Posts Tagged ‘Mathematics’
“A physicist is just an atom’s way of looking at itself”*…
Encouraged by successes in understanding black holes, theoretical physicists are applying what they’ve learned to whole universes. As Matt von Hippel explains, what they’re finding has them questioning fundamental assumptions about how physics ought to be done…
Tinkering at their desks with the mathematics of quantum space and time, physicists have discovered a puzzling conundrum. The arcane rules of quantum theory and gravity let them imagine many different kinds of universes in precise detail, enabling powerful thought experiments that in recent years have addressed long-standing mysteries swirling around black holes.
But when a group of researchers examined a universe intriguingly like our own in 2019, they found a paradox: The theoretical universe seemed to admit only a single possible state. It appeared so simple that its contents could be described without conveying even a single bit of data, not even a choice of a zero or a one. This result clashed with the fact that this type of universe should be capable of hosting black holes, stars, planets — and people. Yet all those rich details were nowhere to be seen.
“We look around, and certainly the world seems more complex than that,” said Rob Myers, a theoretical physicist at the Perimeter Institute for Theoretical Physics in Waterloo, Canada, who has not been directly involved in this research.
Physicists have good reason to trust the calculation, which builds on fundamental physical ideas. The math implies a universe with only one state; our universe is clearly not like that. Now a team of theorists has floated a possible answer. The paradoxical result occurred when physicists sought an objective description of the state of an entire universe. But a description like that might not be possible, even in principle. It implicitly assumes a universe that exists without an observer to observe it. And perhaps without observers, the complexity of the universe loses its meaning…
[von Hippel unpacks the scientific thinking that led to this conundrum. He concludes with an account of recent work by Ying Zhao that hints at a resolution…]
… Quantum mechanics requires a distinction between an observer — such as the scientist carrying out an experiment — and the system they observe. The system tends to be something small and quantum, like an atom. The observer is big and far away, and thus well described by classical physics… Perhaps an observer could do the same to these closed, impossibly simple-seeming universes?
In 2024, Zhao moved to the Massachusetts Institute of Technology, where she began to work on the problem of how to put an observer into a closed universe. She and two colleagues —Daniel Harlow and Mykhaylo Usatyuk — thought of the observer as introducing a new kind of boundary: not the edge of the universe, but the boundary of the observer themself. When you consider a classical observer inside a closed universe, all the complexity of the world returns, Zhao and her collaborators showed.
The MIT team’s paper came out at the beginning of 2025, around the same time that another group came forward with a similar idea. Others chimed in to point out connections to earlier work.
At this stage, everyone involved emphasizes that they don’t know the full solution. The paradox itself may be a misunderstanding, one that evaporates with a new argument. But so far, adding an observer to the closed universe and trying to account for their presence may be the safest path.
“Am I really confident to say that it’s right, it’s the thing that solves the problem? I cannot say that. We try our best,” Zhao said…
“Cosmic Paradox Reveals the Awful Consequence of an Observer-Free Universe,” from @quantamagazine.org.
Apposite: “Biology Might Not Be Quantum, but Its Math Is Quantumlike“
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As we keep an eye out, we might send cosmological birthday greetings to Enrico Fermi; he was born on this date in 1901. A physicist and Nobel Laureate (1938) for his discovery of nuclear reactions brought about by slow neutrons, he is best remembered for (literally) presiding over the birth of the Atomic Age. Fermi was perhaps especially remarkable as the last “double-threat” in his field: a genius at creating both important theories and elegant experiments. As observed before (and illustrated above :), the division of labor between theorists and experimentalists has since been pretty complete.
The novelist and historian of science C. P. Snow wrote that “if Fermi had been born a few years earlier, one could well imagine him discovering Rutherford’s atomic nucleus, and then developing Bohr’s theory of the hydrogen atom. If this sounds like hyperbole, anything about Fermi is likely to sound like hyperbole.”
“Something is happening here / But you don’t know what it is / Do you, Mister Jones?”*…
Further to the post of Septmeber 15: on trying to understand what might lie on the other side of a transformation driven by AI (and related technologies). Microsoft’s Deputy CTO, Sam Schillace, suggests that organizations are about to change completely…
… Here’s a nerdy TL;DR if you want it: I think AI will enable a switch from what is essentially circuit switching today, in all kinds of organizations, to something that will look and feel a lot more like packet switching. This will happen partially because it can now (AI is to this problem what fast processors were to distributed networking protocols), and because networks are almost always more powerful, flexible, and scalable. This idea is going to feel super uncomfortable and be rejected by business leaders, employers and academics in much the same way that network engineers rejected the idea of packet-switched networks…
… In 1965, making a long-distance phone call in America was a feat of engineering coordination. A call from New York to Los Angeles required a dedicated copper path, physically reserved, for the duration of the conversation, and AT&T had to employ armies of switching engineers to manage this. The Bell System was one of the great technical achievements of the twentieth century, and the people who built it were not fools. They worked ferociously for what they were optimizing for: guaranteed, high-quality, reliable connections between two points.
It seemed fine – expensive, but that’s what everyone was used to. If you are my age, you remember when long-distance was even a thing, and those calls cost more and were precious. I still get a tiny bit nervous talking to my parents 3000 miles away in Michigan, that was so deeply ingrained. Part of me still can’t believe that phone calls all cost the same.
Then a small group of researchers, largely funded by ARPA, mostly working in universities, proposed something that sounded to the Bell engineers like a joke. Instead of reserving a dedicated path, you would break messages into small pieces, label each piece with a destination, and release them into the network to find their own way. Different pieces of the same message might travel completely different routes. They might arrive out of order. There was no guarantee that any particular piece would arrive at all. A phone call, under this scheme, would be like mailing a letter by tearing it into a hundred pieces and dropping each one in a different mailbox, hoping they’d all show up. (Kind of like training a giant AI model on lots of small fragments of language (tokens) and hoping that somehow intelligence would emerge from putting them back together randomly – which largely worked).
The Bell engineers had a name for this general idea: chaos. They had a specific critique too, which was that you couldn’t make any guarantees about quality of service. Voice needs low, consistent latency: you can’t have some syllables arriving seconds after others. And of course, data needs reliability: a corrupted file is useless. The engineers were right about all of this. Packet switching, in its early forms, was genuinely worse at the things circuit switching was designed to do.
It also, within a few decades, made the Bell System irrelevant. The internet didn’t improve on the telephone network; it made the telephone network a minor application running on top of something much larger. Today your phone calls travel as packets (which is why they all cost the same – there’s no such thing as a “long circuit” now), just like everything else. The Bell engineers optimized for the wrong thing. I love that this also has a flavor of “worse is better”. The Bell engineers were optimizing for “correct, complete, but complex and rigid” and the internet is more “simple and recoverable beats correct and complex”.
What actually happened was a philosophical shift about where intelligence should live. Circuit switching is a centralized control philosophy. Someone, somewhere, reserves the path before any message travels it. The network takes responsibility for delivery. The endpoints are dumb; they just send and receive. All the complexity, all the guarantee-making, lives in the infrastructure itself. This works, and it scales up to a point, and then it hits a wall, because the central system has to know everything about every connection, and that knowledge becomes impossible to manage as the network grows. (Sound familiar? This is how we build companies these days. We’ll come back to that idea)
Packet switching inverts this entirely. The network is dumb and the endpoints are smart. A router doesn’t understand the content of the packets it handles – it reads a destination address and forwards the packet one hop closer. The intelligence about what the message means, whether it arrived correctly, what to do if it didn’t – all of that lives at the edges, in the devices doing the communicating. The protocol defines what the endpoints must do. The network just moves bits.
You have seen this argument before, in a different domain. Central planning versus markets is the same disagreement. The Soviet planning apparatus tried to do for an economy what circuit switching does for a telephone network: reserve resources, guarantee outcomes, manage the whole system from the center. It failed not because the planners were stupid but because the information required to run a complex system centrally exceeds what any center can hold (it also failed because many of them were corrupt, and centralized systems are vulnerable to this, since there’s no way to build in mechanisms to correct, outcompete or otherwise reverse that corruption).
Prices, in a market, are packets. They carry information about supply and demand from the edges without anyone at the center having to understand or coordinate it. Markets are chaotic, wasteful, frequently unfair, and they consistently outperform central planning at scale for the same reason the internet outperformed the Bell System. The network always wins, eventually, because the network is where the information actually lives.
Let’s think about companies now. Companies love to do re-orgs. Or at least, they have to do them. They’re a lot of work, just like it was a lot of work to organize and change the old Bell network, and for the same reasons. Companies have to deal with the pain of organizational overhead, because to a large extent, they try to do central planning – circuit switching.
A manager, in an org chart, is a leased line. From an organizational perspective, they’re like a circuit between strategy and execution, reserved whether or not signal is flowing. The org chart is the circuit map. It tells you the paths that exist, which means it also tells you the paths that don’t. If you need to get information or work from point A to point B and there’s no line between them on the chart, you have a problem – you need to go up to a common node and back down, or you need to get someone to lay a new line. When a company needs to change how it behaves, it rewires the circuit map, which we call a reorg.
Why do we do this? This is not because companies are stupid or because the people running them lack imagination (hopefully). It’s because circuit-switched organizations are the only kind that could be managed without the infrastructure that packet-switched organizations require (which isn’t unlike what happened with the internet – we needed cheap processors to be able to build routers that could handle packets fast enough. Before we had that, we had no choice but to set up dedicated circuits).
If you can’t do the things an organization needs – observing and judging work, finding capabilities, holding people accountable – without the overhead of a fixed org chart, well, you just have to have it.
The cost of this solution is enormous and mostly invisible because it’s so normalized…
[Schillace expands on this idea…]
… When packet switching was proposed, the experts who objected most loudly were the most qualified people in the room. They were “right” in an engineering sense, but wrong in the systems sense. And they did what most people do, asking a “why not” question (against the old design and constraints) instead of a “what if” one: what if the new processors could enable a new network design?
Most people looking at AI are making the same mistake, and I predict we will see a lot of this from the orthodox “business advisor” community. I hear this from engineers and leaders already, and it sounds just like the Bell folks: You can’t guarantee accountability in a decentralized system. You can’t audit a network that routes around hierarchy. Regulators will never accept this. Sophisticated investors will raise concerns about governance.
All of these objections will be correct in the short term and beside the point in the long term. They are right from some kind of local engineering perspective but wrong from an emergent systems perspective. The question is not whether packet-switched organizations perform better on the metrics that circuit-switched organizations were built for. They won’t, at least at first, just like packet-switched networks didn’t at first. The question is whether the new architecture unlocks things the old one couldn’t do at all. And I think it will – I think we will get levels of scale, creativity and responsiveness by using these new tools to push intelligence to the edge of all kinds of organizations – not just businesses but political, social, educational and creative ones…
…There is a useful heuristic here, which is that disruptive ideas tend to produce a room that’s roughly split in half. Usually something like half the people think the idea is obviously correct (the folks asking “what if”), and half think it’s obviously wrong (the “why not” crowd). I’ve written about this kind of bifurcation before – this is the sign of something that is genuinely disruptive, and the split is between people who are rejecting it because it feels threatening, and people who have internalized the new world view and are beginning to extrapolate it…
[Schillace considers earlier, unsuccessful attempts at “packet-switched organizations,” concluding that “AI is [the] missing piece – cheap enough intelligence to put “everywhere” and enable a bunch of these ideas for real. It’s the equivalent of those cheap processors [that enabled packet switching] finally showing up, except it’s a much more complete stack arriving all at once” and that the next few years will consist in discovering the new protocols that will allow “packet-switched organizations” to succeed…]
… The Bell engineers who resisted packet switching were not, in the end, wrong about what they said. Voice quality did suffer in early VoIP, packets did get dropped a lot, and for a while, quality was worse. But they were wrong about what would matter in ten years, and catastrophically wrong about what would be possible in thirty.
Email was not a better telegram, the web was not a better encyclopedia, and streaming was not a better cable network. Each of these looks, in retrospect, like an obvious evolution, but none of them were conceivable from inside the circuit-switched world (I remember the freak out about how video was going to break even the highly scalable new internet. It wasn’t even thinkable in the circuit switched world). They required the new infrastructure before anyone could imagine what to build on it.
The organizational equivalent is similarly hard to see in advance. It is likely that packet-switched, AI-native organizations will not just do existing organizational things more efficiently, but that they will do things that are not currently possible. There might be organizations that form around specific problems and dissolve when the problem is solved, rather than persisting as structures in search of purpose, or expertise that routes to where it is needed rather than being allocated by headcount. We might have large groups with a degree of strategic flexibility and sophistication that have never existed together – we might get political or creative movements in radically different shapes because of that. We are definitely going to see an evolution in the creation, management and use of institutional memory. Training will probably look very different, as will job descriptions.
And almost certainly: we will get things we cannot name yet. Nobody in 1975 could have described Google, or Facebook, or VOIP, or even Google Docs.
The phone company’s engineers built something extraordinary. The Bell System was a genuine engineering achievement, and the people who designed it were some of the best technical minds of their era. They lost anyway. Networks are more robust, once the infrastructure to support them exists, and the right protocols to create and maintain them are adopted.
My bet is that the organizations of the future, the ones that compete and thrive, will make use of AI to build networks the same way the internet made use of those cheap processors and new protocols to build huge scale and new capabilities.
This will feel controversial, and might even be worse at first, but networks always win. The age of centrally planned organizations is coming to an end…
Eminently worth reading in full. Chaos that scales: “The Network Always Beats the Castle,” via @timoreilly.bsky.social.
See also: Benedict Evans‘ “AI, tools and transformation“:
… With every new technology, we start by using it for the work we already have, and we just do that more and faster. But then, over time, you make entirely new things. We will use AI to automate broad classes of stuff inside existing workflows and existing companies (although, as I’ve outlined above, that will be enormously more trouble and work than just giving everybody a model). But with every previous platform shift, the stuff that actually mattered was the stuff that wasn’t even possible before and that no-one even imagined…
And by way of keeping all of this in perspective, see Henry Farrell‘s “Machine god metaphors eat your brain” and Max Read‘s “Between 5 and 14 thoughts about AI and the discourse cycle.”
(Image above: source)
* Bob Dylan, “Ballad of a Thin Man” (from Highway 61 Revisited, 1965)
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As we try to peer over the horizon, we might send carefully-calculated birthday greetings to a man whose work has helped propel the advances we’re pondering: James Hardy Wilkinson; he was born on this date in 1919. A mathematician, he is best known for his pioneering work in numerical analysis and numerical linear algebra, especially in the context of high-speed digital computing. In 1970 he won the ACM Turing Award.
“It is entirely possible that you get unpredictable behavior out of predictable rules”*…
For centuries, the hypnotic winter flights of starlings were thought to be telepathy. Today, science reveals a much darker reality. Behind one of nature’s most breathtaking visual displays lies a mathematical masterpiece built entirely on fear. From The Great Planet…
During the autumn and winter months, if you look to the skies over Europe and North America about an hour before sunset, you might witness one of nature’s most breathtaking visual spectacles. These massive black clouds of tens of thousands of European starlings (Sturnus vulgaris)—constantly shifting into spheres, planes, and waves—are known in the scientific community as a “murmuration.” Named after the deep, resonant sound produced by thousands of wings beating in unison, the phenomenon looks from afar like a colossal calligraphy brushstroke across the sky or a ball of fire flickering in the wind. Yet, beyond this aesthetic illusion lies a ruthless and flawless algorithm for survival.
For centuries, researchers were baffled by how tens of thousands of birds could fly at speeds of around 45 kilometers per hour without a single collision. This coordination was so inconceivable that in 1931, prominent ornithologist Edmund Selous described it as “a madness in the sky.” To Selous, there was only one logical explanation for this complex behavior: telepathy. “They must think collectively, all at the same time,” he wrote.
Like many others, Selous assumed that complex behavior must stem from equally complex origins. However, groundbreaking observations made in 2005 by the married physicists Andrea Cavagna and Irene Giardina from the rooftop of the Palazzo Massimo in Rome revealed a truth far more striking than telepathy. Over three years of chilly evenings, the team used pairs of cameras to reconstruct the 3D positions of more than 4,000 birds in a single flock, proving that there is no overarching plan and no leader. Each starling in the flock interacts exclusively with its seven closest neighbors—a number researchers suggest might be the maximum capacity their brains can handle. The starlings do not keep track of these constantly changing alliances; they simply align their flight path with whichever seven birds are nearest, remaining close, but never too close.
The birds do not need to engage in complex communication to navigate their massive trajectories. When Charlotte Hemelrijk of the University of Groningen in the Netherlands plugged just three simple rules—avoidance, alignment, and attraction—along with basic aerodynamics into a computer model, the resulting virtual murmuration perfectly matched the real-world data gathered in Rome.
Physicists explain this vast web of interaction through the concept of “scale-free correlation.” When a starling turns, this new alignment influences its neighbors, who in turn influence theirs. Yet, unlike a children’s game of telephone, this transfer of information does not degrade with errors; instead, the errors are entirely washed away in the movement. No matter how large the flock becomes, the information arrives at the furthest edge completely uncorrupted. Even though each bird only pays attention to seven neighbors, its senses effectively extend across the entire murmuration, allowing it to instantly react to a movement hundreds of birds away. The reason the flock looks like a single entity from the outside is that, physically, it truly behaves like one.
Why do starlings expend such an immense amount of energy to constantly shift shape in the sky? For a long time, the scientific consensus leaned toward the “warmer together” hypothesis, assuming the displays served to advertise a roosting site and gather more birds for thermal benefits on cold nights. However, an analysis of over 3,000 murmurations across 23 countries, gathered through citizen science, revealed only a weak negative correlation between temperature and flock duration. The true determining factor was the “safer together” principle: defense against predators.
In approximately 30% of the recorded murmurations, birds of prey such as hawks, falcons, or harriers were present. As predators approached or actively engaged the flock, both the duration and the size of the murmuration increased…
… Millions of years of evolution have granted starlings this colossal collective intelligence, allowing them to outmaneuver even nature’s most apex aerial predators. Yet, the primary threat they face today comes not from the sky, but from human activity on the ground. While European starlings are omnivorous, they rely heavily on insect larvae, such as leatherjackets, that live in the soil beneath grasslands; as these grassy areas vanish, so does their food supply. Their habitats are rapidly shrinking due to urbanization and changing farming practices, such as moving livestock feed indoors…
… For a species to merely exist within an ecosystem, narrowly avoiding physical extinction, does not mean it has truly been saved. Earth’s most beautiful phenomena only emerge when living things can act together in great, thriving numbers. You can evolve a flawless biological algorithm based on scale-free correlation to outwit nature’s most ruthless predators, but when your habitat is suddenly paved over with concrete, evolution affords you no time to design a new defense mechanism. The fact that we have finally decoded the mathematical secret of the starling murmuration just as we are on the verge of losing it forever remains one of the most haunting questions suspended in our skies—a stark reflection of modern human hubris and our destructive relationship with the planet…
A flawless algorithm is fading from our skies: “The Mathematics of Fear: Why Starlings Murmurate.”
See also: “The brain has corridors surpassing / Material place.”
* Frank Heppner (an ornihologist who helped create the first computer simulations of bird flocks in flight)
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As we fly, we might send beautifully-plumed birthday greetings to Elliott Coues; he was born on this date in 1842. An army surgeon, historian, writer, and geographer, he led surveys of the Arizona Territory, first in his army posting there and later as secretary of the United States Geological and Geographical Survey of the Territories.
But we remember him here for his lifelong pursuit of his boyhood passion for birds. As a teen, he met many naturalists at the Smithsonian Institution and published his first ornithological paper in 1859, at age 19. As his army assignments took him to various locations throughout the West, he continued studying the bird life in each new area, and found new species. His Key to North American Birds (1872) was the first work of its kind to present a taxonomic classification of birds according to an artificial key and promoted the systematic study of wild fowl in North America. He founded the American Ornithological Union in 1883, and was editor of its publication, The Auk.
“Every river seems to come with a purpose”*…
The Yukon Delta in Alaska formed where the Yukon River flows into the Bering Sea
A simple scaling law brings order to the chaos of flowing water, rock, and sediment. As Natalie Wolchover reports, new findings have extended the law even further…
A river has my heart. It’s not the austere, black Thames winding through London, where I was born, but a lazy green one 5,000 miles away, where I spent my adolescence: the Blanco River in Texas. My maternal ancestors have dipped into its waters for generations, as I have on countless summer days.
The Blanco is a tributary of the San Marcos, which flows into the Guadalupe, and on into the Gulf of Mexico. You can probably picture how this looks on a map because all river networks look similar, creeping through the landscape, merging into ever wider and longer channels, downhill to the sea. The pattern resembles twigs on branches that connect to trunks of trees (and the branching of their root systems, too), and it likewise resembles the veins of plant leaves, our own systems of blood vessels, and train and highway networks that feed into cities.
There’s something appealing about this ubiquitous pattern, so appealing to me personally that I have it tattooed on my forearm: the silhouette of a tree, with leafless branches reaching upward and roots burrowing downward, almost in mirror image. “The shapes of rivers and leaf vasculature and so on — branching networks — you can just about grasp the pattern, but it’s still chaotic, so there’s something fascinating with that,” said Chris Paola, a river scientist at the University of Minnesota.
Systems that branch in this way are “transport networks”: They transport some fluid substance (water, blood, traffic) from every place to a single place (the sea, a heart, a city center). Of the various examples, rivers are especially revealing, I think, since they arise from neither biological evolution nor urban planning, but rather chaotic Earth processes. Yet they obey simple, universal laws…
…
… In 1957, a U.S. Geological Survey scientist named John Hack discovered the most important law of river networks. In rivers and streams in Virginia and Maryland, Hack measured the length of each stream and the area of the land that slopes toward that stream and therefore drains into it, called its basin or drainage area. What he discovered is now known as Hack’s law: Any stream, from the littlest brook to the mightiest river, has a length that’s proportional to its drainage area raised to the power of 0.6. (In symbolic form: L ~ A0.6.) There’s a bit of variance around that 0.6 value — Earth is, after all, a complicated place — but “the general regularity of the relation is nevertheless remarkable,” Hack wrote. “Stream lengths tend to increase proportionally to the 0.6 power of the drainage area, regardless of the geological or structural characteristics of the area.”
As more and better data has accrued, especially from satellite imagery, Hack’s law has held worldwide. Why this is the case is the essential mystery geomorphologists have grappled with ever since. “Hack’s law is still the big question,” said Hansjörg Seybold, a geomorphologist at the Institute for Interdisciplinary Mountain Research at the Austrian Academy of Sciences.
It’s not so surprising that the bigger the land area of the basin, the longer the stream that drains it. But in a purely mathematical sense, one might expect that stream length would follow a slightly different power law. Imagine a square patch of land. You might guess that regardless of slope or size, in idealized form, the land would drain into a stream that’s the length of one of its sides — a vertical line down the middle, for example. That length is the square root of the area — or A to the power of 0.5.
Under that circumstance, big river basins would have the same proportions as the small river basins that feed the tributaries within them. Their structure would be the same, regardless of size. But that’s not what Hack’s law reveals.
Instead, as a drainage areas get larger, the length of their streams increases faster. “A nice way to phrase it would be that small basins are short and squat, and large basins are long and thin,” said Daniel Rothman, a geophysicist at the Massachusetts Institute of Technology. We unknowingly pick up on this pattern when we look at a network of tributaries on a map; a perfectly self-similar, fractal river network wouldn’t look quite right. Basins and streams become elongated at larger scales, so that river networks have an inherent directionality that stretches toward the sea. One result of that elongation is that neighboring river networks must lie closer together than they would with a 0.5 power law…
…
… Rivers do shift their layouts all the time. In the 1990s, in parallel with the work on optimal channel networks, geomorphologists developed powerful landscape evolution models to capture this constant adjustment and show the mechanism by which Hack’s law etches itself on the landscape. These computer simulations start with water flowing downhill, eroding rock as it goes. Tiny, random irregularities in the topography cause some channels to capture more runoff than others. Those channels in turn erode faster and deepen, which causes them to attract still more water. One streambed might grow toward its neighbor, and thereby intercept some of its runoff. The victorious stream grows longer and carries more water, while the losing stream shrinks or disappears. These sorts of local adjustments like these route water along ever more efficient paths. As the entire drainage network gradually reorganizes over thousands of years or more, it attains and then continues to tweak a configuration that transports water downhill with minimal energy dissipation.
Gravity and friction are the driving forces of this process. Gravity supplies potential energy to flowing water. Friction, the cause of erosion, dissipates that energy. A channel configuration that wastes energy by forcing water along inefficient routes tends to erode rapidly and change. A configuration that routes water more effectively is stabler and therefore more persistent. The network becomes optimal through this dynamic evolution, eventually arriving at a form that adheres to Hack’s law.
That explanation of river network geometry hangs together for me, though geomorphologists still have many questions. Some study rivers that deviate from Hack’s law. Others organize transport networks that follow Hack’s law into one class of optimal transport networks, among a whole family of them. Trees, which branch in three dimensions instead of two, would be in a different class from rivers and follow different optimal scaling laws, for instance.
Now, geomorphologists have a new finding to explain. In April 2026, Tian Dong of the University of Texas, Rio Grande Valley and co-authors made the cover of Science for discovering that Hack’s law holds not only for rivers’ tributary networks, but also for their deltas, the fanlike structures that form where a river meets the sea.
Rivers essentially hit a brick wall when they reach the (nonflowing) ocean. The sudden deceleration of the water causes it to drop the sediments it carries. These pile up to form new land. In the process, the river’s water splits into a different kind of network of channels, which shift locations constantly as sediments build up and wash away.
Scientists told me that they’ve long wondered about the organization of channels in river deltas, but they are hard to study. Unlike the upstream river network, where slope and elevation differences make it easy to calculate the area of land that drains into any given tributary, deltas are flat and especially dynamic. But through a sophisticated analysis of satellite data that allowed them to distinguish land from water, Dong and his collaborators determined that the length of a channel in a river delta scales with the size of its nourishment area — the area that it supplies with sediments — raised to the power of 0.6. Rivers’ tributary networks and distributary networks are opposites — sediments are transported away from one end and deposited at the other — yet they abide by the same math. Geomorphologists are now considering why Hack’s law should apply in this inverse context.
Reflecting on my own question, I think it’s the coexistence of simplicity and determinism with chaos and randomness that makes the optimal structure of rivers so captivating. Natural efficiency is, perhaps, innately appealing to us…
The order in seeming chaos: “Why Are Rivers So Mathematical?” from @nattyover.bsky.social in @quantamagazine.org.
* Haruki Murakami, Kafka on the Shore
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As we go with the flow, we might send carefully-calculated birthday greetings to Moritz Cantor; he was born on this date in 1829. A historian of mathematics, he is best remembered for the four volume work Vorlesungen über Geschichte der Mathematik (“Lectures on the History of Mathematics”) which traces the history of mathematics up to 1799, the year of Gauss‘s doctoral thesis. Modern historians credit Moritz with introducing a new discipline to a field, the history of mathematics, that had hitherto lacked the sound, conscientious, and critical methods of other fields of history.
“Things that are so far removed from our daily experience… are inherently hard to understand”*…
That’s certainly true of numbers. And as the numbers grow, the cognitive challenges grow with them. (Indeed, by way of example: 1 million seconds, is roughly 11.5 days; 1 billion seconds is almost 32 years.)
We’ve looked before at the mysterious extremes of math: zero and infinity [and here]. But as Dan Falk reminds us, the numbers in between can seem pretty strange as well– especially the extremely large ones. In a review of Richard Elwes‘ Huge Numbers: A Story of Counting Ambitiously, From 4½ to Fish 7, Falk spotlights some of the largest numbers humans have ever contemplated…
… Aficionados of huge numbers are called “googologists,” a reference to the number 10100, known as a googol. Such numbers have a peculiar sort of existence. For the vast majority of us, they’re of limited everyday value. Calculations at the supermarket checkout, or at tax time in April, typically involve far more modest figures. Perhaps we’ve read that the U.S. national debt is in excess of $38 trillion — a mind-numbing figure, to be sure, but it’s not as though any one individual needs to count it up in stacks of $20 bills.
And yet, much larger numbers await those who seek them out. Consider the kinds of numbers that crop up in problems involving combinations and permutations. For example, in how many distinct ways can one shuffle a deck of cards? Elwes takes us through the calculation, and we end up with a figure of about 8×1067. Compared to that number, the odds of getting a royal flush when dealt a five-card poker hand seem pretty decent, sitting at a mere 1 in 649,740 (still rare enough that many poker players have never held such a hand). Or consider that famous 1980s cultural touchstone, the Rubik’s cube. In how many ways can one scramble the cube? It turns out that the figure is about 43 quintillion, or 4.3×1019 — but in spite of that ridiculously large figure, people do routinely solve the puzzle, and champions can do it in mere seconds. In fact, as Elwes explains, no Rubik’s cube arrangement is more than 20 moves away from any other arrangement.
Or consider the age of the universe, estimated to be about 13.8 billion years. This may seem like a lengthy span of time, but our cosmic future is where the really big numbers come up. Elwes examines the so-called heat death of the universe, in which all matter has broken down into subatomic particles. We may reach this point in [10 raised to the 10th power, raised again to the 120th power] years — this dizzying figure is 10 raised to the power of 10120 — at which point, Elwes says, the universe will have ballooned up to a diameter of 10 to the power of 10 to the power of 10120 light years. (Yes, that’s [10 raised to the 10th power, again to the 10th power, then to the 120th power] light years.) Elwes adds a footnote: “At this point, the choice of units hardly matters; the distance is so immense that whether we choose to measure it in Planck lengths or giga-light years makes little difference.” Let that sink in!
As mind numbing as such figures are, the highest numbers contemplated by humans come not from physics but from pure mathematics and computer science. Like “Graham’s number” — an immense figure put forward as the upper-bound for solutions to a problem in a branch of mathematics known as Ramsey theory. Some readers may find the ensuing discussion of multi-dimensional hypercubes a bit challenging, but one can enjoy the payoff regardless: We end up with a number that can’t even be expressed in conventional notation, and which earned a mention in the 1980 edition of the “Guinness Book of World Records” as “the highest number ever used in a mathematical proof.”
Reading this book is a little bit like sitting in the back row of an auction house where a rare Picasso (let’s say) is up for grabs: How high is this thing going to go? And indeed, Elwes keeps going. We eventually meet the so-called busy beaver numbers, a set of numbers that crop up in theoretical computer science, when one tries to deduce whether a particular computer program will eventually stop, or keep going forever — a conundrum known as the “halting problem.” As Elwes explains, it’s not at all straightforward to distinguish the two types of programs (and if it was, it would help mathematicians tackle some of the most vexing problems in their field).
The fifth busy beaver number, known as BB(5) — associated with a computer program that can access five internal states — works out to 47,176,870. And that’s as far as we’ve gotten, Elwes explains. No one has worked out the value of BB(6), but he assures us that it’s beyond the range of any physical computer; and BB(16) leaves even Graham’s number in the dust.
But wait, there’s more! “Rayo’s number,” concocted by Agustín Rayo — a dean and professor at MIT — using set theory, is bigger still (here’s a fun video about it); and “Fish 7,” mentioned in the book’s subtitle, named for a Japanese googologist who goes by the pseudonym “Fish,” builds on Rayo’s number, and … well, the details are not easily digested, but the mind-melting nature of these numbers comes across as a feature, not a bug, of Elwes’s story… the narrative is enlivened by explorations of the peculiarities of math history…
… Archimedes tried to estimate how many grains of sand would be needed to fill up the known universe, back in the third century B.C. Did he simply have too much time on his hands? Not at all, insists Elwes: The Greek thinker was articulating an important idea — that no matter how unfathomably large a quantity may be, we can describe it with precision, thanks to mathematics. “Archimedes,” he writes, “was penning a manifesto for the expressive power of large numbers.”…
… [Elwes focuses] on numbers that are ridiculously large and yet finite. In the end, perhaps this is the most mind-boggling fact of all: that these enormous numbers, from Graham’s number to Fish 7 and beyond, fall as far short of infinity as does the humble number 1…
The mysteries of the massive: “The Mind-Boggling Science of Enormous Numbers,” @danfalk.bsky.social on @richardelwes.bsky.social in @undark.org.
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As we enumerate enormity, we might spare a thought for a seminal mathematician, Alan Turing; he died on this date in 1954. He was a foundational computer science pioneer (inventor of the Turing Machine (an influential model for the general-purpose computer), creator of the “Turing Test” (only too relevant in these AI-infected times), inspiration for “The Turing Award” (the “Nobel Prize of computing“), and cryptographer (leading member of the team that cracked the Enigma code during WWII).
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