The most praiseworthy form of painting is the one that most resembles what it imitates.
—Leonardo da Vinci
ike the artist, the scientist is a lover of nature. Just as the artist is restricted only by his imagination and his facility with his chisel or brush, the scientist is restricted only by his imagination and his facility with his mathematics. The artist uses imagery and metaphor; the scientist, numbers and mathematics.1 The artist is more interested in the whole of his composition than in its very fine details. And the scientist is more interested in the generality of nature’s laws than in its particulars. However, it is from scrutiny of a very small section of the universe, the earth, that he tries to explain the whole. A “beautiful law” of nature, one whose fundamental symmetries have been deciphered, one that is simple and yet general, evokes the image of an ornate tapestry, and in physicist Richard Feynman’s words, “Nature uses only the longest threads to weave her tapestry, so each small piece of her fabric reveals the organization of the entire tapestry.”2
For mathematicians and physicists it is undeniable that there exists inherent beauty in mathematics. This is the aesthetics of mathematics. Perspective, proportion, and symmetry in any context are quantifiable. Accordingly, art indeed possesses quantifiable aspects. There is the symmetry expressible in mathematical terms, and then there are “nature’s numbers.” These notions figure into the mathematics of aesthetics. The associated quantification can be formulated at various levels of mathematical sophistication. In this book the mathematical authority—embodied in a modest number of equations—will be relegated to the endnotes. The Fibonacci series gives rise to the notion of dynamic symmetry, the golden section, or the “divine proportion,” which Fibonacci himself could not have anticipated. Three hundred years after Fibonacci formulated his series Leonardo da Vinci illustrated a book called De divina proportione. But the integration of science and art has many more strands than Fibonacci’s mathematics and Leonardo’s art: It also draws in elements of architecture, astronomy, biology, chemistry, geology, engineering, mathematics, philosophy, physics—encompassing the extraordinary range of Leonardo da Vinci’s interests. For him these were branches of the same tree, part of a grand unified structure, the universe.
In nature we observe symmetric shapes at the macroscopic level in both animate and inanimate objects. At the microscopic level and at the supramacroscopic, both beyond the capabilities of our senses, some of the same shapes, symmetries, and regularities emerge. When magnified one hundred thousand times, the spirals seen in the cross-section of the microtubules of the heliozoan resemble, in scale and shape, the spirals seen in the horns of the ram. This shape, magnified another hundred billion billion times, resembles the arms of a spiral galaxy. At one extreme the observing apparatus may be an electron microscope or its more powerful cousin, the scanning-tunneling microscope, and at the other, an optical or radio telescope.
X-ray diffraction technology reveals certain symmetries in crystals that also manifest themselves at the macroscopic level. Crystallographers identify five possible Bravais or space lattice types in two dimensions and fourteen types in three dimensions. All of the two-dimensional and some of the three-dimensional types are found in man’s artistic creations. A millennium before crystallography became a science, Moorish artists—especially Sunni Muslims, forbidden to produce the human likeness—were creating magical calligraphic and geometrical designs displaying intuitive understanding of the space lattices. This is nowhere more dramatically illustrated than in the mosaics and stone carvings at the Alhambra Palace in Granada and at the Great Mosque in Córdoba.
In the physical and mathematical sciences the recognition of symmetries in nature plays a central role in seeking new laws. The physicist observes symmetries in physical laws; however, he is often more interested in partial or incomplete symmetries than in perfect symmetries—because it is in the former that one can look for a deeper story, a more fundamental or profound insight into the laws of nature. The same is true in art, where an off-centered location of the focal points makes the picture more intriguing than would a centered location.
In modern physics one finds that some of the most abstract and arcane areas of pure mathematics, the elements of which often appear entirely unconnected to the laws of nature, in time prove to be the basis of fundamental physical laws. (Linear algebra, differential geometry, and algebraic topology are three such areas of mathematics that have helped theoretical physicists express the fundamental workings of nature.) From this experience emerges an appreciation of the aesthetics of mathematics and an abiding faith among physicists: Mathematics is indeed the tool to unravel those subtly hidden laws, to explain why a particular experiment yielded the results it did or, still better, what is yet to be observed. But why there should exist this remarkable effectiveness of mathematics in describing the laws of nature has never been shown. Princeton’s ineffably wise Eugene Wigner, one of the greatest theoretical physicists of the twentieth century, wrote: “The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift, which we neither understand nor deserve. We would be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.”3
Wigner, seen in my 1978 ink sketch (Figure 2.1), described the failure of science to answer the question of the mathematical nature of the universe as “a scandal … an enormous gap in human understanding.” No one could have spoken with greater authority on the subject. No one had earned the right to be as boastful and haughty, and yet no one was more gentle and humble in his demeanor. His most important work had been the introduction into physics of group theory with its attendant symmetry considerations, a formalism that turned out to be a transcendent contribution to our quest to understand nature.4
Just as symmetry can produce a sense of harmony, balance, and proportion, too much symmetry in certain contexts, such as in an endless line of row houses or identical statues, can have negative emotional impact. Similarly, asymmetry can produce a sense of discord and lack of proportionality. But in some instances, such as in the shape of an egg (as opposed to a smooth sphere), it can generate a positive emotional response, a sense of release, freedom, and mystery. Thus, released from the prejudice of viewing only perfect symmetries as ideal, we have come to regard the Alps as uncommonly beautiful creations of nature.5 Likewise, the finest examples of visual art and music are anything but endlessly regular. Indeed, the notion of “the monotonous” is one of artistic and social aversion. In the seventeenth century, a younger contemporary of Shakespeare’s captured this message:
Figure 2.1. Portrait of Princeton Nobel Laureate Eugene Wigner. Ink drawing by the author, 1978
A sweet disorder in the dress
Kindles in clothes a wantonness.
A careless shoestring, in whose tie
I see a wild civility:
Do more bewitch me, than when art
Is too precise in every part.
—Robert Herrick (1591–1674)
It is not the purpose of this book to present an exhaustive inventory of examples of natural and artistic phenomena that demonstrate the same patterns, but rather to scrutinize the symmetries and patterns at a fundamental level, and to examine possible forces that would produce similar shapes at wildly disparate scales. We will also review the notion of aesthetics, and the mathematics underlying aesthetics. By studying the interdynamics of art and science one gains a sense of the confluence of the two, and to a lesser extent, the psychology underlying the human affinity for symmetry.
The observational skills necessary to perform modern science come from the skills introduced by artists in the Renaissance. It has been said that the greatest discovery of science is science itself.6 Yet it was the artist, in his attempt to mirror nature, who first learned to observe nature as it really presented itself. And, in turn, for the artist the greatest discovery of the Renaissance—actually a rediscovery—was the notion prevailing in classical antiquity that man was the measure of all things.
Twenty-five years before Leonardo was born, a few discerning artists began to develop a new approach to observing and representing nature with correct perspective. Leonardo, however, made a science of linear perspective, and as a part-time artist, he produced among a handful of paintings the two most famous paintings in history (the Last Supper and the Mona Lisa). He also left behind thousands of pages of personal observations, ruminations, and sketches on every subject conceivable. In the last few years, as scientists—physicists, mathematicians, engineers, anatomists, botanists—have undertaken examinations of Leonardo’s work in science and technology, they have returned to pronounce Leonardo the first modern scientist.7 His methodology prefigures Galileo’s work by more than a century. Ultimately, it is the originality of his questions and the prescience of the solutions that still impress us after five hundred years.
An unfortunate banality in philosophy ascribes to science the exclusive process of analysis, and to art, the exclusive process of synthesis. The scientist, this platitude explains, takes apart his subject. The artist, puts its together. Jacob Bronowski brings this misconception to task: in reality the scientist engages in both processes, as does the artist. For each, imagination begins with a very close scrutiny and analysis of nature, and ends in synthesis, putting together a “form by which the creative mind transcends the bare limits, the bare skeleton that nature provides.”8 Bronowski offers sculpture for the model of science. Sculptors from Phidias onward imagined the existence of the statue in the rough-hewn block, the figure beckoning to be released, to be discovered. In Michelangelo’s words,
When that which is divine in us doth try
To shape a face, both brain and hand unite
To give, from a mere model frail and slight,
Life to the stone by Art’s free energy.
The best artists hath us thought to show
Which the rough stone in its superfluous spell
Is all the hand that serves the brain can do.9
Likewise, the scientist operates as if physical laws already exist, in unique form, and only need to be discovered, or to be extricated from nature. But in reality, the physical laws no more exist in unequivocal manner than the statue in that rough block. In the hands of different sculptors the block is destined to yield different forms. And in the hands of different scientists the laws are destined to emerge in different form, although ultimately perhaps susceptible to a demonstration of equivalence. Depending on how exactly the questions are asked, the results can appear differently, but correctly.
Progress in certain areas—in the arts and the sciences especially—involves vicissitudes along the way, bumps, forward surges, occasionally retrograde slippage, before true understanding occurs. At the risk of oversimplifying, offered here are a pair of converse views. James Burke, who starred in the BBC series Connections, claimed that at the roots of the progress of civilization were incremental steps. Bronowski, a one-time colleague of Burke, had argued in an earlier series, The Ascent of Man, that progress came in the form of phase transitions, revolutions. The contention in this book is that both types of change are necessary. Certainly the incremental changes represent the norm, but then a need for a radical reductionism, a synthesis, becomes a periodic necessity in order to put some order into the incremental progress. This type of transformation has occurred in science, in art, and in other fields. Bronowski expressed this message succinctly: “Although science and art are social phenomena, an innovation in either field occurs only when a single mind perceives in disorder a deep new unity.”
Radical reductionism has its purpose in the sciences and in art. In the sciences, one tries to reduce the laws to their bare essence. The unification or synthesis of the laws of physics represents a radical reduction into a single sublime formalism, a Holy Grail of physics. Reductionism in art occurred naturally and in a most healthy manner—progressively. Certainly it was underway with Vermeer in the seventeenth century, gained momentum with the great colorist J.M.W. Turner in the nineteenth, and came to full fruition in the twentieth century with geometric abstractionists like Mark Rothko. This movement, however, started even earlier when Michelangelo, toward the end of his incomparable career, carved the series of sculptures called the Captives, depicting bodies emerging from rough hewn stone, statues that now line the long room leading to his David in the Gallerie dell’Academia in Florence. In another late work, his Rondanini Pieta (c. 1550–53), Michelangelo abandoned realism, depicting his figures with elongated limbs in the fashion of Mannerism, a style in which the work evokes previous art rather than nature. These works are magnificently finished in their seemingly unfinished states, as is Schubert’s Unfinished Symphony. Only half a century after Michelangelo’s death, Rembrandt began to create some of the most powerful portraits in the history of art, having discarded fine brushes and minute close-up detail, while capturing the soul and character of his subjects with his bold multi-layered brushwork.
While science and art share similarities in their patterns of inspiration and creativity, there are also glaring differences. Although there exist instances where the artist consciously draws on some of the symmetries and proportions that we shall express in quantifiable terms in this book, the vast majority of art is ultimately produced intuitively, emotionally, and created with imagination and subjectivity, with no interest in such quantification. And this is as it should be. The scientist, for whom scientific or physical intuition and imagination can be a powerful asset, avoids emotion, constrained by drawing on analysis and objectivity—working within the boundaries dictated by the principles of science, and ultimately by the laws of physics. It is not only acceptable for an artist to draw from religious inspiration, but historically religion has been art’s driving force, providing content and theme (and often the cleric’s patronage). Indeed, it has been in the name of God(s) that virtually all of the most enduring works of art and architecture were created.
The decoupling of art and religion began anew in the Renaissance. When Leonardo painted his incomparable psychological portraits of the “three women”—the Ginevra de’ Benci, Cecilia Gallerani, and the Mona Lisa—it was again the beginning of a break from the accepted norm—religious art. By the early seventeenth century the great Dutch artists, including Rembrandt and Vermeer, were more interested in nonreligious subjects than religious. Today, whether secular or nonsecular, good art is simply good art.
Science, in contrast to art, can at best coexist with religion, even in this presumably more liberal and enlightened age. In the past, the two were at constant loggerheads. Even in antiquity the adversarial nature of the relationship between science and religion was apparent. The natural philosopher Anaxagoras, an advisor to Pericles, would have been lynched by a mob for ridiculing the gods of the pantheon had Pericles not intervened. As it was, he was simply allowed to retreat into exile, becoming one of the first intellectuals persecuted for his religious views. Similar danger existed in the Middle Ages for philosophers challenging the authority of the Church, and this remained the case well into the Renaissance. In modern times, however, mixing the two would be anathema for the serious scientist, and probably discouraging for the religious fundamentalist. Creation science, for instance, is frankly an oxymoron, a contradiction in terms.
When the great scientists of the Scientific Revolution—Newton and Galileo, both religious men—performed their experiments in physics, their mode of questioning was no different than it would be for a scientist today. They did not mix their science and religion. Modern science is performed in a secular manner, detached from religion. Among pure scientists today, one still encounters individuals with a deep spiritual side, but even they do not mix their science with religion. There appears to be no reasonable way to reconcile science and religion. In short, one is not testable by the other. Each requires a certain kind of faith. In the case of science, there is, first, that there exist laws of nature; second, that the laws of nature can be extracted from her, and third, that they can be expressed mathematically. As a metaphor for the order in the universe—the cosmos—scientists have frequently invoked the name of God. Einstein, giving a guest lecture at Princeton University in 1930, expressed this circumstance in his native German, “Raffiniert ist Herr Gott, aber Böshaft ist er nicht!” (Subtle is the Lord, but malicious he is not!) The laws of nature are there for us to decipher. The task may not be easy, but unlike the Mad Queen in Alice in Wonderland, he does not change the rules of the game already in progress. Paul Dirac, a central player in the development of quantum mechanics, was known for his frequent reference to the value of “beautiful equations,” and the pronouncement, “God is the greatest mathematician of them all.” But as we have seen, for Einstein and Dirac, God was the order of the universe.
Once I was in Cleveland, Ohio, for an interview on a CBS affiliate station. I had been speaking about the birth and death of the universe, being mindful not to trample on the religious sensibilities of anyone in the listening audience. The call-in nature of the program brought forth a question that put me in difficult straits, “How does a physicist reconcile science and religion?”
I was there to answer questions about the physicist’s view of the origin of the universe, not about the place of God in the scheme of things. Reflexively, I blurted out, “Physics and religion are orthogonal functions. They are linearly independent.”
The moderators looked puzzled, trying to figure out what I said. And I tried to figure out what I had said. My answer pertained to wave functions, and had come straight out of quantum mechanics, a subject I had taught for many years. Trying to gain some time, or to deflect attention away from what to me was an intractable question, I invoked a story about the great nineteenth-century physicist Joseph Henry, the inventor of the electromagnet.
In 1846 President James K. Polk wrote to the Royal Society of London, asking for the name of the world’s greatest scientist. (The Royal Society at the time functioned as a clearinghouse for matters scientific.) Polk explained, “[W]e would like to offer him the position of Secretary of the Smithsonian Institution in Washington, D.C.” The Royal Society wrote back, identifying Joseph Henry,10 professor of physics at Princeton University, as the leading scientist. (This was also the time Michael Faraday was operating in England, and with a much greater scientific legacy than Joseph Henry, but clearly the British did not want to send their greatest scientist to the hinterland of the United States.) Polk’s assistant was immediately dispatched to Princeton to offer Mr. Henry the job. The man found Henry just getting ready to begin an experiment with one of his electromagnets. Henry invited Polk’s assistant to observe the experiment, but then suddenly stopped, proposing, “Let’s get a few more witnesses.” Then he stepped into the hallway and rounded up some graduate students. Everyone watched curiously, as Henry, just before turning on the power, pronounced, “We are going to ask God a question. Let us pray that we do not miss his answer when He gives it to us.”
Henry had reduced performing scientific experiments to asking God questions. The magnificence and genius of the Creator was ultimately embodied in nature’s laws, and it was precisely by extracting these laws that he was honoring God. Though he thought his answers came from God, he asked his questions according to the same scientific method that had taken shape in the seventeenth century and had been followed by the great early scientists of that era, most notably Galileo and Newton, and Leonardo before them. This mode of questioning nature did not require any appeal to God or a Creator or any other supernatural force or being. Henry was a devout and simple man who happened to be immensely gifted as a scientist. When Henry (or for that matter, Faraday, who was also an intensely religious man) performed his experiments, his mode of questioning was no different than it would be for a scientist today. These men did not mix their science and religion, although ultimately both were convinced that by uncovering the truth and beauty of nature they were glorifying God.
In Rocks of Ages, the evolutionary biologist Stephen Jay Gould compared the domains of science and religion: “The net, or magisterium, of science covers the empirical realm: what is the universe made of (fact) and why does it work this way (theory). The magisterium of religion extends over questions of ultimate meaning and moral value. These magisteria do not overlap … science gets the age of rocks, and religion the rock of ages; science studies how the heavens go, religion how to go to heaven.”11
Leonardo was the most impious of the thinkers of his time (although there exists no evidence that he was not a believer) and operated without ever appealing to divine providence in either his science or his art. Nonetheless, he created in the Last Supper one of the two most important religious works in all of Western art (the other being Michelangelo’s Sistine Ceiling). His working dictum extended to other fields where blind faith ran counter to experimental and mathematical demonstration. He abjured astrology, which even the later scientists Kepler and Galileo did not reject, and he renounced alchemy, which Newton secretly embraced. In this sense, Leonardo’s perspective was even more modern than those of the luminaries of the Scientific Revolution.