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What are the worlds of knowledge inside a bio-acoustic, spectrogram, a mathematical diagram, or an illustrated equation? 

 is a column in which scientists describe the power of drawings.

 in college, 130 years after its publication. By then, the book had long stirred controversy and secured its position among the most important scientific books ever published. From the moment his theory appeared, Darwin’s game-changing vision of how organic life on earth developed had an immense impact on myriad aspects of human culture—from philosophy and religion to literature and the history of art. Reading 

 for the first time, it wasn’t hard to see why. A fledgling humanist, I recall being struck by the efficiency and elegance of Darwin’s prose and by the book’s mix of concrete detail (pigeon breeding, instinctual behavior in ants and bees, animal life on islands) and grand theory. But I do not remember noticing the book’s one illustration, a diagram often referred to as the “Tree of Life,” lithographed by the London-based printmaker William West and inserted as a double-page fold-out towards the end of chapter 4, the chapter devoted to Darwin’s theory of natural selection. A minimalist array of upward-branching dashed and dotted lines annotated by letters and numbers and bisected by solid horizontal lines, it resembles the silken threads spun by a spider or the delicate craquelure of an antique vase.

for the second time almost two decades later, as a scholar of art history and visual culture, studying works of art and other kinds of images in Europe and the Americas from the eighteenth century to the present. Much of my research considers how art-making and scientific inquiry intersected in this period—how knowledge in the sciences shaped the ways artists and their audiences envisioned their world, and how scientists used images to generate and communicate novel ideas and theories. This time around, I noticed Darwin’s diagram and paid close attention. A more careful reader in my forties than I was in my twenties, I understood that Darwin’s explanation of the diagram elucidates one of the most important principles of his theory: divergence of character. If natural selection describes how species originate, diverge, and become extinct—thereby accounting for the history and diversity of organic life—divergence of character explains, first, how small differences among varieties of organisms in a group can eventually lead to the existence of new species and, second, the critical importance of significant or extreme variation for those species’ survival. The finches of the Galápagos Islands most famously demonstrate divergence of character, even if they did not initially inspire the idea. Closely related but distinct species, the finches are differentiated by the size and depth of their beaks. These differences arose through the selection of specific traits that decreased competition among the birds and enabled many varieties of finch to coexist. Because each variety of bird consumed a different type of seed, as befitting the form of its beak, there was enough food to go around.

Darwin introduced the section on divergence of character in chapter 4 by announcing that his diagram would aid the reader “in understanding this rather perplexing subject.” On the face of it, however, the diagram hardly appears up to the task. The tree has an elegant, even fragile beauty that does not appear commensurate with the weight of Darwin’s thought. Nor does the diagram look much like other scientific illustrations of the period, which frequently referred directly to the observable world, either through visual representations of natural specimens or, in the case of charts and diagrams, robust verbal annotation. Darwin’s diagram, hypothetical rather than empirical, strips away all references to the real and cuts to the chase. The spare configuration that results is charged with demonstrating the immensely complex operations of divergence of character, but it is so abstract that it appears inconsequential, even superfluous. No wonder I missed it on the first pass.

Billed as an aid to understanding a “perplexing” subject, it is not immediately clear just what Darwin’s diagram does to augment and illuminate what he painstakingly explains in the text. His description of divergence of character runs for 16 dense pages and necessitates constant toggling between the fold-out image and text, a lengthy and disjointed process that runs the risk of obscuring the diagram’s meaning rather than revealing it. The book’s only illustration, the “Tree of Life” can feel extraneous, if not illegible. Even the dotted lines, a conventional means in nineteenth-century scientific illustration for indicating the hypothetical, make the diagram seem especially flimsy. Normally, dotted lines in a scientific image accompanied empirical information, as in the hypothesized anatomy of a dinosaur known through partial fossil remains. In Darwin’s diagram, however, nothing exists to anchor his theory to 

. The diagram is commonly called the “Tree of Life,” but it does not really look like a tree and Darwin never characterized it as such, nor did he fully embrace the tree metaphor for the mechanisms of natural selection, despite his poetic description of an endlessly branching tree at the close of chapter 4.

A case might be made for abstraction, which Darwin used before it became par for the course for scientific visualization in the twentieth century. Darwin wrote 

after drafting a lengthy manuscript, titled 

 that offered a thorough elucidation of his theory, which he abandoned in 1858 after learning that the British naturalist Alfred Russel Wallace had independently arrived at many of the same conclusions. 

was written quickly in the aftermath, as an “abstract” (Darwin’s own term) of the whole. Perhaps the diagram itself served as an abstract, this one illustration presenting the overall gist of Darwin’s theory in visual form, its expedience matching that of his swiftly written book.

Even so, the diagram undertakes an impossible task—impossible because evolution as Darwin conceived it presented a near-insurmountable problem for representation in the nineteenth century. Science in Darwin’s time gave rise to radically new ideas about existence, and this new knowledge created something like a crisis of picturing, with tried-and-true methods of visualization, conventionally mimetic in nature, coming up short. Time presented a special problem for scientific images. Just as the Copernican Revolution had displaced Earth from the center of the universe and Darwin’s own theory of evolution would go on to remake the human as just another animal among countless other beasts, geology and the closely related disciplines of paleontology and archaeology radically expanded the timescale of the earth while drastically reducing humanity’s share in its chronology. Deep time, in turn, introduced a very big problem of visibility. “The mind cannot possibly grasp,” Darwin wrote in 

, “the full meaning of the term a hundred million years; it cannot add up and perceive the full effects of many slight variations, accumulated during an almost infinite number of generations.” What is more, “we see nothing of these slow changes in progress, until the hand of time has marked the long lapse of ages, and then so imperfect is our view into long past geological ages, that we only see that the forms of life are now different from what they formerly were.” So many tiny changes occurred over so much time, Darwin says here, that no mortal could begin to isolate and identify them, let alone capture them in an image.

, Darwin never hesitated to point out gaps in science and in his knowledge—or as he put it, forms of understanding one could only “dimly forsee.” One gets the sense that, in the aftermath of a series of paradigm shifts in the sciences and in the wake of his own groundbreaking discoveries, he felt he knew even 

 about existence than before. Or, more accurately, that what he had learned about existence was monumental, even as it raised innumerable new questions and brought to light vast tracts of unexplored scientific terrain—what he referred to in the conclusion of 

as a “grand and almost untrodden field of inquiry.” Perhaps Darwin’s diagram, a sparse and tenuous graphic array, stood in stark contrast to other scientific imagery in the period because he wished to resist implying that he had all the answers (not least because the book’s nearly five hundred pages of detailed explication and data, along with the overall confidence of Darwin’s rhetoric, could easily have suggested to the reader that Darwin believed he had it all in the bag).

Given this, I cannot help but think that Darwin’s diagram bears the burden of Darwin’s doubt. It begins in thin air, with a collection of unanchored, dangling lines. And it ends, at the top of the fold-out page, with a spare collection of lines, dashes, letters, and numbers. The spaces between each of the dots in the branching lines indicate a temporal interval. But as a series of empty spaces or blanks, they also conjure absence, as in the absence of seeing or knowing.

The diagram wipes the slate clean, dismantling and destroying convention to clear a path for new modes of seeing.

Even Darwin’s detailed, multi-page explanation of the diagram stumbles in places, registering notes of misgiving. Each of the intervals between the diagram’s horizontal lines, Darwin noted, represent a thousand generations, but “it would have been better if each had represented ten thousand generations.” Here, Darwin pauses amid his explanation to undercut his own design, signaling to the reader that all is not said and done, and that more knowledge—and a better diagram for articulating this knowledge—are yet to come. He does something similar on the following page. “I must here remark,” he wrote, “that I do not suppose that the process ever goes on so regularly as is represented in the diagram, though in itself made somewhat irregular.” A few sentences later: “In our diagram, the line of succession is broken at regular intervals by small numbered letters marking the successive forms which have become sufficiently distinct to be recorded as varieties. But these breaks are imaginary, and might have been inserted anywhere.” As Darwin made clear in 

, natural selection was unpredictable and followed no strict script, differing according to geography and climate, arising from any number of causes, happening at vastly different rates, and yielding inconsistently and variably differentiated new species. Only a diagram simple and pliable enough to mean many different things could accommodate such variety and range.

Darwin’s self-conscious hedges suggest that he understood the diagram’s opacity, and wished to ensure the reader knew he was aware of this trouble. In this way, the diagram offers to the reader a kind of pedagogy, a primer on how to encounter the knowledge presented in Darwin’s book. With excitement, of course, and the feeling of having been convinced by Darwin’s claims, but also with a real sense of what was left to know, and what, in the end, might always exceed human grasp. That said, the diagram also succeeds, fantastically so, by wiping the slate clean, dismantling and destroying convention to clear a path for new modes of seeing and knowing and by generating, through abstraction, the possibility of an elastic and polyphonic mode of scientific visualization. In this way, Darwin tells us something about the history of life on our planet, but he also has something profound to say about the exertions of science and its verbal and visual languages, including the questionable capacity of any expressive form to delineate the nature of existence. ♦

Rachael Z. DeLue is the Christopher Binyon Sarofim ’86 Professor in American Art and Director of the Humanities Initiative at Princeton University.

Darwin's iconic "Tree of Life" illustration, depicting the branching pattern of evolution where all species descend from a common ancestor, showcasing his theory of descent with modification as described in 

Picture This: Darwin's Diagram

On June 29, 2023, I gathered alongside my colleagues at the National Science Foundation’s Headquarters to announce evidence for an astronomical signal likely originating from millions of supermassive black holes simultaneously merging. If we could hear it, the signal—known as a gravitational wave background—might sound like a low-frequency hum, generated by the cosmic population of these supermassive black hole mergers. My colleagues and I found this signal by monitoring galactic pulsars—dense cores of dead stars—which send regular flashes of radio waves to the Earth.

The gravitational wave background’s hum also leaves a distinct imprint in pulsars. The waves change the distance between these pulsars and the Earth, causing their lighthouse flashes to arrive a little earlier or later as the spacetime is stretched and squeezed. I study supermassive black hole mergers as part of a team known as NANOGrav, which has data providing evidence of this gravitational wave background. The chances of the signal being random are about 0.3%. For a definitive detection, as accepted by most of the scientific community, this should be 0.00006%. We expect to make a detection at this level in the next few years.

Our discovery that supermassive black holes merge changed everything. For the last 40 years, leaders in the field met and openly debated the question. I have been working on experiments like NANOGrav for almost 15 years, while others have been working in the field for over 20 years. Now, for the first time, we have compelling evidence that supermassive black holes—millions to billions of times the mass of the sun—merge to create even more gargantuan black holes. This means that the simplest assumptions about how a pair of supermassive black holes evolve as they sink to the bottom of a new galaxy—and stall in their evolution—are wrong. We now know that gas and stars help these massive black holes merge, and that they form part of an astronomical ecosystem at the center of galaxies, where the black holes slingshot out stars to get closer together, and twist and turn as vast quantities of hot gas push them closer together. It is, in fact, big news that these supermassive black holes merge at all. This discovery shows that our models of how the universe works pass another test: the current paradigm is that galaxies form in the early universe and grow through mergers, getting larger with each merger. Each galaxy has a central supermassive black hole, so galaxy mergers should lead to supermassive black hole mergers. We now have evidence that this final step happens too—that the supermassive black holes merge. Before our NANOGrav result, we only knew that black holes about 30 times the mass of the sun merged. It took a whole new experiment, NANOGrav, and 15 years of data to show that supermassive black holes merge too.

Now, for the first time, we have compelling evidence that supermassive black holes—millions to billions of times the mass of the sun—merge to create even more gargantuan black holes.

If I could time travel and tell my younger self that this was the case, I’m not sure I would have believed the good news. Reflecting back, I remember how amazed I was when I learned that gravitational waves had nothing to do with light. I had mistakenly imagined that beyond the known spectrum of light, past optical light, past the infrared, I would eventually find gravitational waves somewhere after radio waves. But, in fact, gravitational waves are not light. They are waves in the fabric of spacetime. All gravitational waves change the distances between objects, stretching and squeezing this fabric of spacetime as if it were lycra, pulling objects closer together and then further apart in predictable patterns. In the same way that different kinds of light require different tools for measurement, we need different telescopes to detect different gravitational wavelengths. NANOGrav and its precisely timed pulsars are needed to search for the low-frequency hum of gravitational waves with wavelengths that are light-years long.

In 1983, physicists Ron Hellings and George Downs published a seminal paper noting that the incoming signals from pulsars should be advanced or delayed by gravitational waves as they stretch and squeeze the distance between the Earth and the pulsars. The graph above is known as the Hellings and Downs curve. You don’t need to be fluent in data analysis to get a sense of its content. The curve encapsulates a prediction: A pair of pulsars that are close together, hence have a small angle on the sky (left hand side of the figure) are bobbing back and forth in the gravitational wave sea together, in sync. Ones that are further apart, at around 90 degrees, are out of sync.

When I first read the paper, I wondered if we would ever measure it. Also… why did it matter? What assumptions were baked in, and how did they write down the final equation? To answer these questions, we need to go back 40 years to NASA’s Jet Propulsion Laboratory. Hellings and Downs expected that the deviation in the pulsar’s timing would be so small (about 100 nanoseconds in a decade) that they had to be very clever about how to measure these tiny differences. What they discovered was that by cross-correlating pulsar data streams, anything common to both data streams gets amplified. The weaker the signal, the more pulsar data streams are needed to sufficiently amplify the signal and make it detectable. The more cross-correlations, the more amplified the signal.

The Hellings and Downs detection strategy introduced a new way of understanding whether or not the signal we would eventually measure originates from gravitational waves, which would result in making the Hellings and Downs pattern after cross-correlation. Nothing else in nature that we know of can create the distinctive Hellings and Downs correlation pattern between pairs of pulsars.

Fast forward about 30 years, to 2014. When I was a Marie Curie Fellow at the California Institute of Technology (Caltech), and a visiting scientist at NASA’s Jet Propulsion Laboratory, some of the most famous scientists I knew told me that maybe one day NANOGrav would measure just one number—the amplitude of the gravitational wave background—if we were lucky. But it was unlikely, since everyone 

 that supermassive black holes don’t merge. They advised me to work on another experiment, LIGO, that detects gravitational waves from drastically smaller black holes. My chances of having a career were much better if I developed the gravitational-wave theory for LIGO instead of NANOGrav. It was good advice: LIGO was a juggernaut with an almost $2 billion investment from the National Science Foundation, and there was a very large group at Caltech dedicated to its development. NANOGrav had just secured its first $15 million grant, over five years, to support its 100 scientists. That is to say, still a very small experimental budget. But I couldn’t give up on pulsar timing arrays. Hellings and Downs assumed that the gravitational wave background was arriving equally from all directions in space, but I was not convinced. Surely, I thought, some parts of the sky could have a nearby pair of supermassive black holes, or that clusters of them could exist, and then what happens to the curve?

It was as if they were all chanting “Om” together at the end of their galaxy merger process, incredibly thrilling—as if everything finally came together to make this gravitational wave sound.

This brings us back to June of last year, at the National Science Foundation Headquarters. After timing an array of pulsars every two weeks for 15 years with some of the best telescopes on the planet, we finally saw the Hellings and Downs curve emerge in the cross-correlated NANOGrav data (the blue points in the figure). When I saw the curve emerging and could see with my own eyes that the curve was really there, I was filled with disbelief and excitement. The blue dots and their error bars were finally overlapping with the purple line, which is the Hellings and Downs curve. There before us was evidence that supermassive black holes merge, and not just a few, but the cosmic population of supermassive black holes humming in unison. It was as if they were all chanting “Om” together at the end of their galaxy merger process, incredibly thrilling—as if everything finally came together to make this gravitational wave sound.

NANOGrav showed that the Hellings and Downs curve exists, but what I was able to show was that this should just be the first smooth layer of the gravitational wave background—a layer which many presumed did not exist. I predicted that there are many layers we can measure, and, in fact, we could measure more than just these maps. We could probe different models of the early universe, test Einstein’s Theory of General Relativity, explore and probe exotic forms of dark matter, and more. Now many teams are searching for the subtler layers in the background and we are hoping to eventually make maps of the entire sky of gravitational waves. It’s an exciting time. Many colleagues have now shown interest in making these gravitational wave background maps, and a whole new subfield of astrophysics related to finding and measuring these layers of the gravitational wave background is thriving.

We may soon be able to make these maps, and to locate the closest or most massive supermassive black hole pairs. Finding these pairs would be irrefutable evidence that supermassive black holes merge, and that our current theories of how galaxies form and evolve, through mergers, hold. We will be able to constrain exotic theories of dark matter, theories of the early universe, and be on the lookout for new, unexpected humming signals which we haven’t yet predicted or anticipated. What else is out there? I have no idea. That’s what’s so exciting.♦

Chiara Mingarelli is a Professor of Physics at Yale University and a Public Voices Fellow, OpEd Project.

Picture This: Making Waves

This blackboard hung in Stephen Hawking’s office at the University of Cam­bridge. After Hawking’s passing, it was acquired for the nation by the London Science Museum Group, where it’s now displayed in the Exploring Space gallery. Filled with inscrutable doodles, drawings, inside jokes, and half-finished equations, this blackboard is a glimpse into the abstract universe of theoretical physicists. There is a floppy-nosed squid climbing over a brick wall, but also a tin can labeled “EXXON SUPERGRAVITY” and ample references to hypothetical species of particles. Hawking himself is mysteriously drawn in the center near the bottom, with his back toward us.

I first saw this blackboard in 1998 as a beginning PhD student of Hawking’s. At the time, I wondered whether these scribbles could be some of his last hand-scrawls. This isn’t the case, as it turns out. The writings are from participants of a month-long workshop that Hawking and his Cambridge colleagues convened in the summer of 1980, called 

. The blackboard is a memento from this workshop. Quite a few of the drawings were made by Hawking’s co-organizer and postdoc at the time, Martin Roček, whose face can be seen, sketched in chalk, near the board’s center. As a matter of fact, a picture of this blackboard decorated the cover of the conference proceedings, published after the workshop’s end. Hawking refers to it in his foreword, before summarizing the meeting as “an instructive way to spend four weeks.”

In that same year, as incumbent of the Lucasian Chair of Mathematics at the University of Cam­bridge, Hawking delivered his inaugural lecture, “Is the End in Sight for Theoretical Physics?” In this lecture, at the zenith of his confidence in the power of physical theory, he predicted that physicists would find the final theory of everything by the end of the twentieth century. Moreover he famously—and controversially—put forward the theory of supergravity as the prime candidate for such a theory. “The best hope for a unifying quantum theory of gravity seems to lie in an extension of general relativity called supergravity,” he said, before adding a key caveat: that “a complete theory includes besides a theory of dynamics, such as supergravity, also a set of boundary conditions.” Elaborating on this point, he added, “Many people would claim that the role of science was confined to the first of these and that theoretical physics will have achieved its goal when we have obtained a set of local dynamic laws. They would regard the question of the boundary conditions of the universe as belong­ing to the realm of metaphysics or religion. But we shall not have a com­plete theory until we can do more than merely say that things are as they are because they were as they were.”

This is an important point. Ever since Galileo and Newton, physics has been based on a dualism of sorts, in that it has relied on a fundamental separation between two distinct sources of information. First, there are laws of evolution, mathematical equations such as those of supergravity, that prescribe how physical systems change in time from one state to another. Second, there are boundary conditions, a concise description of the state of a specific system at a given moment in time. The laws of dynamics take that state and evolve it, either backward or forward in time, to determine what the system was like at an earlier moment or what it will be like at a later moment. It is the combination of the laws of evo­lution and the boundary conditions that yields the framework for predic­tion on which physics prides itself.

"We shall not have a com­plete theory," Hawking said, "until we can do more than merely say that things are as they are because they were as they were."   

For example, imagine we want to predict where and when the next solar eclipse will occur. To do so, we can apply Newton’s laws of motion and gravity to describe the future trajectories of Earth and the moon. To put these laws to use, however, one must first specify the position and velocity of Earth and the moon relative to the sun (and to Jupiter) at one particular moment in time. These data are boundary conditions. They describe the state of these two celestial bodies in the solar system at one specific mo­ment in time. No one expects Newton’s laws to explain why these positions are what they are at that moment. Instead, we measure what they are. With this information at hand, we then solve Newton’s equations to determine their positions at future times, in order to predict when and where solar eclipses will occur, or at earlier times, to retrodict documented eclipses.

But this separation between universal, law-like dynamics and ad hoc boundary conditions can become a major embarrassment in cosmology, when we embed our experiments and ex­perimenters—our planet, the stars, and the galaxies—in the much larger evo­lution of the universe as a whole. When we do this, boundary conditions on the original experiment are subsumed into the law-like evolution of the larger system, together with boundary conditions on the latter. Returning to the example of a solar eclipse, a holistic cosmologist could argue that the ve­locities and positions of Earth and the moon at any given moment—the original boundary conditions—follow from their past histories, and that our plane­tary system itself is the outcome of the formation history of the solar sys­tem, which in turn arose from the condensation of remnants of prior stellar systems, whose seeds ultimately grew out of minute density variations in the primeval universe that came from… what?

When we arrive at the beginning we reach a paradox. What determines the ultimate boundary conditions at the origin of time? Clearly these are not up to us to choose and we can’t try out different conditions to see what kind of universes they produce. That is, the beginning of the uni­verse poses a problem of boundary conditions that we do 

 control. In­stead, very interestingly, the conditions at the big bang appear to be dragged into the laws we seek to understand. Yet dualism in physics holds that boundary conditions aren’t part of the physical laws.

In his Lucasian lecture in 1980, Hawking put his finger on this paradox. More than any other physicist of his generation, he felt that the deeper questions in cosmology required us to rethink the centuries-old framework for prediction in physics. His idea was to take physics beyond its stubborn dualism of eternal laws versus conditions. Such dualism, he felt, was too narrow a way of thinking about the world if we were to truly understand our cosmic origins.

Working with Jim Hartle, Hawking went on to develop an ingenious theory of the ultimate boundary conditions at the universe’s birth. Their model involved turning time into an extra dimension of space back at the beginning, thus quite literally morphing dynamics into state. Next they bent space into a four-dimensional version of a round sphere to close the “past,” thereby excising the notion of a beginning altogether. “Asking what came before the big bang would be like asking what lies south of the South Pole,” Hawking summed up their model, and he went on to refer to their theory as the 

 hypothesis, since a sphere has no edge or boundary.

In what may well have been an intended resonance, Hawking first pro­posed his cosmogony at a meeting of the Pontifical Academy of Sciences in the Vatican in Octo­ber 1981. Earlier in the week, Pope John Paul II had told the assembled scientists that “every scientific hypothesis on the beginning of the world … requires knowledge be­yond physics.” As if he were responding to the pope’s allocution, Hawking put forward the bold idea that there might not quite have been a beginning. “There ought to be something very special about the boundary conditions of the universe and what can be more special than the condition that there is no boundary.”