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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.

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.”

The enigma of the big bang has been one of these rare instances in science where broader philosophical considerations have come to the forefront. Such occasions teach us something not only about nature, but also about the nature of science and what it is that physics ultimately finds out about the world. Over the years, the Vatican and physicists alike have had to revise their positions on this.

A theory of boundary conditions, we came to realize, isn’t just a law of the beginning but also the beginning of laws. 

But what fresh philosophy of physics flows from Hartle and Hawking’s cosmogenesis that replaces dualism? For years this has remained mysterious. Worse, the original version of the no-boundary theory didn’t work: it predicted an empty universe devoid of galaxies and life. Starting in the early 2000s, however, Hawking, Hartle, and I set out to rethink their earlier ideas on the big bang from a proper quantum viewpoint. We were led to a fundamentally evolutionary understanding of these earliest stages of the universe, in which the physical laws coevolve with the emerging universe. A theory of boundary conditions, we came to realize, isn’t just a law of the beginning but also the beginning of laws. This was way more radical than what even Hawking initially had in mind.

The younger Hawking sought a final theory of physics that was hovering above the universe—or even the multiverse—like an eternal truth. He assumed there was a fundamental causal explanation for the origin of the cosmos that was to be found deep in the mathematical underpinnings of physics. The later Hawking rejected the idea that the universe is like a machine gov­erned by unconditional laws with some sort of prior existence. The theory we developed toward the end of his life states that physics itself fades away back into the big bang. It is not the laws as such but their capacity to change and evolve that has the final word. One might say our theory provides physics with an epistemic horizon.

While it took many years to develop this new vision, the first inklings in Hawking’s mind go back to the workshops at Cambridge in the early 1980s. Indeed, if he did predict the end of dualism on the blackboard, whatever he wrote with the remaining strength in his hands got eaten by the squid. ♦

To read more about Hawking’s final theory, read Hertog’s new book, 

Thomas Hertog is a former student & key collaborator of Stephen Hawking, and the author of 

On the Origin of Time: Stephen Hawking’s Final Theory

Stephen Hawking's Blackboard

Picture This: Stephen Hawking's Blackboard

When Francis Crick and James Watson were sure they had adduced the correct molecular structure of the DNA molecule in 1953, Crick barrelled into the Eagle, their favorite pub in Cambridge, “to tell everyone within hearing distance that we had found the secret of life.” Or so Watson recounts in 

—his 1968 account of their remarkable discovery.

Watson has now admitted he made the story up. It’s just one among the many distortions and fabrications in the book, but you can perhaps understand his reasons. That elegant double helix didn’t just describe the shape of the molecule believed to be the gene carrier in our cells. It also suggested how genes—the hitherto mysterious entities responsible for inherited traits—actually work. DNA, Watson and Crick deduced, is like a string of letters: the twin strands are chains of interlinked molecules that each contain one of four so-called nucleotide bases: A, T, C and G. Genes were then distinct segments of these chains with particular base sequences.

Watson and Crick described the implications of DNA’s structure in a May 1953 paper in 

. It “seems likely,” they wrote, “that the precise sequence of the bases is the code which carries the genetic information.” Here was the key to the modern theory of genetics: it was all about information encoded in molecules, an idea that melded perfectly with the emerging technology of digital computers, which was all about information encoded in data strands of magnetic tape.

In a 1957 lecture in London, Crick spelled out “the main function of the genetic material,” which, he said, “is to control . . . the synthesis of proteins.” Proteins—folded-up chains of amino acids—were, in the form of enzymes, already known to facilitate particular biochemical processes. Somehow, the base sequence on DNA was “decoded” and converted into the amino-acid sequence of a protein. Neither Crick nor anyone else understood the details yet, but the process was known to involve the intermediary molecule RNA, which was also a string of nucleotide bases. In his lecture, which became the basis of an article in 

 the same year, Crick presented this picture in terms of the 

He had sketched the concept the previous year in a note titled “Ideas on protein synthesis,” where he declared, “Once information has got into a protein it can’t get out again.” In other words, information was passed from DNA to RNA (a process called transcription) and then to protein (translation). It might flow back from RNA to DNA, Crick said, but it could not go back from protein to the nucleic acids.

He gave the scheme a striking name: the Central Dogma. It was a curious choice, given that science is meant to be always open to correction rather than dogmatically stuck to any preconceived idea. Crick later explained that he simply didn’t know, until Nobel laureate biologist Jacques Monod pointed it out to him, what “dogma” meant:

I used the word in the way I myself thought about it, not as most of the rest of the world does, and simply applied it to a grand hypothesis that, however plausible, had little direct experimental support.

That was Crick for you: an error became simply his personal choice. “I have never seen Francis Crick in a modest mood,” was how Watson began 

, to his professional partner’s irritation. But it was perhaps an unfortunate confusion, because there is nothing like calling an idea a “dogma” to motivate some to defend it trenchantly, and others to challenge it.

The status of Crick’s Central Dogma today, now that we have learnt so much more about how genetics and molecular biology work, still excites passionate debate. Some think that the Dogma is obsolete; others assert that it remains intact, despite originally being a guess way beyond what the knowledge then available could support. One might even be tempted to suppose from the intensity of the arguments that the integrity of modern biology rests upon it.

There are very few statements one can make about biology for which exceptions can’t be found: the living world is a messy place with little respect for general rules.

But it really doesn’t. The Central Dogma, however poorly named, states something that is important and true. If one can adduce possible exceptions to it, that’s not such a big deal in itself, for there are very few statements one can make about biology for which exceptions can’t be found: the living world is a messy place with little respect for general rules. What perhaps matters more is what these arguments represent. The Central Dogma itself encodes a hidden message about how life works—and it is this implicit message that needs updating. For not only is the Central Dogma not a dogma, but it is not really very central.

It doesn’t help matters that the Dogma was garbled by Watson in his influential 1965 textbook, 

. Presaging the sloppy relationship to rigor that he displayed in 

, he presented it as a simple linear sequence: DNA → RNA → protein. This is unambiguously wrong—not just because it ignores the ability of DNA to copy itself but also because it was found in 1970 that viruses that encode their genomes in RNA use an enzyme called reverse transcriptase to write their genes into the DNA of the host cell: the kind of possibility Crick had tentatively but sagely included in 1956.

Responding to the discovery of reverse transcriptase (which prompted 

to call the Central Dogma “a considerable over-simplification”), Crick clarified his views in 1970. He now presented the Dogma in its more iconic form as a triangular series of information transfers. He explained that the only way information could flow back from proteins to nucleic acids was if “the cell had evolved an entirely separate set of complicated machinery for back translation, and of this there was no trace, and no reason to believe that it might be needed.” The discovery of one of these forbidden information transfers, he said, “would shake the whole intellectual basis of molecular biology.” All the same, he was sensibly cautious about what surprises biological research might yet have in store, saying of the Dogma that “our knowledge of molecular biology . . . is still far too incomplete to allow us to assert dogmatically [!] that it is correct.”

Crick’s Central Dogma articulates the central principles of protein synthesis and of enzyme function. The reason your body contains the same enzymes as your parents’ did is that the instruction for making them was passed along in the parental genes to the sperm and egg cells that combined at the start of your being. These enzymes all do the same things because they have the same shapes—and those shapes are themselves specified by the amino-acid sequences.

However, the simple formula that informed the Central Dogma—the notion that a gene encodes a particular protein and that protein plays some well-defined role in building and maintaining the organism—is now known to be inadequate. For one thing, genes often encode more than one protein. On average, each human gene encodes around six, and our 20,000 or so protein-coding genes can give rise to around 80,000-400,000 proteins. This is because the base sequence of the RNA that encodes a protein is typically reshuffled before it is translated. Which variant of a protein is produced in a given instance depends on many things, such as the type of cell in which the gene is transcribed—the result might be different, say, in a heart cell to a kidney cell. What’s more, proteins interact with one another in complex networks, and for some proteins—especially ones that feature as key connecting hubs of the networks—it is impossible to associate the molecule with any specific trait or feature of the organism. And RNA molecules do much more than just encode proteins. For example, some of them instead regulate which genes get transcribed, helping to turn that process on or off, or influence the reshuffling that determines how RNAs are translated to proteins. The organism is not, then, a transparent and linear “readout” of the genome sequence.

Does any of this threaten the Central Dogma itself? Not obviously—for Crick wisely left space for RNA to have more roles, and the reshuffling of protein sequences doesn’t alter the fact that the information ultimately encoded in a given protein cannot make its way back into an RNA or DNA sequence. One component of the Central Dogma that 

 been challenged is the assertion that proteins can’t inform other proteins. In particular, proteins called prions can misfold into aberrant forms that stick together in tangles and cause disease—they can trigger the death of neurons, leading to neurodegenerative conditions such as bovine spongiform encephalopathy (BSE) in cattle, scrapie in sheep, or the human equivalent, Creutzfeldt-Jakob disease (CJD). Crucially, a misfolded prion can induce another to misfold too, making the disease transmissible. (Some people have contracted CJD by eating beef afflicted with BSE.) This is arguably a kind of information transfer from protein to protein. But it doesn’t happen in the way Crick specified in the Central Dogma, by a letter-by-letter transfer of sequence information. All the same, Crick presciently alluded to scrapie in his 1970 paper as a possible fly in the ointment.

There is no reason to suppose that the Dogma is inviolable in principle. It’s possible to imagine designing enzymes that can selectively edit another protein’s sequence. Though nothing like this is known to happen in nature, it could probably be done synthetically in the lab. Even so, it would merely be an exception that proves the rule.

Rather than seeking doggedly to disprove the Central Dogma, it’s probably more helpful to put it in its rightful place. The Central Dogma seems to impute a hierarchy of importance that gives primacy to the gene. Perhaps that is the subtext to Watson’s misconceived one-way information flow: it makes the gene the origin of all biological information. This is surely the wrong way to see things. A DNA sequence has no intrinsic meaning—it only becomes informative in the context of a living cell. It’s not just that the cell is needed to enable a gene to be “read”; the very notion that there is anything to be read and decoded is only meaningful when genes and their products are part of the information economy of living organisms. Further, many developmental biologists now believe that theories of how organisms evolve must take better account of how they grow.

Even then, genes don’t (contrary to common belief) control the cell. To some extent, it’s the other way round: genes are regulated in response to the overall state of the cell, which in turn depends on its surroundings: on signals received from the outside, perhaps from other cells or from the environment. And even when a gene is activated, what actually gets made from it (and what role the product performs) is also often determined by top-down control of RNA reshuffling. Sure, this top-down information flow doesn’t alter a genetic sequence, but it may control what effect a gene has on, say, metabolism or development. Some microbiologists even argue that cells in effect have some say in their DNA sequences. Whether or not that’s so, biological information flow is not purely bottom-up, but omnidirectional.

Increasingly, genes look ever less like the elements of a blueprint for life, and more like source material that a cell consults for growth and metabolism. Crick’s Central Dogma summarized, well before the data were in, the story of how cells mobilize those resources. Keep it in perspective and it remains a startling intuition. ♦

Philip Ball is a science writer. His work appears in

The Central Dogma

Picture This: The Central Dogma

It looms over every chemistry classroom and lecture theatre, two towers bookending serried ranks of compartments rather like the British Houses of Parliament—and with at least as much authority. The Periodic Table is chemistry’s icon, a codification of the entire chemical universe expressing the relationships between the elements from which all ordinary matter is constituted. There are just 92 or so of these natural elements (occasional oddities like technetium, element 43, barely exist in nature because they are so unstable), but scientists have now appended to the roster a gaggle of extra ones, from neptunium (element 93) to oganesson (118), that are wrought artificially in nuclear reactions, the most massive of them living for just an instant before decaying.

I must once have known the Periodic Table by heart. It’s too long ago for me to be sure, but I have to surmise as much because chemistry students at Oxford University weren’t given the table for their exams. In a gesture of characteristically perverse exceptionalism, we were expected to memorize it. But don’t ask me now where to locate rhenium or iridium: all those obscure transition metals in the long central block of the table are a blur. If, however, you do know where an element goes—if you can assign it to the right row and column of the table—you can deduce a lot about it. You can figure out how the electrons in its atoms are arranged into shells, and make good guesses about the types of compounds it forms, its melting and boiling points and its propensity to react with other elements. In my finals, I wrote an entire essay about niobium—niobium!—on that basis. Goodness knows what it said.

The Periodic Table was conceived as a scheme for bringing order to the elements. When there were deemed to be only four of these—the earth, air, fire, and water of the Greek philosopher Empedocles (it was just one of the elemental systems proposed in ancient times, but enjoyed the weighty advocacy of Plato and Aristotle)—things seemed simple enough. But during the Renaissance, natural philosophers were increasingly forced to accept that the metals then known—copper, iron, lead, tin, mercury, silver and gold—were not as interconvertible as the alchemists believed, but seemed to have an elemental primacy about them, too. More and more of these became recognized—zinc, bismuth, cobalt, and others—along with other new elements such as sulfur, phosphorus, carbon, and, in the late eighteenth century, gaseous elements like nitrogen, hydrogen and oxygen. When the French chemist Antoine Lavoisier (who named those latter two) drew up a list of known elements for his seminal textbook 

 in 1789, he counted 33—including light and heat, which he called caloric.

The list didn’t seem to be arbitrary though. In the early nineteenth century, several scientists noted that some elements seemed to come in families, resembling one another in the kinds of reactions they engaged in and the compounds they formed. Some claimed to see triads: the halogens chlorine, bromine and iodine for example, or the reactive metals sodium, potassium (both discovered by English chemist Humphry Davy in 1807) and lithium (identified in 1817). Was there a hidden pattern to the elements?

The Russian chemist Dmitri Mendeleev, working at Saint Petersburg University, is usually credited with discovering that pattern. A Siberian by birth, with Rasputin-like dishevelled hair and an irascible manner, he published his first Periodic Table in 1869. It is “periodic” because, if you list the elements in order of their mass, certain chemical properties seem to recur periodically along the list. The table is produced by folding that linear list so that elements with shared properties sit in vertical columns (although Mendeleev’s first table had them instead in rows, effectively turning today’s table on its side).

Mendeleev’s insight wasn’t unique; by then the existence of a periodic structure that organized the known elements was clear to others too. In particular, the German chemist Julius Lothar Meyer drew up a table almost identical to Mendeleev’s in 1868, but he didn’t get it published until later—and so missed out on the accolades, to his immense chagrin. Mendeleev, however, had the foresight to see that his table only worked if he left some slots empty: elements presumably yet to be discovered. When some of these were found soon after and had just the properties he predicted, he was vindicated.

Even now, there’s no consensus about how to draw the Periodic Table.

Still, it’s a weird kind of periodicity. At first, chemical properties seemed to recur every eight elements. But in the row that starts with potassium, there’s an interlude of ten metals—the transition metals—and so it continues thereafter, creating a periodicity of 18. And after lanthanum (element 57), chemists discovered a whole series of 14 metallic elements with almost identical properties that have to be squeezed in too—frankly, these elements, called the lanthanides after the first of their ilk, all seem a bit redundant. There’s another block like this after radioactive actinium (element 89), called the actinides. In most Periodic Tables, the lanthanide and actinide blocks are left floating freely underneath so the table doesn’t get stretched beyond the confines of the page. (Some insist that this long-form table is the only proper one.) Why this odd structure?

The answer became clear with the invention of quantum mechanics in the early twentieth century. The chemical properties of elements are mostly determined by how the electrons in their atoms are arranged. New Zealander Ernest Rutherford showed that atoms comprise a central, very dense nucleus with a positive electrical charge, surrounded by enough negatively charged electrons to perfectly balance that charge. Rutherford imagined the electrons orbiting the nucleus like moons, but in the quantum-mechanical description they occupy nebulous, smeared-out clouds called orbitals. Using quantum mechanics to describe the disposition of electrons shows that they are arrayed in shells. The first of these can contain just two electrons—this is the only shell possessed by hydrogen and helium, the two lone elements at the tops of the towers—while the next has eight, and then 18. The shape of the periodic table thus encodes the character of the quantum atom.

All clear? Not quite. Even now, there’s no consensus about how to draw the Periodic Table. Hydrogen, the first and lightest element, has always been awkward: it tends to get plonked on top of the first column (the alkali metals), but it doesn’t really fit there—it’s not a metal, after all. Some prefer to see it float freely above the rest, a hydrogen balloon over the edifice of elements. And representing the rather awkward nuances of the quantum shell structure in a two-dimensional diagram involves compromises, which have prompted the invention of all manner of ingenious alternatives to the traditional block format: spiral and circular tables, loops and stadium shapes, tiered ziggurats, three-dimensional models, dizzyingly imaginative cartographies of elements. None has caught on.

Some of the fiercest arguments involve the lanthanides and actinides, which begin in the third column of the Table. Which elements truly belong in those two slots? Older tables put lanthanum (symbol La) and actinium (Ac) there, with the rest of the series relegated to those disconnected basement blocks. Others instead assign those two positions to the last of the lanthanides and actinides: lutetium (Lu) and lawrencium (Lr). Some leave the position undefined, labeled only ‘La–Lu’ and ‘Ac–Lr.’ The problem is that the arguments for one choice are chemical—which elements are chemically most similar to scandium and yttrium higher up in column three?—while the other option is preferred quantum-mechanically, based on how the electrons are configured. In some ways this is a dispute about authority. Which has the final say on the periodic table: chemistry or physics?

Put that way, you can see the potential for acrimony. I was for a time a member of a group tasked by the International Union of Pure and Applied Chemistry—the authority on chemical nomenclature and systematization—to make recommendations that might resolve the matter. But the group couldn’t agree, and so the argument continues. Or to give it a more positive spin: you’re free to choose the Periodic Table you like best. ♦

Philip Ball is a science writer. His work appears in 

The Periodic Table

Picture This: The Periodic Table

What did it mean? How could it do that? It was June 1st, 1984, when I first made this picture of what I called Rule 30. And the more I looked at the picture, the more surprising it seemed. How could a rule so simple (shown above) make something so complex? My intuition from science—and engineering—was that, to make something complex, one had to do complex things according to some kind of complex procedure. But yet, here was a very simple rule making what seemed to be immense complexity.

The more I studied it, the more I realized that this was something fundamentally different from what existing science normally talked about. And instead, it was a first clue towards a new kind of science.

That was all nearly 40 years ago. In the intervening decades, what began as one picture has indeed grown into a whole new kind of science, with new kinds of methods, new kinds of intuition, and a whole new kind of way of thinking about the world.

For me, seeing the picture of Rule 30 was my personal "Galileo moment." Nearly four centuries earlier—on January 2, 1610—Galileo had turned his telescope to the sky and seen "three little stars" lined up near Jupiter—which he soon realized were moons that orbit Jupiter much like planets orbit the Sun. It was a discovery nobody expected, but it was a clue towards a new, more abstract way to think about science. And from it grew what over the course of three centuries was to become the whole edifice of modern mathematical science.

My discovery of Rule 30 was the result of turning not an optical telescope at the astronomical universe, but instead a "computational telescope" at the computational universe—of observing not a pattern of light coming from a distant body, but instead a pattern of bits made by a simple program out in the computational universe of possible programs.

Galileo's discovery in 1610 was the direct result of the availability of new technology: an improved telescope that made it possible to resolve things like the moons of Jupiter. My discovery of Rule 30 was also the direct result of new technology: in this case, the availability of a high-resolution printer that let me easily see many steps in the behavior of Rule 30. As it turns out, I had actually produced small versions of the Rule 30 picture, and even published them in scientific papers a couple of years before June 1st, 1984. But what Rule 30 does just didn't fit into my previous way of thinking, so I never internalized what I was seeing. It was only through the “you-can't-ignore-this” force of the picture above—with its obvious complexity—that I came to internalize the "Rule 30 phenomenon."

It didn't take long before I began to see some of the significance of what was going on. And in particular, before I began to understand the phenomenon of computational irreducibility. But before talking about this, let me describe just what the Rule 30 picture is showing.

Let’s start with the top row of cells, all white except for a single black one in the middle. Then we go line by line down the page, at each step applying Rule 30 to each cell. The rule looks at a cell and its left and right neighbors, then determines the new color of the cell. It's like a very simple program that takes as input one configuration of colors, and outputs another. I called it "Rule 30" because if one treats the sequence of output cell values 00011110 in the rule (with black being 1 and white being 0) as a number in base 2, then when converted to ordinary base 10, one gets 30.

Computational irreducibility means that we can’t just solve for our future and know the outcome will be; we actually have to live out our years, in effect playing out the irreducible computation that is our lives.

So there's a simple rule for generating the picture. You might think that, given such a simple rule, you should immediately be able to predict everything about what will happen. Indeed, most of the great achievements of the mathematical approach to science launched by Galileo's discovery have at their core the idea that we can derive a mathematical formula for how something in nature behaves, and then use this formula to predict what the thing will do. An idealized planet might make a million revolutions around an idealized star, but by using the mathematical formula for the two-body problem, we can immediately predict the outcome of those million revolutions.

But what about Rule 30? Can we predict the outcome of a million steps of its evolution? Clearly we can just run it—say on a computer—for a million steps, and see what happens. But the essence of computational irreducibility is that we can't expect to do much better than Rule 30. Unlike for our idealized planet, there is no "mathematical formula" for what will happen. There is no "reduced computation" that will let us predict what the system will do. Instead, we must think about the system as doing a certain irreducible amount of computation.

It's an important idea—it suggests that in a sense science defines its own limits. Yes, we can know what the underlying rule is. But we can't know its consequences except by doing an irreducible amount of computational work. And while this might seem bad, in terms of limiting our foresight about what will happen, it's also good at making the passage of time seem meaningful. Computational irreducibility means that we can't just solve for our future and know the outcome will be; we actually have to live out our years, in effect playing out the irreducible computation that is our lives.

Seeing the Rule 30 picture, I could have just said, "Unfortunately, the science I know doesn't really say anything about this" and ignored it. But I feel very fortunate to have realized that the challenge of Rule 30 was not an end, but a beginning: a beginning of what would define a whole new kind of science.

In ancient Greek times, science mostly just described the explicit structure of things. The great shift initiated by Galileo's discovery was the idea that one could also base science on mathematics, and describe things in nature by mathematical formulas. What began with Rule 30 is a newer and different paradigm for science, based not on mathematics and mathematical equations, but rather on computation and computational rules.

Some of what we see in nature can be readily described at just a structural level. For some, we can derive mathematical formulas. But a lot of what goes on in nature just seems too complex for that. Now that we have the computational paradigm, we can begin to see that the secret that nature has long harbored which allows it, seemingly so effortlessly, to create such complexity is just the same kind of thing we see in the picture of Rule 30—a manifestation of the fundamental phenomenon of computational irreducibility.

In a sense, the most striking thing about the Rule 30 picture is how much one gets out from the little one puts in. And thinking about this makes me wonder about our whole universe. Could it work like Rule 30? With only a very little being put in, but everything we experience eventually coming out? Starting 30 years ago, I began to formulate how such ideas might relate to fundamental physics and the construction of our universe.

About three years ago, all this began to come together. And, yes, it really does seem that there is a lower-level "machine code" from which the core results of physics emerge. The details aren't like Rule 30. But the concept that something like this might be possible arose entirely from that one great surprise of the Rule 30 picture.

Nowadays I talk about the ruliad—the entangled limit of all possible computations—and how physics, as well as mathematics, are the result of our human observation of this underlying formal structure. The science of ruliology is concerned with studying what particular rules do; in effect, exploring individual places in the ruliad, the space of all rules. Rule 30 was the first glimpse of the amazing structure that we are now beginning to understand as the ruliad.

In the end it all came from that one picture, my all-time favorite scientific discovery. A timeless image of the consequences of a particular, simple computational rule. A new way of thinking about science and much beyond: Rule 30. ♦

Rule 30

Picture This: Rule 30

Of Euclid’s five postulates, which form the basis of Euclidean geometry, the fifth is the most controversial. Called the Parallel Postulate, it’s a pretty basic idea. Take a flat piece of paper, draw a line segment and then draw two lines that are perpendicular to it. The Parallel Postulate simply says that the lines will remain forever parallel, staying precisely the same distance apart even if we extend them to infinity on either side of the line segment.

For a long time mathematicians strived to prove the Parallel Postulate using the first four (more) basic postulates, and so elevate it from a postulated axiom to a proven theorem. It turns out that you can’t, and for good reason—the postulate isn’t true for all geometries. Euclidean geometry, represented by the flat piece of paper, and on which lines that start off parallel never intersect, is just one kind of geometry, but there are others. You can imagine replacing the Parallel Postulate with some alternative: maybe lines that are parallel initially draw closer together and touch eventually, or maybe they always move apart. The first option is the case with 

, which describes shapes drawn on the surface of a sphere (think of lines of longitude that start of parallel at the equator but converge at the poles), while the second defines 

, which is more like the surface of a saddle or a Pringle’s potato chip.

These non-Euclidean alternatives weren’t noticed for millennia. Euclid himself flourished around 300 BCE, and the first alternatives to the Parallel Postulate were proposed in the early 1800s by Russian mathematician Nikolai Lobachevsky and Hungarian mathematician János Bolyai. Interestingly, both studied the hyperbolic alternative to flat Euclidean geometry.

This raises two questions: First, why did it take so long? And second, why did the mathematicians study hyperbolic geometry before spherical? Everyone knew about spheres. How hard could it have been to imagine drawing some lines on them, and exploring the geometric consequences of doing so?

The questions share a common answer. Yes, spheres were commonplace, but that’s exactly the point. Nobody thought that you needed an entirely new kind of geometry to deal with them. The two-dimensional surface of a sphere might be curved, but it’s easy enough to construct one in ordinary three-dimensional space: Pick some origin point, and consider the set of all points some fixed distance from that origin; that set is a sphere. You could draw lines on a sphere’s 2D surface, but mathematicians thought “real” lines live in 3D space, and so ignored the curved lines on the sphere.

You cannot construct hyperbolic geometry, however, in ordinary 3D space. One can glibly say things like “saddle shaped,” but every saddle has a special point in the middle where the curvature is highest, and the curvature fades as you move away from the middle. So, in regular 3D space, unlike a sphere, all points on a saddle are not created equal. The new hyperbolic geometry of Lobachevsky and Bolyai is special: there, too, every point is treated the same.

The reason mathematicians took so long to invent hyperbolic geometry is that the perfect form of such a geometry exists only in our minds. You can’t build it, the way you can build a sphere or a saddle, or even imagine building it. It’s an example of an imaginary space that human beings can postulate and study mathematically, without ever being able to reach out and touch it.

You can, nevertheless, try to visualize hyperbolic space as honestly as possible, keeping in mind that some distortions are inevitable when going from hyperbolic to Euclidean space. One of the most interesting visualizations is the hyperbolic disk (shown above), also known as the Poincaré disk. It was first used by Italian mathematician Eugenio Beltrami (1835-1900) and only later by French mathematician Henri Poincaré (1854-1912), but Poincaré is more famous, and famous people often get their names attached to things.

The image highlighted above is a hyperbolic disk tessellated by triangles. It represents an infinitely big two-dimensional hyperbolic geometry, but clever mathematical techniques have been used to compress the infinite space down to a finite disk-shaped region. That manipulation introduces a difference between what we see in 2D Euclidean space and the “true” hyperbolic geometry being represented. In particular, distances are compressed as we approach the boundary of the disk; in reality, both the radius of the disk and the circumference of its perimeter should be thought of as infinitely long.

In the 2D image, there are straight lines running at various angles through the center of the disk. But in the true geometry, all of the lines depicted are “straight,” although they appear to us as arcs stretching from one point on the edge to another one. And all of the triangles thus described have precisely identical geometries in hyperbolic space, despite appearances to the contrary. It’s just that our compression technique has squeezed and distorted them, especially near the boundary, analogous to how flat maps of Earth’s curved surface distort straight lines and measures of lengths.

This representation of the hyperbolic disk famously made an impression on artist M.C. Escher. Inspired by a figure in a paper by British mathematician H. S. M. Coxeter, Escher used it as the basis for a series of woodcuts titled Circle Limit, employing fish and angels and demons to divide up the disk.

Hyperbolic space played a crucial role in the development of non-Euclidean geometry. It was the first explicit example of a non-Euclidean space that mathematicians could only conceive of in the abstract, freeing their minds to imagine other kinds of wild geometries. Soon thereafter, German mathematician Bernhard Riemann developed a more general theory of non-Euclidean spaces, ones where arbitrary kinds of curvature could exist in spaces of arbitrary dimensionality. It is this Riemannian geometry that serves as the basis for general relativity, Einstein’s formulation of gravitation as a manifestation of the curvature of spacetime.

The hyperbolic disk continues to astound us. As a simple and concrete example of a non-Euclidean space, it pops up in different contexts within modern physics. Today, a great deal of effort is being expended studying something called the “AdS/CFT correspondence,” a relationship between a physical theory without gravity in 