with the reclusiveness of some natural scientists. Although they may sometimes be unsociable, most of them, like Leonardo (page 26), are not especially introspective (page 90). Their attention is focused on external objects other than people.

Religion and Science

Creation science is an oxymoron (page 22), but what of the more subtle, insidious concept of intelligent design? The Templeton prize rewards the sort of thinking that encourages this concept. Freeman Dyson promulgates this sort of thinking, and exacerbates the problem. Other physicists, from Newton (page 240) to Schroedinger, have verged on mysticism in their thinking about religion. (It seems quite plausible that Newton's mysticism was part of his madness, and was caused by mercury poisoning (page 290). This topic has contemporary relevance because of the mercury emissions from the coal-fired power plants that are being built instead of nuclear plants. Also, there is ongoing controversy about the use of mercury amalgams in dentistry.)

The statement that physics and religion are orthogonal functions (page 23) is provocative, and, if true, perhaps profound. Factor analysis in psychology is the analog of the eigenvalue problem in physics, and identifies independent psychological traits. Maybe religion and science are the result of independent modes of addressing the world.. Stephen Jay Gould, an entertaining writer (page 99) whose columns were the high spot of “Natural History” magazine, and who rarely addressed religion (page 25), did say that science and religion do not overlap, but is this so? One must accept that evolution occurs or it doesn't, and the world was created a few thousand years ago or it wasn't, unless one adopts a position of extreme subjective solipsism. Of course, it may not be possible to determine with certainty what is true. Feynman said that he wasn't really certain about much of anything.

Physics doesn't provoke as much conflict between science and religion as does biology, or as does psychology with its puzzlement at “free will,” but occasionally one of my students would broach the topic, and spark an argument in class. It was easy to debunk ignorant misconceptions, such as the commonly supposed inconsistency between evolution and the second law of thermodynamics (although elsewhere I have heard complicated arguments that a simple reading of the second law does not readily dispel). Some claims are more subtle. I have silenced both sides of an argument by saying that I believed the world was created on that very day, with individual memories and the fossil record intact. No student could prove me wrong. I also told them that physics has almost nothing to say about miracles. By definition, a supernatural power could intervene in the world by overriding natural laws and natural causality. I reminded my students about different degrees of uncertainty. There is a difference between direct observations of an event that a student can make in an experiment and inferences that the student can make about events in the distant past. Then I showed the students a few optical illusions (page 145) to convince them that caution is needed even in making direct observations.

I wonder if a tendency to believe in religion may be a survival mechanism that has been hard-wired into the human brain by evolution. Humans acquired a highly developed brain that became aware of its ultimate destruction. This awareness mocked the individual drive to survive honed by millions of years of evolution. Religion provided a means of displacing concerns about the ultimate futility of life of the individual and the species.. The historian W. H. McNeill has remarked on the loss of comfort endured by those who have abandoned the religion of their youth. Studies have confirmed the common perception that physical scientists tend to be less religious than the general population. Theorists tend to be somewhat less so than experimentalists and engineers, and perhaps slightly less conventional in general. Steven Weinberg (note 12-2), at the end of “The First Three Minutes,” concisely and expressively articulates the world of many of the more secularly minded physicists.

Pseudoscience (page 60) is currently rampant, as evidenced by the popularity of overnight talk-show host Art Bell and his ilk on Clear Channel radio. “New Age” occultists make much of the Piri Reis maps (page 36), claiming that they show the coast of Antarctica without ice, and that they therefore give evidence of an ancient worldwide civilization of humans or extraterrestrials. Feynman (note 12-2), when mentioning “flying saucers” in “The Character of Physical Law,” expresses the scientist's skeptical judgement in assessing unsubstantiated claims. As Carl Sagan liked to say, exceptional claims require exceptional proof. (I would be inclined to be even more skeptical than I am, except that I myself saw a “UFO,” a light that moved across the sky in an unremarkable way, and then suddenly shot straight up and vanished in the distance in a second or so.) “The Amazing Randi,” a magician turned debunker, claims that scientists are the people least able to investigate hoaxes because they are used to natural phenomena that are neutral rather than adversarial. (This recalls Einstein's statement that nature is subtle but not malicious (page 23).)

Popular sociology (page 258) contributes to the spread of nonsense when it sloppily misinterprets statements of quantum uncertainty and duality (page 139), just as when it loosely and inappropriately generalizes the concept of the relativity of reference frames. (See Allan Bloom's “The Closing of the American Mind” for a conservative thinker's appraisal of the importance of philosophy in education. Bloom attributes the ills of education and society to the deleterious effects of philosophical relativism.) It is a difficult task to sort the wheat from the chaff, and this is not always simply a matter of doing an experiment. Not long after the discovery of X rays, laboratories were confirming the existence of “N rays,” until a debunker exposed the self-deception practiced by the experimenters. A more recent example of self deception may be the belief in “cold fusion.”

If string theory and other such conjectures of modern physics remain untestable, physical theory itself may degenerate into mysticism, just as “pure” mathematics, removed from the requirements of practicality, becomes increasingly baroque, except when it is subjected to the tasteful judgement of a master such as Gauss.

Physical Models

Physical models have been used with varying success in the social sciences (page 47), from the “social physics” of the middle of the twentieth century to the “econophysics” of today. Sociobiologist E. O. Wilson provides the quotation “physics envy is the curse of biology.” Perhaps this complaint is even more true of the social sciences.

The question of “indefinables” (page 48) is profound. I called the attention of my students to the fact that Goldstein, in the second paragraph of the very first chapter of at least the second edition of his “Classical Mechanics,” saw fit to comment on undefined terms, and that Symon, in the first chapter of at least the third edition of his “Mechanics,” discussed problems with the definition of terms in Newton's laws of motion. Lindsay and Margenau also discuss this in “Foundations of Physics.” I told my students that homework problems provided a means of “learning through doing,” and some of what they learned in this way was just what physicists mean by the terms that they use. Incidentally, it is interesting that physicists adopt simple words and apply exact meanings to them, whereas many other scientists, along with professional “educators,” coin polysyllabic neologisms, perhaps in a misguided attempt to sound as impressive as the lawyers with their Latin.


It is interesting to consider the different origins (page 100) of different spirals in nature (page 45), and even in the malfunctioning Trident missile (page 146). Spirals are ubiquitous in nature (page 14), but once this is recognized, it is possible to overemphasize them to the exclusion of other shapes. Once noticed, spirals seem to be everywhere, the way a newly learned word suddenly seems to appear everywhere in print.

The notion of “hidden images” (page 142) relates to why some people see a woman to the left of Jesus in Leonardo's “Last Supper” (page 167).

The upside-down face (page 152) does demonstrate what people take for granted in perception. It is amusing to have someone lie on the floor, to kneel at the person's head, facing the feet, and then to stare at the person's upside-down face. After a while, the face will seem strange and funny, with the hair looking like a beard, and the mouth appearing in the “forehead.”

MRI patterns (page 154) differentiate between artists and non-artists. According to a recent New York Times article, MRI patterns differentiate between Republicans and Democrats. I think that individual philosophies are largely a matter of temperament, and I sometimes feel that philosophy should be a branch of abnormal psychology. Art has value in helping to see things from different points of view (page 140). (How many impressionists (page 139) in art were in fact myopic?) Certainly surroundings - such as mountains (note 2-6) - can affect the point of view. Some people have said that the world's highest-performing cars have come from within a couple of hundred miles of the Alps, because of the need for suspensions, steering, brakes, and engines that could cope with the Alpine roads. I liked the nimble British cars, such as the MG that I had, that were designed for the twisting roads of the English countryside.


Leonardo (page 163) called music the younger sister of painting. Most physicists, especially theorists, prefer music to art. Feynman was a notable exception. Theorists, more than experimentalists and engineers, have some similarities to psychologists, who are often musically inclined. Experimentalists and engineers, more than theorists, have some similarities to visually oriented biologists, who are likely to feel more affinity for art than for music. Note that representational art is necessarily related to concrete or mundane objects, and is therefore in a sense less abstract than music, and is closer to geometry than to arithmetic.

Theorists tend to resemble social scientists in having an interest in verbally intensive subjects, including history. Mendelssohn (page 60) is not the only example of a physicist interested in archaeology. Murray Gell-Mann retains an early interest in archaeology and linguistics. (In general, though, even most theorists dislike learning languages, finding them illogical and boring.)

Leonardo was a military engineer (page 197). Gauss, “Prince of Mathematicians,” and other great mathematicians, did not distinguish between pure and applied mathematics. Einstein did his most important work when enjoying employment as a patent examiner. During the Second World War, the most talented physicists of the time, including Feynman (note 11-10), worked on weapons, but then returned to more congenial surroundings and more fundamental or intrinsically interesting pursuits, leaving military engineering largely to those with more modest abilities.

Physicists are early achievers (page 254). Physics, like mathematics, music, and formal poetry, requires mental athletics involving non-verbal abilities, which decline with age. Theorists, especially, are most creative before the age of thirty. A theorist need learn a few basic principles, and can then, by mental effort, deduce new conclusions, or formulate new principles. An experimentalist must conduct painstaking, and often time-consuming and expensive operations. A biologist often is much older than a physicist before making a significant contribution to science. The biologist must learn many facts or make many observations before making an inductive synthesis.

It is interesting that, in dealing with quantum phenomena, Schroedinger, who was no youngster, drew on his years of experience with classical wave motion to formulate wave mechanics, while Heisenberg, who was still youthful, conceived the more radical matrix mechanics. Nevertheless, Schroedinger was capable of novel insights. In “What is Life,” he considers haploid gametes and the diploid fertilized egg to comprise alternating haploid and diploid generations. This view at first may seem ludicrous in the case of humans, where there is such a disparity between a gamete and the adult individual into which an egg can grow. But in the case of other organisms, the reasonableness of this view is more apparent. Schroedinger observes that the drone bee and the leafy part of a moss are technically haploid gametes. According to this view, some human lives - the members of haploid generations - do not begin at conception. This view should shock both liberals who oppose the death penalty for adults, and conservatives who oppose the killing of unborn children.

Mention of the Princeton Institute for Advanced Study (page 267) reminds me of Newton's comment about standing on the shoulders of giants. Dirac said that, in his early years, second-rate physicists could do first-rate work, but later even first-rate physicists could do only second-rate work, because the ongoing exploitation of quantum mechanics eventually led to diminishing returns. If Newton had not existed, could others have stood in his place? The controversy about whether Newton and Leibnitz invented calculus independently mirrors the disagreements between those who accept the “great man” theory of history and those who believe that discoveries will happen when the times are right. (Examples such as the Bernoulli family (note 10-3) suggest additional consideration of the “great family” theory.) Another notable example of nearly simultaneous but probably independent creation of an idea was the conception by Darwin and by Wallace of the theory of evolution by natural selection.

The Third Culture

Physics departments often required graduate students to demonstrate a reading knowledge of both French and German, which were once considered appropriate and relevant research tools. (Romans were builders, Greeks were theorists (page 36). Americans and Europeans can be contrasted similarly.) Such a requirement is now anachronistic. (Just ask George W. Bush about the current importance of France and Germany, and he'll tell you!) Language requirements are indeed disappearing (page 271). Many departments have allowed the use of a programming language to satisfy the language requirement. Virginia Tech's physics department dropped the language requirement in the year after I left Blacksburg. Some of the old guard among the faculty objected to this loss of a supposed liberalizing influence on the students. Personally, I never considered either ordinary languages or computer languages to be intellectually broadening in themselves.

Kelvin (page 26) said that when you can measure something, you know about it. Aside from being dismissive of almost every subject except experimental exact science, this attitude confuses information with knowledge. Rutherford (page 56) was similarly condescending, but with a different perspective, when he said that all science is either physics or stamp collecting. These days, we don't even have to measure something in order to think that we understand it. We shut up and calculate. That is, we compute. Educators attempt to replace real experiments by virtual ones in the classroom, but the real world is not one of our simulations. As Alfred Korzybski emphasized, the map is not the terrain.

Recognition of the true nature of the digital divide (page 272) and the emergence of a “third culture” is very perceptive, and addresses a contemporary matter of great practical importance. The digital divide is usually taken to mean a socioeconomic gap between those who “have” a computer and can use it, and those who “have not,” but the difference is more profound. In physics, the third culture is represented by the emergence of computational physics, and its ascendance over the traditional specialties of experimental and theoretical physics. (I did not mention mathematical physics. In a recent issue of “Physics Today,” David Mermin provides a joke: “ ‘Theoretical physics is done by physicists who lack the necessary skills to do real experiments; mathematical physics is done by mathematicians who lack the necessary skills to do real mathematics.' Mathematical physicists tend not to like this joke, but other physicists seem to. Nonphysicists, of course, are entirely immune to its charms.”)

Science fiction had the right idea, but the wrong details, about the robots taking over the world. The Golem and Frankenstein's monster have taken over by making us pieces of peripheral equipment. The digital divide is the triumph of process over content, of appearance over substance. Computers are a wonderful tool, but too often they have become an end in themselves rather than a means to an end. Many engineers and programmers now spend their days learning about the latest hardware and software, or reprogramming all of their codes from FORTRAN to C.++, without ever using the codes for anything. Such efforts are considered to be contributing to measures of productivity. Computers enable us to produce beautiful, if empty, visual presentations. Word processors make writing or rewriting much easier, but often result in too much writing rather than better writing. (Is my commentary here an example?) The pervasiveness of computers has affected our view of the world. Deep thinkers have conjectured that the universe is a great computer executing some program. At a more modest level, some thinkers have proposed that the life is but a container for a message embodied in the genetic code. In accord with Moore's Law, computers have been developing very rapidly, so anything to do with them is soon obsolescent. It is my observation that young people, who have been familiar with computers all of their lives, generalize their attitude about computers to other aspects of their world, and have little patience with tradition or respect for experience.

Extraterrestrial Life

The environment which computers ultimately dominate may be “outer space.” Space-faring robots may be the form into which of intelligent life, in a sense, evolves in the course of adapting to that new environment.

Extraterrestrial life (page 252) may or may not be common. The Drake equation is useful conceptually but not practically, because we don't know all of the factors that should be included, and we don't know the probabilities for some of the factors that are already included. Abandonment of the Ptolemaic earth-centered view of the universe (page 214) eventually led to the view that the earth was not unique in any special way, so that if life existed here, it should exist elsewhere. Physicists, who are able to estimate probabilities in the Drake equation for such factors as the percentage of stars with earth-like planets, tend to optimism about finding evidence of extraterrestrial life. Biologists are more skeptical of extraterrestrial intelligence than are physicists. Biologists are more aware of the astounding complexity and fragility of living structures, and of the many contingent features of the earth that make life possible here. Ward and Brownlee are well-known proponents of this skeptical view, which embraces many of the arguments advanced in support of the weak form of the anthropic principle. Then there is Fermi's famous question. “Where is everybody?” Either relativity and energetics impose an effective quarantine on intelligent extraterrestrial life, or it is very rare or nonexistent.

Of course, space-faring intelligence need not exist in an organic being that is alive in the usual sense. There is now an ongoing debate between advocates of manned exploration of space and proponents of the use of robots in space. There is little doubt that robots will be more “intelligent” in the future. There seems to be a widespread assumption that very intelligent machines would be conscious, but I think that this assumption is unwarranted. Is consciousness an epiphenomenon of structure, or of function, or of material? Consciousness may depend, not simply on structure, or even on function alone, but also on the properties of the particular materials composing the brain.

The assumption that a robotic brain necessarily would be conscious seems to me to a manifestation of a Platonist mind-set. Platonism, or “realism” as opposed to “nominalism” in a medieval sense, actually asserts the reality of ideas. The form of a structure, as opposed to the materials of which it is made, is an abstract idea that can be embodied in a variety of substances. Platonism also is related to issue of symmetry in the abstract, apart from the materials involved. Dirac's search for beautiful equations is another example of this mind-set. Solid-state physicist P. W. Anderson and others who think that the emergent properties of complex systems are not inherent in the constituents provide yet another example. The statement (page 94) that mathematical shapes “have already been replicated by nature,” as though shapes have a prior existence in the abstract, seems to verge on Platonism.

Art and Creativity

Ruskin advocated sketching (note 1-10). Roger Tory Peterson convincingly advocated the use of drawings instead of photographs to illustrate field guides for birds. A drawing can capture, in a way that no single photograph can, all of the essential features that could be extracted from numerous observations under various conditions, summarized into a single representation, and simplified by the omission of extraneous distractions.

Stoppard (note 7-19) denigrated contemporary art as showing imagination without skill. At the other extreme, critics sneer at makers of representational art as lacking creativity. They point to skillful forgers who skillfully can copy masterpieces, but who never could conceive them. Scientists struggle with creativity (page 238). Steven Weinberg has commented on the balance required in physics, where creativity must be constrained by reality.

If talent and creativity are different, does creativity arise from traits of temperament, or is it a peculiar, special kind of talent? Feynman and Einstein were both talented and creative. Feynman (note 11-2), with what I suspect was reverse snobbery, used to claim that, according to tests, he was not smart enough to join Mensa. Some persons have claimed that Einstein would not have done as well on tests as might be expected. Someone with a photographic memory, the ability to do lightning-fast calculations, and to perform quickly, can do well on standard tests. But idiot savants have some of these abilities. Einstein (note 13-7) may not have been the fastest thinker, but he was a profound thinker. Feynman (note 10-1) was so famously quick-witted that I might be tempted to attribute his allegedly poor early performance on tests to some sort of dyslexia or “attention deficit disorder,” but I suspect that it was the result of indifference and obstinacy.


Math and the Mona Lisa is the most thought provoking book I have read.    

Professor Atalay lucidly explains the mathematics of the Golden Mean which Leonardo both used in his art and eloquently wrote about in his notes which contain a fascinating array of scientific and engineering endeavors.

The Golden mean was a formula used directly or understood intuitively by almost all of the Renaissance artists. Although other Renaissance artists did not carry dissection of human anatomy to the extreme that Leonardo da Vinci did, Renaissance artists were the scientists of their day.

But "Math and the Mona Lisa" is not just another Leonardo da Vinci book. Dr. Atalay uses the Mona Lisa as a literary device to roam through the history of math and science as well as the history of art. As a scientist himself, Professor Atalay's descriptions make great scientists like Galileo, Newton, and Einstein seem real and understandable.

Yet Dr. Atalay is not a stranger to art. He has grown up simultaneously as an accomplished artist. Once the Queen of England asked him to publish a book of drawings of the English countryside.

 Dr. Atalay's education includes Eton, St. Andrews, Georgetown, Oxford and Princeton.

Dr. Atalay was born in Istanbul. His home town was the capital of the Roman Empire,Constantinople, Byzantium itself, and then the capital of the Ottoman Empire at the very border of Europe and Asia.

He recounts with awe, reverence and without prejudice the history of mathematics, art and science as they survived and flourished in the great seats of knowledge and learning in both the Europe and the Moslem world.

       — Ken Richards, Jr., Denver, Colorado, January 13, 2005


Private communiqué regarding Math and the Mona Lisa

I finally got some time — on 11 hours of airplane time, making a one-day turnaround trip to California — to go over the draft ... I can only try to describe my reaction to this iteration of the book.  It is so much cleaner, clearer, crisper, more tightly reasoned and written that I was beyond awe: I was in something like shock.  (The 25-year-old surfer … sitting next to me — who turned out to be one of the two creators of the PBS documentary and Random House book Roadtrip Nation — finally blurted out, one time when I came up for air, "Please, please, can I ask what you're working on??")

This is simply one of the great intellectual histories. There are numberless histories of science, and probably even more of art. But yours is not merely a history of both, but — as I'm sure was your goal — an organic history of both, anchored in the genius of Leonardo, which this general reader has never found either so accessible or so comprehensible.

I have, by my own limited lights, a pretty thrilling life. But being able to contribute in some very small way to this astonishing work is one of the most thrilling things I've ever done, and I cannot thank you enough for the opportunity.

       — Nick Murray, author of Simple Wealth, Inevitable Wealth, and several
other successful books on investment. New York, NY, July 25, 2003

 “MWC physics professor is a true Renaissance man”

I thoroughly enjoyed the story, "Art and Arithmetic," about Bulent Atalay, the distinguished professor of physics at Mary Washington. It is good to see genius recognized even when it's cloaked in humility.

I've been in Mr. Atalay's company only a half a dozen times, yet character isn't hard to notice. It's unfortunate that while much of our society is steeped in the superficial self-congratulatory rhetorical rubbish masquerading as "substance," the Bill Atalays of life go unnoticed.

[Bulent] is the quintessential Renaissance man. To me, that is the greater essence of the man: his civility.

— Richard Ford, Stafford, Virginia, March 23, 2004 

Dear Professor Atalay:

This letter is to tell you how much I enjoyed Math and the Mona Lisa. I have long been interested in that wonderful area where art and physics meet, and your book was another big step in my education. A few years ago I had a memorable lunch with Dr. Leonard Schlain [author of Art and Physics] in San Francisco and can recall a fine remark about surgery: “If an operation doesn’t look elegant, it probably isn’t much good. If it looks well, it will probably succeed.” A most interesting man, and ever since visiting the Leonardo Museum in Vinci, I have been fascinated by the possibilities of making models of his devices. Interestingly, one of the few fields where he was not on the cutting edge of the technology of his time (or mostly well beyond it) was architecture. Of the famous designs in his notebooks, all were possible with the technology of the time. One, in fact was built at Todi and finished after his death: S. Maria della Consolazione. You have probably visited the place.

The foregoing in no way lessens my profound admiration for your work. I was especially delighted with the distinction between the old and new quantum mechanics.

— Leonard K. Eaton   October 30, 2004
Emil Larch Professor of Architecture Emeritus, University of Michigan

Private Communication, Sherwin Nuland, M.D., F.A.C.S.
Clinical Professor Emeritus of Surgery at Yale University

Leonardo da Vinci
The Mysteries Within: A Surgeon Reflects on Medical Myth
Doctor: The Biography of Medicine
The Wisdom of the Body
Lost in America
How We Die: Reflections on Life's Final Chapter, Winner of the National Book Award for Nonfiction;

Dear Professor Atalay:

I am honored to sign this small book [Leonardo da Vinci] to Professor Bulent Atalay, whose fertile intellect will soon give us a monumental gift to Leonardo scholarship.

— Sherwin Nuland, March 23, 2003

Dear Professor Atalay:

I don't know which pleases me more: the very kind things you have said about my book or the trust you have placed in me by sending your manuscript, which a monumental gift to Leonardiana. I am honored to have the two beautiful lithographs and the short bio of your fascianating life.... May this important book achieve all the recognition it so richly deserves.

— Sherwin Nuland, March 23, 2003

Dear Professor Atalay:

It's wonderful to have your landmark book. As I skim through its pages on a first look, I am once again convinced that word "essential" is indeed the proper one for such a magnificent contribution to Leonardo studies. Aand I also admire the physical characteristics of the volume, especially the high quality with whcih the illustrations are reproduced and the appropriate beauty of the Leonardian chapter headings. The Smithsonian has served you scholarship well.

I wish I could be there with you on the day of the publication [at the Smithsonian Institution Lecture] to see the glow of admiration that this glorious work elicits from those who are present as it makes its debut. But, that will not be porssible. I will be showing Math and the Mona Lis to many a friend and colleague in the coming years, with the pleasure that comes from a real discovery. ,

— Sherwin Nuland, MD April 7, 2004


Review: Math and the Mona Lisa by Bulent Atalay

This is an analysis of the life and works of one of history's certifiable geniuses, a man who epitomizes the rubric of "Renaissance man." Atalay, a professor of physics at [the University of ] Mary Washington... in Virginia and an accomplished artist whose lithographs have appeared in several periodicals and have been on display in various exhibitions including one at Buckingham Palace, provides insightful analyses of the contributions of Leonardo to both the artistic and scientific worlds. He examines what he calls the science behind Leonardo's art and the art underlying his science. He begins by citing noted 20th century British scientist and author, C.P. Snow. During a lecture in the 1950's Snow had posited an essential dichotomy between the intellectual cultures of the humanists (including writers and artists) and that of the scientists (natural scientists as well as mathematicians) which placed each at odds with the other. Atalay takes issue with this construct. He suggests that an essential unity exists between art and science in terms of structure, methods, modes of analyses and in other ways. The book presents Leonardo as one of the two pre-eminent artists of his day (along with Michelangelo) and as the "first modern scientist" because of his use of the methodology of empirical science (even though others such as Galileo and Newton may have left more profound scientific legacies) many years before the great empiricists came into vogue. Atalay notes the "cross-fertilization" of art and science and the "seamless integration" of each into the other in Leonardo's works, informing virtually everything he attempted and making him, therefore, the embodiment of this unity. Indeed, the author suggests that Leonardo's life's work was devoted to establishing the connections and the symmetries between everything he attempted, which incorporated an enormous breadth of interest in the scientific, artistic and engineering fields. He calls this the "Leonardo model" — the "cross-semination" and "transcendent unity" of science and art. Atalay organizes the book to follow a basically chronological scheme. The first eight chapters follow the development of the fundamental sciences that Leonardo was interested in and also examine the nature of artistic expression. In these chapters, the author covers the achievements of many other individuals but always relates their work back to Leonardo's "model." The purpose of these chapters is to explore the differing approaches scientists and artists have taken throughout history to explain and describe nature. Next are three chapters that focus exclusively on Leonardo and his specific achievements and legacy. A final chapter gathers all together as a conclusion to the author's thesis of a synthesis between the artistic and scientific cultures. Throughout the book are numerous charts, graphs, formulae and Leonardo's black and white drawings. There is a center section of color plates. The book ends with a notes section incorporating the author's sources and a general index.

Victoria College, Library VC/UHV University of Houston Victoria, December 2004

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