Sean Fleming: The Water Year in Review

The top five water-related news stories of 2017—and what to expect for 2018

FlemingThe thing about water is that something’s always happening, and the implications of that fact are growing – fast.  What are the top five water-related news stories of 2017?  Read on to see, along with a little context and some implications for next year and beyond.

Oops!  (The Oroville Dam evacuation)

Possibly the most obvious water story of 2017 happened right after the New Year: nearly 200,000 Californians were evacuated beneath Oroville Dam as it threatened to fail under record flooding, which in turn ended a historic drought that had cost the state billions of dollars.  Previously of little note to most living outside the region, Oroville is in fact the tallest dam in the US.  It’s located on the Feather River, a headwater basin to the Sacramento River that drains the western slopes of the snow-laden Sierra Nevada and Cascades in the wet, northern part of California.  Oroville Dam is a key component the California State Water Project, shifting water into the California Aqueduct to help irrigate the Central Valley, which produces about 25% of the food consumed in the US, and to transport water to southern Californian urban centers.  Critics charge that in spite of its size and status as a cornerstone of the civil works in a heavily populated but largely arid state where water is everything, dam maintenance and upgrading lagged far behind, setting the stage for problems.  Record rains in February provided the trigger, and the main spillway failed – which might in turn have undermined the dam as a whole, sending the entirety of massive Lake Oroville downstream all at once in a wave of destruction and death.  Disaster was averted, but the costs were tremendous and the risks were real.  For thoughts on improving America’s river infrastructure, see my recent Scientific American post.

Water goes bang on the India-China border

The most exciting, yet perhaps most under-reported water story of 2017 took place on the India-China border.  A military buildup and tense standoff over disputed ownership of a Himalayan frontier area shared by China, Nepal, Bhutan, and India this summer may have cooled off, but India charges that China followed up by using water as a weapon – withholding key data that India needs to manage lethal monsoon flooding on transboundary rivers.  Violent international conflict solely over water is extremely rare because it usually doesn’t work strategically, though it does happen from time to time.  For instance, in 1965, when Syria was building an upstream diversion of a tributary to the River Jordan that would deeply reduce Israel’s water supply – a catastrophe for a desert nation – Israel responded with air strikes against the facility.  And water has been used as a weapon in wars that were being fought for other reasons: Chiang Kai-shek’s Nationalist government in China opened the dikes on the Yellow River in 1938 in an effort to hold back the invading Imperial Japanese army. The action was only partially successful and had a disastrous humanitarian cost.  The soaring mountain ranges wrapping around the Tibetan Plateau – including the Hindu Kush, Karakoram, and Himalayas, spanning China, India, Pakistan, and  several other countries – host one of the world’s largest remaining icefields and are the source of the Indus, Yangtze, Yellow, Ganges, Brahmaputra, and Mekong Rivers among others, and thus help provide water to a full quarter of the global human population.  Perhaps nowhere else on Earth is it more important for nations to cooperate over water.

Two inter-state water lawsuits go to the US Supreme Court

The volume was turned up in the country’s water wars, with SCOTUS announcing this fall it will hear both Texas’s lawsuit against New Mexico over Rio Grande water rights, and Florida’s lawsuit against Georgia over the Apalachicola.  Rivers and aquifers don’t respect borders.  The geophysics of where water comes from and how and where it flows is complex, fascinating, and full of surprises, such as flash floods, alternating drought and flood sequences, and abrupt and catastrophic changes in river channel location.  And those are just the natural aspects – the engineering and management part can be just as complicated for some basins, and a high ratio of demand to supply, as we have in the increasingly heavily populated deserts of the Southwest for instance, exacerbates these issues.  Originating from snowy headwaters in the mountains of southern Colorado and northern New Mexico, the Rio Grande flows south through increasingly arid country and then turns southeastward, forming the US-Mexico border until emptying in the Gulf of Mexico.  Water projects abound on the Rio Grande, and each influences the other in some way.  For example, the San Juan-Chama project diverts water from the Colorado River into the Rio Grande, municipal groundwater pumping in Albuquerque interacts with Rio Grande flows through subterranean geologic pathways, and a series of dams withdraws water from the river for agriculture, reducing what’s left for downstream users.  Water law is complicated.  Texas says New Mexico is taking more than its fair share of Rio Grande water; New Mexico says it isn’t.  The potential for disagreement over water will only continue to grow in the Southwest, though there are success stories as well: after some earlier missteps, Las Vegas has invented one of the most advanced and successful water conservation programs around, reportedly reducing its water consumption by almost a quarter over a ten-year period while its population grew by half a million.

Saying goodbye to the Paris Agreement on climate change

Why is climate change important to rivers?  Lots of natural processes and human activities affect how high rivers run and how much water arrives at your tap, and climate variables like precipitation and temperature rank high among these influences.  While the new administration’s withdrawal from the Paris Agreement in 2017 was obviously a setback for action on climate change, it was also a democratic response to widespread sentiment.  And this fact suggests that explaining climate change may be turning into the greatest science communication failure in history.  As scientists, we clearly need to adjust course – but in what direction?  Consider a recent article by a multi-disciplinary team in the respected research journal, Global Environmental Change.  Applying complex network theory (kind of a mathematical formalization of the seven degrees of Kevin Bacon) to social media feeds about climate change, they demonstrated the dominance of so-called echo chambers, and that constructive progress is made only when groups with opposing views actually talk with each other.  Consider also that populism – which is by nature skeptical around the competence and integrity of designated experts – has been growing over the last decade on both the left and right, as evidenced by the mayoralties of Rob Ford in Toronto and Boris Johnson in London, the Tea Party and Occupy movements, Brexit, and Bernie and The Donald.  If there is a silver lining to withdrawal from the Paris accords, it’s that it may teach us valuable lessons around communicating about climate change: reach out to people who don’t believe us yet, treat them with respect, and focus on just explaining our science.

Houston, we have a problem

Hurricane Harvey hit Houston hard.  In late August, the fourth largest city in the US, with over 4 million residents counting Harris County, was at the epicenter of what some are saying will be the costliest natural disaster in US history.  Though no hurricane is to be trifled with, why was the flooding so intense in this case?  To be sure, the rainfall generated by this particular storm was unusually heavy.  But risk is, by definition, what you get when you take the probability that something bad will happen (like record rainfall under a hurricane) and multiply it by the impact it will have if it does happen (like flooding and the associated economic cost and human suffering).  In the case of Harvey’s visit to Houston, it had a lot to do with local-scale choices that affected the second part of that equation.  In fact, parts of the greater Houston metropolitan area have seen a spate of floods over the last few years, and they weren’t all associated with huge storms.  The region has experienced an explosion of population growth and urban sprawl.  Lots of residences were built in low-lying, flood-prone areas, which is the single best of way of increasing flood risk.  And urbanization – the conversion of wild or agricultural land to rooftops, parking lots, and roadways – is another powerful flood risk factor.  Soils and wetlands hold on to rainwater for a while, and then gently release it to natural drainage systems like aquifers and rivers.  If you pave and build over these things, their ability to attenuate flooding is removed.  While these effects are particularly noticeable in Houston, and especially so when the city gets hit by a major hurricane, they’re ubiquitous; increased flooding in the UK over the last decade has been attributed to exactly the same causes.

What will 2018 have in store for us?  If we can be sure about one thing, it’s to expect the unexpected.  But the larger trends are clear.  Global water demand will increase 55% in the next few decades, urbanization will spread, tens of millions more will congregate in floodplain-located megacities, the climate will subtly but profoundly shift overhead, and cooperation and conflict over water will vie for supremacy.  We can, in short, expect that water stories will make the news with increasing frequency and force.

Sean W. Fleming has two decades of experience in the private, public, and nonprofit sectors in the United States, Canada, England, and Mexico, ranging from oil exploration to operational river forecasting to glacier science. He holds faculty positions in the geophysical sciences at the University of British Columbia and Oregon State University. He is the author of Where the River Flows: Scientific Reflections on Earth’s Waterways.

Monarch butterflies: Out of sight, but not out of mind!

By Anurag Agrawal


The annual migratory calendar for monarch butterflies in eastern North America.

As winter approaches, monarch butterflies are not in sight for most Americans. Beginning in the fall, hundreds of millions of butterflies east of the Rocky Mountains oriented south and began their migration. And indeed the story of how they navigate is truly remarkable: the little insect uses a sun compass that is adjustable depending on the time of day to find its way. Details of the migration and much more are in Monarchs and Milkweed: A Migrating Butterfly, a Poisonous Plant, and their Remarkable Story of Coevolution. And 2017 was a spectacular fall season for monarch butterflies. As far as most monarch biologists can remember, this was perhaps the biggest summer season on record, with monarchs in epic numbers congregating and flying south.


Southward migrating monarchs in Ontario during autumn. Although monarchs are usually dispersed in the summer, as the fall migration takes hold, butterflies congregate in larger clusters.

As the holiday season approaches, it is useful to keep in to keep in mind where monarchs are and what they are doing.  Cool and concentrated, they huddle en masse for nearly five months.  Will the numbers of butterflies overwintering in Mexico this year show a rebound from their precipitous decline?  If the migration was successful, yes, we all expect (hope!) the numbers to be up.  But only time will tell, as the official numbers are typically announced each February by World Wildlife Fund Mexico.  The monitoring of these unimaginable aggregations of butterflies has been a critical piece in the conservation puzzle for monarchs.


The state license plate in Michoacán State, Mexico.

In November around the Day of the Dead and leading to American Thanksgiving, monarchs arrive to their overwintering grounds in the highlands Michoacán, Mexico. And legend has it that the butterflies are the returning souls of loved ones. They form clusters that are so dense, they weigh down the Oyamel Fir trees they inhabit above 10,000 feet of elevation in these exquisite sites. The sites are terribly small, with all of them fitting into area smaller than New York City.


A congregation of monarchs within the Monarch Butterfly Biosphere Reserve, a UNESCO World Heritage Site. Most wings are closed, but look for the orange spots of open butterfly wings.

But before 1975, there was no conservation conversation about monarchs, because scientists simply did not know where monarchs went in the winter (of course native Mexicans of the region have known for centuries).  More importantly, we didn’t know how restricted and sensitive their overwintering sites are. The story of how the monarchs were found is too lengthy to recount here, but it is an astonishing story. In short, Professor Urquhart from the University of Toronto was hot on the trail, and knew that they flew south into Mexico during the fall.  Nora and Fred Urquhart marshaled a citizen science campaign that included a massive effort to engage folks far and wide in the search for the overwintering grounds.  In fact, in 1973, they wrote an article in an English language newspaper in Mexico City requesting help in finding the monarch overwintering sites.



I obtained this reproduction of the original article outlining monarch butterfly biology and requesting help finding the overwintering grounds from the Library of Congress. It came on microfiche and was a treasure to hold and read.

Still, it was another two years before the overwintering colonies were found and reported to the world. After thirty years of tagging butterflies, enlisting thousands of citizen scientists, and much speculation, shortly after new year’s day in January 1975, the great discovery was made. The Urquharts wrote to their thousands of volunteers: “We now wish to announce to our associates, that, after these many years of intensive study, after having tagged thousands of migrants, we have, finally located the exact area where they overwinter, with the very able assistance of Ken Brugger and Cathy Brugger of Mexico City”. And the rest is history.

Anurag Agrawal is a professor in the Department of Ecology and Evolutionary Biology and the Department of Entomology at Cornell University. He lives in Ithaca, New York.


Matthew J. Salganik on Bit by Bit: Social Research in the Digital Age

In just the past several years, we have witnessed the birth and rapid spread of social media, mobile phones, and numerous other digital marvels. In addition to changing how we live, these tools enable us to collect and process data about human behavior on a scale never before imaginable, offering entirely new approaches to core questions about social behavior. Bit by Bit is the key to unlocking these powerful methods—a landmark book that will fundamentally change how the next generation of social scientists and data scientists explores the world around us. Matthew Salganik has provided an invaluable resource for social scientists who want to harness the research potential of big data and a must-read for data scientists interested in applying the lessons of social science to tomorrow’s technologies. Read on to learn more about the ideas in Bit by Bit.

Your book begins with a story about something that happened to you in graduate school. Can you talk a bit about that? How did that lead to the book?

That’s right. My dissertation research was about fads, something that social scientists have been studying for about as long as there have been social scientists. But because I happened to be in the right place at the right time, I had access to an incredibly powerful tool that my predecessors didn’t: the Internet. For my dissertation, rather than doing an experiment in a laboratory on campus—as many of my predecessors might have—we built a website where people could listen to and download new music. This website allowed us to run an experiment that just wasn’t possible in the past. In my book, I talk more about the scientific findings from that experiment, but while it was happening there was a specific moment that changed me and that directly led to this book. One morning, when I came into my basement office, I discovered that overnight about 100 people from Brazil had participated in my experiment. To me, this was completely shocking. At that time, I had friends running traditional lab experiments, and I knew how hard they had to work to have even 10 people participate. However, with my online experiment, 100 people participated while I was sleeping. Doing your research while you are sleeping might sound too good to be true, but it isn’t. Changes in technology—specifically the transition from the analog age to the digital age—mean that we can now collect and analyze social data in new ways. Bit by Bit is about doing social research in these new ways.

Who is this book for?

This book is for social scientists who want to do more data science, data scientists who want to do more social science, and anyone interested in the hybrid of these two fields. I spend time with both social scientists and data scientists, and this book is my attempt to bring the ideas from the communities together in a way that avoids the jargon of either community.  

In your talks, I’ve heard that you compare data science to a urinal.  What’s that about?

Well, I compare data science to a very specific, very special urinal: Fountain by the great French artist Marcel Duchamp. To create Fountain, Duchamp had a flash of creativity where he took something that was created for one purpose—going to the bathroom—and turned it a piece of art. But most artists don’t work that way. For example, Michelangelo, didn’t repurpose. When he wanted to create a statue of David, he didn’t look for a piece of marble that kind of looked like David: he spent three years laboring to create his masterpiece. David is not a readymade; it is a custommade.

These two styles—readymades and custommades—roughly map onto styles that can be employed for social research in the digital age. My book has examples of data scientists cleverly repurposing big data sources that were originally created by companies and governments. In other examples, however, social scientists start with a specific question and then used the tools of the digital age to create the data needed to answer that question. When done well, both of these styles can be incredibly powerful. Therefore, I expect that social research in the digital age will involve both readymades and custommades; it will involve both Duchamps and Michelangelos.

Bit by Bit devotes a lot attention to ethics.  Why?

The book provides many of examples of how researchers can use the capabilities of the digital age to conduct exciting and important research. But, in my experience, researchers who wish to take advantage of these new opportunities will confront difficult ethical decisions. In the digital age, researchers—often in collaboration with companies and governments—have increasing power over the lives of participants. By power, I mean the ability to do things to people without their consent or even awareness. For example, researchers can now observe the behavior of millions of people, and researchers can also enroll millions of people in massive experiments. As the power of researchers is increasing, there has not been an equivalent increase in clarity about how that power should be used. In fact, researchers must decide how to exercise their power based on inconsistent and overlapping rules, laws, and norms. This combination of powerful capabilities and vague guidelines can force even well-meaning researchers to grapple with difficult decisions. In the book, I try to provide principles that can help researchers—whether they are in universities, governments, or companies—balance these issues and move forward in a responsible way.

Your book went through an unusual Open Review process in addition to peer review. Tell me about that.

That’s right. This book is about social research in the digital age, so I also wanted to publish it in a digital age way. As soon as I submitted the book manuscript for peer review, I also posted it online for an Open Review during which anyone in the world could read it and annotate it. During this Open Review process dozens of people left hundreds of annotations, and I combined these annotations with the feedback from peer review to produce a final manuscript. I was really happy with the annotations that I received, and they really helped me improve the book.

The Open Review process also allowed us to collect valuable data. Just as the New York Times is tracking which stories get read and for how long, we could see which parts of the book were being read, how people arrived to the book, and which parts of the book were causing people to stop reading.

Finally, the Open Review process helped us get the ideas in the book in front of the largest possible audience. During Open Review, we had readers from all over the world, and we even had a few course adoptions. Also, in addition to posting the manuscript in English, we machine translated it into more than 100 languages, and we saw that these other languages increased our traffic by about 20%.

Was putting your book through Open Review scary?

No, it was exhilarating. Our back-end analytics allowed me see that people from around the world were reading it, and I loved the feedback that I received. Of course, I didn’t agree with all the annotations, but they were offered in a helpful spirit, and, as I said, many of them really improved the book.

Actually, the thing that is really scary to me is putting out a physical book that can’t be changed anymore. I wanted to get as much feedback as possible before the really scary thing happened.

And now you’ve made it easy for other authors to put their manuscripts through Open Review?

Absolutely. With a grant from the Sloan Foundation, we’ve released the Open Review Toolkit. It is open source software that enables authors and publishers to convert book manuscripts into a website that can be used for Open Review. And, as I said, during Open Review, you can receive valuable feedback to help improve your manuscript, feedback that is very complimentary to the feedback from peer review. During Open Review, you can also collect valuable data to help launch your book. Furthermore, all of these good things are happening at the same time that you are increasing access to scientific research, which is a core value of many authors and academic publishers.

SalganikMatthew J. Salganik is professor of sociology at Princeton University, where he is also affiliated with the Center for Information Technology Policy and the Center for Statistics and Machine Learning. His research has been funded by Microsoft, Facebook, and Google, and has been featured on NPR and in such publications as the New Yorker, the New York Times, and the Wall Street Journal.

Browse Our Earth Science 2018 Catalog

Our new Earth Science 2018 catalog ranges from the northernmost reaches of the globe to the unfathomable depths of its oceans, while also covering essential techniques and concepts in the fields of complexity and predictive ecology. 

If you will be attending the American Geophysical Union 2017 meeting in New Orleans this weekend, please stop by booth 730, where you can pick up a copy of the catalog in person and see our full range of books in Earth Science.

In the forthcoming Brave New Arctic, Mark Serreze details the history and the science of the precipitous warming of the Arctic, and its potentially devastating consequences for the planet as a whole. Drawing on his own work, as well as that of pioneering climate scientists, Brave New Arctic is a fascinating account of the not-so-frozen North. 

Brave New Arctic, by Mark Serreze

Eelco Rohling’s The Oceans traces the history of the planet’s oceans from the Earth’s formation to the present day, demonstrating the critical role they play in the Earth’s climate system. Concise but comprehensive, The Oceans is an essential introduction to paleoceanography, from one of AGU’s newest fellows.  

The Oceans, by Eelco Rohling

Drawing on simple computational models, Natural Complexity by Paul Charbonneau analyzes the emergence of complex behaviors and structure in natural phenomena from forest fires to epidemic diseases. Including complete source code in Python, Natural Complexity is a straightforward introduction to complexity in all its forms.

Natural Complexity, by Paul Charbonneau

What is involved in making ecology a more predictive science? In Ecological Forecasting, Michael Dietze covers the cutting-edge techniques that are driving modern ecology, complete with case studies and hands-on examples using R.

Ecological Forecasting, by Michael C. Dietze


Eelco J. Rohling on The Oceans: A Deep History

It has often been said that we know more about the moon than we do about our own oceans. In fact, we know a great deal more about the oceans than many people realize. Scientists know that our actions today are shaping the oceans and climate of tomorrow—and that if we continue to act recklessly, the consequences will be dire. In this timely and accessible book, Eelco Rohling traces the 4.4 billion-year history of Earth’s oceans while also shedding light on the critical role they play in our planet’s climate system. An invaluable introduction to the cutting-edge science of paleoceanography, The Oceans enables you to make your own informed opinions about the environmental challenges we face as a result of humanity’s unrelenting drive to exploit the world ocean and its vital resources. Read on to learn more about the ideas in Eelco Rohling’s new book.

How/Why did you become a specialist in past ocean and climate change?
When I was a boy, I actually wanted to become a brain surgeon. But I did not pass the lottery to get into medical school when I went to university. So I thought about what else to study for a year before trying again. I ended up doing geology, and never looked back—I pushed on with that instead of trying medical school again. In geology, I developed a fascination with the past environments in which animals and plants lived that we now find as fossils. So after my BSc, I did an MSc with a major in microfossils and palaeo-oceanography/-climatology, supported by minors in sedimentary systems and physical oceanography/climatology. Things started to really come together when I started my PhD project, for which I started to truly integrate these streams in a research context. That’s when my interest in past ocean and climate change became much deeper and more specific.

Why did you choose to write a book about the history of the oceans?
I discussed a few ideas with my editor Eric Henney, and we gradually brought the various ideas together into this book concept. We strongly felt that the vast existing knowledge about the past oceans (and past climate) needed to be better articulated, and placed in context of modern changes in these systems, and in the life that they sustain.

Why do we need to understand the history of the oceans?
The oceans’ past holds many fascinating pieces of information about how the ocean/climate system works, and how it interacts with life and the planet itself. No other field can bring that information to the table. The oceans’ history also holds important clues about how Earth may recover from human impact, and on what timescales such a recovery may be expected. This brings important context to the discussion about modern human impact.

Does the history of the oceans give any relevant information about their future?
Oh, yes. It illustrates the key processes by which carbon-cycle changes have occurred over Earth history, and whet the timescales were for these changes. It also illustrates which processes we might try to accelerate to drive atmospheric carbon-removal on timescales useful to humankind. Moreover, the history of the oceans provides insight into the developments (and extinctions) of life on Earth, which again gives context about the severity and rapidity of current changes on Earth.

Why does a book about the oceans contain so much about climate?
The oceans are an integral part of the climate system. The climate system is a complex beast that spans the atmosphere, hydrosphere (all forms of water), cryosphere (all forms of ice), lithosphere (the rocks), and biosphere (all forms of life, be it living or dead). The oceans are a vital link in all this, and one cannot talk about ocean changes without touching upon climate changes, or the other way around.

The oceans appear to have gone through very large changes in the past. How do the changes cause by humanity compare?
The human-caused changes are large, but not among the largest that have ever happened. But the human-caused changes are unique with respect to the rates of change: modern changes are 10 to 100 times faster than the fastest-ever natural changes any time before humans appeared on the scene. And, also, human-made changes have significant impacts from many different sides: warming, ocean acidification, physical (e.g., plastic) pollution, chemical pollution, eutrophication, overfishing, etc. Natural changes were not that all-encompassing. So modern changes are very scary in relation to the natural changes that have occurred, even when including major extinction events.

Are humans really causing damage to the enormous oceans and the life they contain?
Yes, for sure.Humans have trouble imagining how their (often little) actions can add up over time, and across the massive population numbers. But we’re on this planet with well over 7 billion people, all of whom at least partly rely on the ocean as a key resource for such things as: dumping waste/pollution from plastics to oil and from radioactive materials to chemical waste and fertilizers; transportation (with spillages), food production/fisheries; war-mongering, exploration/mining, energy production, etc. Added up over our massive human population and increasing technical infrastructure, all of these aspects alone have devastating impacts already, but taken together they are heading down a particularly terminal route.

OceansEelco J. Rohling is professor of ocean and climate change in the Research School of Earth Sciences at the Australian National University and at the University of Southampton’s National Oceanography Centre Southampton.

Pariah Moonshine Part II: For Whom the Moon Shines

by Joshua Holden

This post originally appeared on The Aperiodical. We republish it here with permission. 

HoldenI ended Part I with the observation that the Monster group was connected with the symmetries of a group sitting in 196883-dimensional space, whereas the number 196884 appeared as part of a function used in number theory, the study of the properties of whole numbers.  In particular, a mathematician named John McKay noticed the number as one of the coefficients of a modular form.  Modular forms also exhibit a type of symmetry, namely if F is a modular form then there is some number k for which

Figure 1

for every set of whole numbers a, b, c, and d such that adbc=1.  (There are also some conditions as the real part of z goes to infinity.)

Modular forms, elliptic curves, and Fermat’s Last Theorem

In 1954, Martin Eichler was studying modular forms and observing patterns in their coefficients.  For example, take the modular form

Figure 2

(I don’t know whether Eichler actually looked at this particular form, but he definitely looked at similar ones.)  The coefficients of this modular form seem to be related to the number of whole number solutions of the equation

y2 = x3 – 4 x2 + 16

This equation is an example of what is known as an elliptic curve, which is a curve given by an equation of the form

y2 = x3 + ax2 + bx + c

Note that elliptic curves are not ellipses!  Elliptic curves have one line of symmetry, two open ends, and either one or two pieces, as shown in Figures 1 and 2. They are called elliptic curves because the equations came up in the seventeenth century when mathematicians started studying the arc length of an ellipse.  These curves are considered the next most complicated type of curve after lines and conic sections, both of which have been understood pretty well since at least the ancient Greeks.   They are useful for a lot of things, including cryptography, as I describe in Section 8.3 of The Mathematics of Secrets.

Figure 1

Figure 1. The elliptic curve y2= x3 + x has one line of symmetry, two open ends, and one piece.

Figure 2

Figure 2. The elliptic curve y2 = x3 – x has one line of symmetry, two open ends, and two pieces.


In the late 1950’s it was conjectured that every elliptic curve was related to a modular form in the way that the example above is.  Proving this “Modularity Conjecture” took on more urgency in the 1980’s, when it was shown that showing the conjecture was true would also prove Fermat’s famous Last Theorem.  In 1995 Andrew Wiles, with help from Richard Taylor, proved enough of the Modularity Conjecture to show that Fermat’s Last Theorem was true, and the rest of the Modularity Conjecture was filled in over the next six years by Taylor and several of his associates.

Modular forms, the Monster, and Moonshine

Modular forms are also related to other shapes besides elliptic curves, and in the 1970’s John McKay and John Thompson became convinced that the modular form

J(z) = e -2 π i z + 196884 e 2 π i z + 21493760 e 4 π i z  + 864299970 e 6 π i z  + …

was related to the Monster.  Not only was 196884 equal to 196883 + 1, but 21493760 was equal to 21296876 + 196883 + 1, and 21296876 was also a number that came up in the study of the Monster.  Thompson suggested that there should be a natural way of associating the Monster with an infinite-dimensional shape, where the infinite-dimensional shape broke up into finite-dimensional pieces with each piece having a dimension corresponding to one of the coefficients of J(z).   This shape was (later) given the name V♮, using the natural sign from musical notation in a typically mathematical pun.  (Terry Gannon points out that there is also a hint that the conjectures “distill information illegally” from the Monster.) John Conway and Simon Norton formulated some guesses about the exact connection between the Monster and V♮, and gave them the name “Moonshine Conjectures” to reflect their speculative and rather unlikely-seeming nature. A plausible candidate for V♮ was constructed in the 1980’s and Richard Borcherds proved in 1992 that the candidate satisfied the Moonshine Conjectures.  This work was specifically cited when Borcherds was awarded the Fields medal in 1998.

The construction of V♮ turned out also to have a close connection with mathematical physics.  The reconciliation of gravity with quantum mechanics is one of the central problems of modern physics, and most physicists think that string theory is likely to be key to this resolution.  In string theory, the objects we traditionally think of as particles, like electrons and quarks, are really tiny strings curled up in many dimensions, most of which are two small for us to see.  An important question about this theory is exactly what shape these dimensions curl into.  One possibility is a 24-dimensional shape where the possible configurations of strings in the shape are described by V♮.  However, there are many other possible shapes and it is not known how to determine which one really corresponds to our world.

More Moonshine?

Since Borcherds’ proof, many variations of the original “Monstrous Moonshine” have been explored.  The other members of the Happy Family can be shown to have Moonshine relationships similar to those of the Monster.  “Modular Moonshine” says that certain elements of the Monster group should have their own infinite dimensional shapes, related to but not the same as V♮.  (The “modular” in “Modular Moonshine” is related to the one in “modular form” because they are both related to modular arithmetic, although the chain of connections is kind of long. )  “Mathieu Moonshine” shows that one particular group in the Happy Family has its own shape, entirely different from V♮, and “Umbral Moonshine” extends this to 23 other related groups which are not simple groups.  But the Pariah groups remained outsiders, rejected by both the Happy Family and by Moonshine — until September 2017.

Joshua Holden is professor of mathematics at the Rose-Hulman Institute of Technology. He is the author of The Mathematics of Secrets: Cryptography from Caesar Ciphers to Digital Encryption.

Pariah Moonshine Part I: The Happy Family and the Pariah Groups

by Joshua Holden

This post originally appeared on The Aperiodical. We republish it here with permission. 

HoldenBeing a mathematician, I often get asked if I’m good at calculating tips. I’m not. In fact, mathematicians study lots of other things besides numbers. As most people know, if they stop to think about it, one of the other things mathematicians study is shapes. Some of us are especially interested in the symmetries of those shapes, and a few of us are interested in both numbers and symmetries. The recent announcement of “Pariah Moonshine” has been one of the most exciting developments in the relationship between numbers and symmetries in quite some time. In this blog post I hope to explain the “Pariah” part, which deals mostly with symmetries. The “Moonshine”, which connects the symmetries to numbers, will follow in the next post.

What is a symmetry?

First I want to give a little more detail about what I mean by the symmetries of shapes. If you have a square made out of paper, there are basically eight ways you can pick it up, turn it, and put it down in exactly the same place. You can rotate it 90 degrees clockwise or counterclockwise. You can rotate it 180 degrees. You can turn it over, so the front becomes the back and vice versa. You can turn in over and then rotate it 90 degrees either way, or 180 degrees. And you can rotate it 360 degrees, which basically does nothing. We call these the eight symmetries of the square, and they are shown in Figure 1.


Figure 1. The square can be rotated into four different positions, without or without being flipped over, for eight symmetries total.

If you have an equilateral triangle, there are six symmetries. If you have a pentagon, there are ten. If you have a pinwheel with four arms, there are only four symmetries, as shown in Figure 2, because now you can rotate it but if you turn it over it looks different. If you have a pinwheel with six arms, there are six ways. If you have a cube, there are 24 if the cube is solid, as shown in Figure 3. If the cube is just a wire frame and you are allowed to turn it inside out, then you get 24 more, for a total of 48.

Figure 2

Figure 2. The pinwheel can be rotated but not flipped, for four symmetries total.

Figure 3

Figure 3. The cube can be rotated along three different axes, resulting in 24 different symmetries.

These symmetries don’t just come with a count, they also come with a structure. If you turn a square over and then rotate it 90 degrees, it’s not the same thing as if you rotate it first and then flip it over. (Try it and see.) In this way, symmetries of shapes are like the permutations I discuss in Chapter 3 of my book, The Mathematics of Secrets: you can take products, which obey some of the same rules as products of numbers but not all of them. These sets of symmetries, which their structures, are called groups.

Groups are sets of symmetries with structure

Some sets of symmetries can be placed inside other sets. For example, the symmetries of the four-armed pinwheel are the same as the four rotations in the symmetries of the square. We say the symmetries of the pinwheel are a subgroup of the symmetries of the square. Likewise, the symmetries of the square are a subgroup of the symmetries of the solid cube, if you allow yourself to turn the cube over but not tip it 90 degrees, as shown in Figure 4.

Figure 4

Figure 4. The symmetries of the square are contained inside the symmetries of the cube if you are allowed to rotate and flip the cube but not tip it 90 degrees.

In some cases, ignoring a subgroup of the symmetries of a shape gets us another group, which we call the quotient group. If you ignore the subgroup of how the square is rotated, you get the quotient group where the square is flipped over or not, and that’s it. Those are the same as the symmetries of the capital letter A, so the quotient group is really a group. In other cases, for technical reasons, you can’t get a quotient group. If you ignore the symmetries of a square inside the symmetries of a cube, what’s left turns out not to be the symmetries of any shape.

You can always ignore all the symmetries of a shape and get just the do nothing (or trivial) symmetry, which is the symmetries of the capital letter P, in the quotient group. And you can always ignore none of the nontrivial symmetries, and get all of the original symmetries still in the quotient group. If these are the only two possible quotient groups, we say that the group is simple. The group of symmetries of a pinwheel with a prime number of arms is simple. So is the group of symmetries of a solid icosahedron, like a twenty-sided die in Dungeons and Dragons. The group of symmetries of a square is not simple, because of the subgroup of rotations. The group of symmetries of a solid cube is not simple, not because of the symmetries of the square, but because of the smaller subgroup of symmetries of a square with a line through it, as shown in Figures 5 and 6. The quotient group there is the same as the symmetries of the equilateral triangle created by cutting diagonally through a cube near a corner.

Figure 5

Figure 5. The symmetries of a square with line through it. We can turn the square 180 degrees and/or flip it, but not rotate it 90 degrees, so there are four.

Figure 6

Figure 6. The symmetries of the square with a line through it inside of the symmetries of the cube.

Categorizing the Pariah groups

As early as 1892, Otto Hölder asked if we could categorize all of the finite simple groups. (There are also shapes, like the circle, which have an infinite number of symmetries. We won’t worry about them now.)  It wasn’t until 1972 that Daniel Gorenstein made a concrete proposal for how to make a complete categorization, and the project wasn’t finished until 2002, producing along the way thousands of pages of proofs. The end result was that almost all of the finite simple groups fell into a few infinitely large categories: the cyclic groups, which are the groups of symmetries of pinwheels with a prime number of arms, the alternating groups, which are the groups of symmetries of solid hypertetrahedra in 5 or more dimensions, and the “groups of Lie type”, which are related to matrix multiplication over finite fields and describe certain symmetries of objects known as finite projective planes and finite projective spaces. (Finite fields are used in the AES cipher and I talk about them in Section 4.5 of The Mathematics of Secrets.)

Even before 1892, a few finite simple groups were discovered that didn’t seem to fit into any of these categories. Eventually it was proved that there were 26 “sporadic” groups, which didn’t fit into any of the categories and didn’t describe the symmetries of anything obvious — basically, you had to construct the shape to fit the group of symmetries that you knew existed, instead of starting with the shape and finding the symmetries. The smallest of the sporadic groups has 7920 symmetries in it, and the largest, known as the Monster, has over 800 sexdecillion symmetries. (That’s an 8 with 53 zeros after it!) Nineteen of the other sporadic groups turn out to be subgroups or quotient groups of subgroups of the Monster. These 20 became known as the Happy Family. The other 6 sporadic groups became known as the ‘Pariahs’.

The shape that was constructed to fit the Monster lives in 196883-dimensional space. In the late 1970’s a mathematician named John McKay noticed the number 196884 turning up in a different area of mathematics. It appeared as part of a function used in number theory, the study of the properties of whole numbers. Was there a connection between the Monster and number theory? Or was the idea of a connection just … moonshine?

Joshua Holden is professor of mathematics at the Rose-Hulman Institute of Technology. He is the author of The Mathematics of Secrets: Cryptography from Caesar Ciphers to Digital Encryption.

Michael Ruse on On Purpose

Can we live without the idea of purpose? Should we even try to? Kant thought we were stuck with purpose, and even Darwin’s theory of natural selection, which profoundly shook the idea, was unable to kill it. Indeed, teleological explanation—what Aristotle called understanding in terms of “final causes”—seems to be making a comeback today, as both religious proponents of intelligent design and some prominent secular philosophers argue that any explanation of life without the idea of purpose is missing something essential. In On Purpose, Michael Ruse explores the history of the idea of purpose in philosophical, religious, scientific, and historical thought, from ancient Greece to the present. Read on to learn more about the idea of “purpose,” the long philosophical tradition around it, and how Charles Darwin fits in.

On Purpose?  So what’s with the smart-alecky title?

It was a friend of Dr. Johnson who said that he had tried to be a philosopher, but cheerfulness always kept breaking in.  Actually, that is a little bit unfair to philosophers.  Overall, we are quite a cheerful group, especially when we think that we might have been born sociologists or geographers.  However, our sense of humor is a bit strained, usually—as in this case—involving weak puns and the like.  My book is about a very distinctive form of understanding, when we do things in terms of the future and not the past.

In terms of the future?  Why not call your book On Prediction?

I am not talking about prediction, forecasting what you think will happen, although that is involved.  I am talking about when the future is brought in to explain things that are happening right now.  Purposeful thinking is distinctive and interesting because normally when we try to explain things we do so in terms of the past or present.  Why do you have a bandage on your thumb?  Because I tried to hang the picture myself, instead of getting a grad student to do it.  Purposeful thinking—involving what Aristotle called “final causes” and what since the eighteenth century has often been labeled “teleological” thinking—explains in terms of future events.  Why are you studying rather than going to the ball game?  Because I want to do well on the GRE exam and go to a good grad school.

Why is this interesting?

In the case of the bandaged thumb, you know that the hammer hit you rather than the nail.  In the case of studying, you may decide that five to ten years of poverty and peonage followed by no job is not worth it, and you should decide to do something worthwhile like becoming a stockbroker or university administrator.  We call this “the problem of the missing goal object.”  Going to grad school never occurred, but it still makes sense to say that you are studying now in order to go to grad school.

Is this something that you thought up, or is it something with a history?

Oh my, does it ever have a history.  One of the great things about my book, if I might show my usual level of modesty, is that I show the whole problem of purpose is one with deep roots in the history of philosophy, starting with Plato and Aristotle, and coming right up to the modern era, particularly the thinking of Immanuel Kant.  In fact, I argue that it is these three very great philosophers who set the terms of the discussion—Plato analyses things in terms of consciousness, Aristotle in terms of principles of ordering whatever that might mean, and Kant opts for some kind of heuristic approach.

If these thinkers have done the spadework, what’s left for you?

I argue that the truth about purposeful thinking could not be truly discovered until Charles Darwin in his Origin of Species (1859) had proposed his theory of evolution through natural selection.  With that, we could start to understand forward-looking thinking about humans—why is he studying on such a beautiful day?  He wants to go to grad school.  About plants and animals—why does the stegosaurus have those funny-looking plates down its back?  To control its temperature.  And why we don’t use such thinking about inanimate objects?  Why don’t we worry about the purpose of the moon?  Perhaps we should.  It really does exist in order to light the way home for drunken philosophers.

Why is it such a big deal to bring up Darwin and his theory of evolution?  Surely, the kind of people who will read your book will have accepted the theory long ago?

Interestingly, no!  The main opposition to evolutionary thinking comes from the extreme ends of the spectrum: evangelical Christians known as Creationists—biblical literalists—and from professional philosophers.  There are days when it seems that the higher up the greasy pole you have climbed, the more likely you are to deny Darwinism and be a bit iffy about evolution generally.  This started just about as soon as the Origin appeared, and the sinister anti-evolutionary effect of Bertrand Russell and G. E. Moore and above all Ludwig Wittgenstein is felt to this day.  A major reason for writing my book was to take seriously Thomas Henry Huxley’s quip that we are modified monkeys rather than modified mud, and that matters.

Given that you are a recent recipient of the Bertrand Russell Society’s “Man of the Year” Award, aren’t you being a bit ungracious?

I have huge respect for Russell.  He was a god in my family when, in the 1940s and 50s, I was growing up in England.  One of my greatest thrills was to have been part of the crowd in 1961 in Trafalgar Square listening to him declaim against nuclear weapons.  But I think he was wrong about the significance of Darwin for philosophy and I think I am showing him great respect in arguing against him.  I feel the same way about those who argue against me.  My proudest boast is that I am now being refuted in journals that would never accept anything by me.

One of the big problems normal people today have about philosophy is that it seems so irrelevant. Initiates arguing about angels on the heads of pins?  Why shouldn’t we say the same about your book?

Three reasons.  First, my style and approach.  It is true that most philosophy produced by Anglophone philosophers today is narrow and boring.  Reading analytic philosophy is like watching paint dry and proudly so.  Against this, on the one hand I am more a historian of ideas using the past to illuminate the present.  That is what being an evolutionist is all about.  Spending time with mega-minds like Plato and Aristotle and Kant is in itself tremendously exciting.  On the other hand, I have over fifty years of teaching experience, at the undergraduate level almost always at the first- and second-year level.  I know that if you are not interesting, you are going to lose your audience.  The trick is to be interesting and non-trivial.

Second, I don’t say that my book is the most important of the past hundred-plus years, but my topic is the most important.  Evolution matters, folks, it really does.  It is indeed scary to think that we are just the product of a random process of change and not the favored product of a Good God—made in His image.  Even atheists get the collywobbles, or at least they should.  It is true all the same.  Fifty years ago, the geneticist and Nobel laureate Hermann J. Muller said that a hundred years without Darwin is enough.  That is still true.  Amen.

Third, deliberately, I have made this book very personal.  At the end, I talk about purpose in my own life.  Why, even though I am a non-believer, I have been able to find meaning in what I think and do.  This ranges from my love of my wife Lizzie and how with dedication and humor we share the challenges of having children—not to mention our love of dogs, most recent addition to the family, Nutmeg a whippet—through cooking on Saturday afternoons while listening to radio broadcasts of Metropolitan Opera matinees, to reading Pickwick Papers yet one more time.  I suspect that many of my fellow philosophers will find this all rather embarrassing.  I mean it to be.  Philosophy matters.  My first-ever class on the subject started with Descartes’ Meditations.  Fifteen minutes into the class, I knew that this was what I was going to do for the rest of my life.  Nearly sixty years later I am still at it and surely this interview tells you that I love it, every moment.

So, why should we read your book?

Because it really does square the circle.  It is cheerful and philosophical.  It is on a hugely important topic and there are some good jokes.  I am particularly proud of one I make about Darwin Day, the celebration by New Atheists, and their groupies of the birthday of Charles Darwin.

Which is?

Oh, hell no.  I am not going to tell you.  Go out and buy the book.  And while you are at it, buy one for your mum and dad and one each for your siblings and multi-copies for your students and….  I am seventy-seven years old.  I need a bestseller so I can retire.  You need a bestseller so I can retire.

RuseMichael Ruse is the Lucyle T. Werkmeister Professor of Philosophy and Director of the Program in the History and Philosophy of Science at Florida State University. He has written or edited more than fifty books, including Darwinism as Religion, The Philosophy of Human Evolution, and The Darwinian Revolution.

Craig Bauer: Attacking the Zodiac Killer

While writing Unsolved! The History and Mystery of the World’s Greatest Ciphers from Ancient Egypt to Online Secret Societies, it soon became clear to me that I’d never finish if I kept stopping to try to solve the ciphers I was covering. It was hard to resist, but I simply couldn’t afford to spend months hammering away at each of the ciphers. There were simply too many of them. If I was to have any chance of meeting my deadline, I had to content myself with merely making suggestions as to how attacks could be carried out. My hope was that the book’s readers would be inspired to actually make the attacks. However, the situation changed dramatically when the book was done.

I was approached by the production company Karga Seven Pictures to join a team tasked with hunting the still unidentified serial killer who called himself the Zodiac. In the late 1960s and early 70s, the Zodiac killed at least five people and terrorized entire cities in southern California with threatening letters mailed to area newspapers. Some of these letters included unsolved ciphers. I made speculations about these ciphers in my book, but made no serious attempt at cracking them. With the book behind me, and its deadline no longer a problem, would I like to join a code team to see if we could find solutions where all others had failed? The team would be working closely with a pair of crack detectives, Sal LaBarbera and Ken Mains, so that any leads that developed could be investigated immediately. Was I willing to take on the challenge of a very cold case? Whatever the result was, it would be no secret, for our efforts would be aired as a History channel mini-series. Was I up for it? Short answer: Hell yeah!

The final code team included two researchers I had corresponded with when working on my book, Kevin Knight (University of Southern California, Information Sciences Institute) and David Oranchak (software developer and creator of Zodiac Killer Ciphers. The other members were Ryan Garlick (University of North Texas, Computer Science) and Sujith Ravi (Google software engineer).

My lips are sealed as to what happened (why ruin the suspense?), but the show premieres Tuesday November 14, 2017 at 10pm EST. It’s titled “The Hunt for the Zodiac Killer.” All I’ll say for now is that it was a rollercoaster ride. For those of you who would like to see how the story began for me, Princeton University Press is making the chapter of my book on the Zodiac killer freely available for the duration of the mini-series. It provides an excellent background for those who wish to follow the TV team’s progress.

If you find yourself inspired by the show, you can turn to other chapters of the book for more unsolved “killer ciphers,” as well challenges arising from nonviolent contexts. It was always my hope that readers would resolve some of these mysteries and I’m more confident than ever that it can be done!

BauerCraig P. Bauer is professor of mathematics at York College of Pennsylvania. He is editor in chief of the journal Cryptologia, has served as a scholar in residence at the NSA’s Center for Cryptologic History, and is the author of Secret History: The Story of Cryptology. He lives in York, Pennsylvania.

Joel Brockner: The Passion Plea

This post originally appears on the blog of Psychology Today

BrocknerIt’s tough to argue with the idea that passion is an admirable aspect of the human condition. Passionate people are engaged in life; they really care about their values and causes and being true to them. However, a big minefield of passion is when people use it to excuse or explain away unseemly behavior. We saw this during the summer of 2017 in how the White House press secretary, Sarah Huckabee Sanders, responded to the infamous expletive-laced attack of Anthony Scaramucci on his then fellow members of the Trump team, Steve Bannon and Reince Priebus. According to The New York Times, (July 27, 2017),  “Ms. Sanders said mildly that Mr. Scaramucci was simply expressing strong feelings, and that his statement made clear that ‘he’s a passionate guy and sometimes he lets that passion get the better of him.’ ” Whereas Ms. Sanders acknowledged that Mr. Scaramucci behaved badly (his passion got the better of him), her meta-message is that it was no big deal, as implied by the words “mildly” and “simply” in the quote above.

The passion plea is by no means limited to the world of politics. Executives who are seen as emotionally rough around the edges by their co-workers often defend their behavior with statements like, “I’m just being passionate,” or “I am not afraid to tell it like it is,” or, “My problem is that I care too much.”

The passion plea distorts reality by glossing over the distinction between what is said and how it is said. Executives who deliver negative feedback in a harsh tone are not just being passionate. Even when the content of the negative feedback is factual, harsh tones convey additional messages – notably a lack of dignity and respect. Almost always, there are ways to send the same strong messages or deliver the same powerful feedback in ways that do not convey a lack of dignity and respect. For instance, Mr. Scaramucci could have said something like, “Let me be as clear as possible: I have strong disagreements with Steve Bannon and Reince Priebus.” It may have been less newsworthy, but it could have gotten the same message across. Arguably, Mr. Scaramucci’s 11-day tenure as White House director of communications would have been longer had he not been so “passionate” and instead used more diplomatic language.

Similarly, executives that I coach rarely disagree when it is made evident that they could have sent the same strong negative feedback in ways that would have been easier for their co-workers to digest. Indeed, this is the essence of constructive criticism, which typically seeks to change the behavior of the person on the receiving end. Rarely are managers accused of coming on “too strong” if they deliver negative feedback in the right ways. For example, instead of saying something about people’s traits or characters (e.g., “You aren’t reliable”) it would be far better to provide feedback with reference to specific behavior (e.g., “You do not turn in your work on time”). People usually are more willing and able to respond to negative feedback about what they do rather than who they are. Adding a problem-solving approach is helpful as well, such as, “Some weeks you can be counted on to do a good job whereas other weeks not nearly as much. Why do you think that is happening, and what can we do together to ensure greater consistency in your performance?” Moreover, the feedback has to be imparted in a reasonable tone of voice, and in a context in which people on the receiving end are willing and able to take it in. For instance, one of my rules in discussing with students why they didn’t do well on an assignment is that we not talk immediately after they received the unwanted news. It is far better to have a cooling-off period in which defensiveness goes down and open-mindedness goes up.

If our goal is to alienate people or draw negative attention to ourselves then we should be strong and hard-driving, even passionate, in what we say as well as crude and inappropriate in how we say it. However, if we want to be a force for meaningful change or a positive role model, it is well within our grasp to be just as strong and hard-driving in what we say while being respectful and dignified in how we say it.

Joel Brockner is the Phillip Hettleman Professor of Business at Columbia Business School.

Two PUP Books Longlisted for the 2018 AAAS/Subaru SB&F Prizes

We are delighted that Monarchs and Milkweed by Anurag Agrawal and Welcome to the Universe by Neil DeGrasse Tyson, Michael Strauss, and J. Richard Gott have been longlisted for the AAAS/Subaru SB&F Prizes for Excellence in Science Books!

The Prizes celebrate outstanding science writing and illustration for children and young adults and are meant to encourage the writing and publishing of high-quality science books for all ages. AAAS believes that, through good science books, this generation, and the next, will have a better understanding and appreciation of science.


Welcome to the Universe

Steven S. Gubser: Thunder and Lightning from Neutron Star mergers

As of late 2015, we have a new way of probing the cosmos: gravitational radiation. Thanks to LIGO (the Laser Interferometer Gravitational-wave Observatory) and its new sibling Virgo (a similar interferometer in Italy), we can now “hear” the thumps and chirps of colliding massive objects in the universe. Not for nothing has this soundtrack been described by LIGO scientists as “the music of the cosmos.” This music is at a frequency easily discerned by human hearing, from somewhat under a hundred hertz to several hundred hertz. Moreover, gravitational radiation, like sound, is wholly different from light. It is possible for heavy dark objects like black holes to produce mighty gravitational thumps without at the same time emitting any significant amount of light. Indeed, the first observations of gravitational waves came from black hole merger events whose total power briefly exceeded the light from all stars in the known universe. But we didn’t observe any light from these events at all, because almost all their power went into gravitational radiation.

In August 2017, LIGO and Virgo observed a collision of neutron stars which did produce observable light, notably in the form of gamma rays. Think of it as cosmic thunder and lightning, where the thunder is the gravitational waves and the lightning is the gamma rays. When we see a flash of ordinary lightning, we can count a few seconds until we hear the thunder. Knowing that sound travels one mile in about five seconds, we can reckon how distant the event is. The reason this method works is that light travels much faster than sound, so we can think of the transmission of light as instantaneous for purposes of our estimate.

Things are very different for the neutron star collision, in that the event took place about 130 million light years away, but the thunder and lightning arrived on earth pretty much simultaneously. To be precise, the thunder was first: LIGO and Virgo heard a basso rumble rising to a characteristic “whoop,” and just 1.7 seconds later, the Fermi and INTEGRAL experiments observed gamma ray bursts from a source whose location was consistent with the LIGO and Virgo observations. The production of gamma rays from merging neutron stars is not a simple process, so it’s not clear to me whether we can pin that 1.7 seconds down as a delay precisely due to the astrophysical production mechanisms; but at least we can say with some confidence that the propagation time of light and gravity waves are the same to within a few seconds over 130 million light years. From a certain point of view, that amounts to one of the most precise measurements in physics: the ratio of the speed of light to the speed of gravity equals 1, correct to about 14 decimal places or better.

The whole story adds up much more easily when we remember that gravitational waves are not sound at all. In fact, they’re nothing like ordinary sound, which is a longitudinal wave in air, where individual air molecules are swept forward and backward just a little as the sound waves pass them by. Gravitational waves instead involve transverse disturbances of spacetime, where space is stretched in one direction and squeezed in another—but both of those stretch-squeeze directions are at right angles to the direction of the wave. Light has a similar transverse quality: It is made up of electric and magnetic fields, again in directions that are at right angles to the direction in which the light travels. It turns out that a deep principle underlying both Maxwell’s electromagnetism and Einstein’s general relativity forces light and gravitational waves to be transverse. This principle is called gauge symmetry, and it also guarantees that photons and gravitons are massless, which implies in turn that they travel at the same speed regardless of wavelength.

It’s possible to have transverse sound waves: For instance, shearing waves in crystals are a form of sound. They typically travel at a different speed from longitudinal sound waves. No principle of gauge symmetry forbids longitudinal sound waves, and indeed they can be directly observed, along with their transverse cousins, in ordinary materials like metals. The gauge symmetries that forbid longitudinal light waves and longitudinal gravity waves are abstract, but a useful first cut at the idea is that there is extra information in electromagnetism and in gravity, kind of like an error-correcting code. A much more modest form of symmetry is enough to characterize the behavior of ordinary sound waves: It suffices to note that air (at macroscopic scales) is a uniform medium, so that nothing changes in a volume of air if we displace all of it by a constant distance.

In short, Maxwell’s and Einstein’s theories have a feeling of being overbuilt to guarantee a constant speed of propagation. And they cannot coexist peacefully as theories unless these speeds are identical. As we continue Einstein’s hunt for a unified theory combining electromagnetism and gravity, this highly symmetrical, overbuilt quality is one of our biggest clues.

The transverse nature of gravitational waves is immediately relevant to the latest LIGO / Virgo detection. It is responsible for the existence of blind spots in each of the three detectors (LIGO Hanford, LIGO Livingston, and Virgo). It seems like blind spots would be bad, but they actually turned out to be pretty convenient: The signal at Virgo was relatively weak, indicating that the direction of the source was close to one of its blind spots. This helped localize the event, and localizing the event helped astronomers home in on it with telescopes. Gamma rays were just the first non-gravitational signal observed: the subsequent light-show from the death throes of the merging neutron stars promises to challenge and improve our understanding of the complex astrophysical processes involved. And the combination of gravitational and electromagnetic observations will surely be a driver of new discoveries in years and decades to come.


BlackSteven S. Gubser is professor of physics at Princeton University and the author of The Little Book of String TheoryFrans Pretorius is professor of physics at Princeton. They both live in Princeton, New Jersey. They are the authors of The Little Book of Black Holes.