Website Editors Note: This website is about a Single Cycle Integrative Effect Topology, the SCIET. It does what this article describes for all of nature, providing a means to identify each point in space uniquely in full relationship to all others. This is why this article is included on this website.

In 1891, when the German biologist Hans Driesch split two-cell sea urchin embryos in half, he found that each of the separated cells then gave rise to its own complete, albeit smaller, larva. Somehow, the halves “knew” to change their entire developmental program: At that stage, the blueprint for what they would become had apparently not yet been drawn out, at least not in ink.

Since then, scientists have been trying to understand what goes into making this blueprint, and how instructive it is. (Driesch himself, frustrated at his inability to come up with a solution, threw up his hands and left the field entirely.) It’s now known that some form of positional information makes genes variously switch on and off throughout the embryo, giving cells distinct identities based on their location. But the signals carrying that information seem to fluctuate wildly and chaotically — the opposite of what you might expect for an important guiding influence.

“The [embryo] is a noisy environment,” said Robert Brewster, a systems biologist at the University of Massachusetts Medical School. “But somehow it comes together to give you a reproducible, crisp body plan.”

The same precision and reproducibility emerge from a sea of noise again and again in a range of cellular processes. That mounting evidence is leading some biologists to a bold hypothesis: that where information is concerned, cells might often find solutions to life’s challenges that are not just good but optimal — that cells extract as much useful information from their complex surroundings as is theoretically possible. Questions about optimal decoding, according to Aleksandra Walczak, a biophysicist at the École Normale Supérieure in Paris, “are everywhere in biology.”

Biologists haven’t traditionally cast analyses of living systems as optimization problems because the complexity of those systems makes them hard to quantify, and because it can be difficult to discern what would be getting optimized. Moreover, while evolutionary theory suggests that evolving systems can improve over time, nothing guarantees that they should be driven to an optimal level.

Yet when researchers have been able to appropriately determine what cells are doing, many have been surprised to see clear indications of optimization. Hints have turned up in how the brain responds to external stimuli and how microbes respond to chemicals in their environments. Now some of the best evidence has emerged from a new study of fly larva development, reported recently in Cell.

Cells That Understand Statistics

For decades, scientists have been studying fruit fly larvae for clues about how development unfolds. Some details became apparent early on: A cascade of genetic signals establishes a pattern along the larva’s head-to-tail axis. Signaling molecules called morphogens then diffuse through the embryonic tissues, eventually defining the formation of body parts.

Particularly important in the fly are four “gap” genes, which are expressed separately in broad, overlapping domains along the axis. The proteins they make in turn help regulate the expression of “pair-rule” genes, which create an extremely precise, periodic striped pattern along the embryo. The stripes establish the groundwork for the later division of the body into segments.

How cells make sense of these diffusion gradients has always been a mystery. The widespread assumption was that after being pointed in roughly the right direction (so to speak) by the protein levels, cells would continuously monitor their changing surroundings and make small corrective adjustments as development proceeded, locking in on their planned identity relatively late. That model harks back to the “developmental landscape” proposed by Conrad Waddington in 1956. He likened the process of a cell homing in on its fate to a ball rolling down a series of ever-steepening valleys and forked paths. Cells had to acquire more and more information to refine their positional knowledge over time — as if zeroing in on where and what they were through “the 20 questions game,” according to Jané Kondev, a physicist at Brandeis University.

Such a system could be accident prone, however: Some cells would inevitably take the wrong paths and be unable to get back on track. In contrast, comparisons of fly embryos revealed that the placement of pair-rule stripes was incredibly precise, to within 1 percent of the embryo’s length — that is, to single-cell accuracy.

That prompted a group at Princeton University, led by the biophysicists Thomas Gregor and William Bialek,to suspect something else: that the cells could instead get all the information they needed to define the positions of pair-rule stripes from the expression levels of the gap genes alone, even though those are not periodic and therefore not an obvious source for such precise instructions.

And that’s just what they found.

Over the course of 12 years, they measured morphogen and gap-gene protein concentrations, cell by cell, from one embryo to the next, to determine how all four gap genes were most likely to be expressed at every position along the head-to-tail axis. From those probability distributions, they built a “dictionary,” or decoder — an explicit map that could spit out a probabilistic estimate of a cell’s position based on its gap-gene protein concentration levels.

Around five years ago, the researchers — including Mariela Petkova, who started the measurement work as an undergraduate at Princeton (and is currently pursuing a doctorate in biophysics at Harvard University), and Gašper Tkačik, now at the Institute of Science and Technology Austria — determined this mapping by assuming it worked like what’s known as an optimal Bayesian decoder (that is, the decoder used Bayes’ rule for inferring the likelihood of an event from prior conditional probabilities). The Bayesian framework allowed them to flip the “unknowns,” the conditions of probability: Their measurements of gap gene expression, given position, could be used to generate a “best guess” of position, given only gap gene expression.

The team found that the fluctuations of the four gap genes could indeed be used to predict the locations of cells with single-cell precision. No less than maximal information about all four would do, however: When the activity of only two or three gap genes was provided, the decoder’s location predictions were not nearly so accurate. Versions of the decoder that used less of the information from all four gap genes — that, for instance, responded only to whether each gene was on or off — made worse predictions, too.

According to Walczak, “No one has ever measured or shown how well reading out the concentration of these molecular gradients … actually pinpoints a specific position along the axis.”

Now they had: Even given the limited number of molecules and underlying noise of the system, the varying concentrations of the gap genes was sufficient to differentiate two neighboring cells in the head-to-tail axis — and the rest of the gene network seemed to be transmitting that information optimally.

“But the question always remained open: Does the biology actually care?” Gregor said. “Or is this just something that we measure?” Could the regulatory regions of DNA that responded to the gap genes really be wired up in such a way that they could decode the positional information those genes contained?

The biophysicists teamed up with the Nobel Prize-winning biologist Eric Wieschaus to test whether the cells were actually making use of the information potentially at their disposal. They created mutant embryos by modifying the gradients of morphogens in the very young fly embryos, which in turn altered the expression patterns of the gap genes and ultimately caused pair-rule stripes to shift, disappear, get duplicated or have fuzzy edges. Even so, the researchers found that their decoder could predict the changes in mutated pair-rule expression with surprising accuracy. “They show that the map is broken in mutants, but in a way that the decoder predicts,” Walczak said.

“You could imagine that if it was getting information from other sources, you couldn’t trick [the cells] like that,” Brewster added. “Your decoder would fail.”

These findings represent “a signpost,” according to Kondev, who was not involved with the study. They suggest that there’s “some physical reality” to the inferred decoder, he said. “Through evolution, these cells have figured out how to implement Bayes’ trick using regulatory DNA.”

How the cells do it remains a mystery. Right now, “the whole thing is kind of wonderful and magical,” said John Reinitz, a systems biologist at the University of Chicago.

Even so, the work provides a new way of thinking about early development, gene regulation and, perhaps, evolution in general.

A Steeper Landscape

The findings provide a fresh perspective on Waddington’s idea of a developmental landscape. According to Gregor, their work indicates that there’s no need for 20 questions or a gradual refinement of knowledge after all. The landscape “is steep from the beginning,” he said. All the information is already there.

“Natural selection [seems to be] pushing the system hard enough so that it … reaches a point where the cells are performing at the limit of what physics allows,” said Manuel Razo-Mejia, a graduate student at the California Institute of Technology.

It’s possible that the high performance in this case is a fluke: Since fruit fly embryos develop very quickly, perhaps in their case “evolution has found this optimal solution because of that pressure to do everything very rapidly,” said James Briscoe, a biologist at the Francis Crick Institute in London who did not participate in this study. To really cement whether this is something more general, then, researchers will have to test the decoder in other species, including those that develop more slowly.

Even so, these results set up intriguing new questions to ask about the often-enigmatic regulatory elements. Scientists don’t have a solid grasp of how regulatory DNA codes for the control of other genes’ activities. The team’s findings suggest that this involves an optimal Bayesian decoder, which allows the regulatory elements to respond to very subtle changes in combined gap gene expression. “We can ask the question, what is it about regulatory DNA that encodes the decoder?” Kondev said.

And “what about it makes it do this optimal decoding?” he added. “That’s a question we could not have asked before this study.”

“That’s really what this work sets up as the next challenge in the field,” Briscoe said. Besides, there may be many ways of implementing such a decoder at the molecular level, meaning that this idea could apply to other systems as well. In fact, hints of it have been uncoveredin the development of the neural tube in vertebrates, the precursor of their central nervous system — which would call for a very different underlying mechanism.

Moreover, if these regulatory regions need to perform an optimal decoding function, that potentially limits how they can evolve — and in turn, how an entire organism can evolve. “We have this one example … which is the life that evolved on this planet,” Kondev said, and because of that, the important constraints on what life can be are unknown. Finding that cells show Bayesian behavior could be a hint that processing information effectively may be “a general principle that makes a bunch of atoms stuck together loosely behave like the thing that we think is life.”

But right now, it is still only a hint. Although it would be “kind of a physicist’s dream,” Gregor said, “we are far from really having proof for this.”

From Wires Under Oceans to Neurons in the Brain

The concept of information optimization is rooted in electrical engineering: Experts originally wanted to understand how best to encode and then decode sound to allow people to talk on the telephone via transoceanic cables. That goal later turned into a broader consideration of how to transmit information optimally through a channel. It wasn’t much of a leap to apply this framework to the brain’s sensory systems and how they measured, encoded and decoded inputs to produce a response.

Now some experts are trying to think about all kinds of “sensory systems” in this way: Razo-Mejia, for instance, has studied how optimally bacteria sense and process chemicals in their environment, and how that might affect their fitness. Meanwhile, Walczak and her colleagues have been asking what a “good decoding strategy” might look like in the adaptive immune system, which has to recognize and respond to a massive repertoire of intruders.

“I don’t think optimization is an aesthetic or philosophical idea. It’s a very concrete idea,” Bialek said. “Optimization principles have time and again pointed to interesting things to measure.” Whether or not they are correct, he considers them productive to think about.

“Of course, the difficulty is that in many other systems, the property being decoded is more difficult than one-dimensional position [along the embryo’s axis],” Walczak said. “The problem is harder to define.”

That’s what made the system Bialek and his colleagues studied so tantalizing. “There aren’t many examples in biology where a high-level idea, like information in this case, leads to a mathematical formula” that is then testable in experiments on living cells, Kondev said.

It’s this marriage of theory and experiment that excites Bialek. He hopes to see the approach continue to guide work in other contexts. “What’s not clear,” he said, “is whether the observation [of optimization] is a curiosity that arises in a few corners, or whether there’s something general about it.”

If the latter does prove to be the case, “then that’s very striking,” Briscoe said. “The ability for evolution to find these really efficient ways of doing things would be an incredible finding.”

Kondev agreed. “As a physicist, you hope that the phenomenon of life is not just about the specific chemistry and DNA and molecules that make living things on planet Earth — that it’s broader,” he said. “What is that broader thing? I don’t know. But maybe this is lifting a little bit of the veil off that mystery.”

Correction added on March 15: The text was updated to acknowledge the contributions of Mariela Petkova and Gašper Tkačik.

In the latest campaign to reconcile Einstein’s theory of gravity with quantum mechanics, many physicists are studying how a higher dimensional space that includes gravity arises like a hologram from a lower dimensional particle theory.

Physicists have discovered a jewel-shaped geometric object that challenges the notion that space and time are fundamental constituents of nature.

The new object dramatically simplifies calculations of particle interactions and challenges the idea that space and time are fundamental components of reality.

“This is completely new and very much simpler than anything that has been done before,” said Andrew Hodges, a mathematical physicist at Oxford University who has been following the work.

The revelation that particle interactions, the most basic events in nature, may be consequences of geometry significantly advances a decades-long effort to reformulate quantum field theory, the body of laws describing elementary particles and their interactions. Interactions that were previously calculated with mathematical formulas thousands of terms long can now be described by computing the volume of the corresponding jewel-like “amplituhedron,” which yields an equivalent one-term expression.

“The degree of efficiency is mind-boggling,” said Jacob Bourjaily, a theoretical physicist at Harvard University and one of the researchers who developed the new idea. “You can easily do, on paper, computations that were infeasible even with a computer before.”

The new geometric version of quantum field theory could also facilitate the search for a theory of quantum gravity that would seamlessly connect the large- and small-scale pictures of the universe. Attempts thus far to incorporate gravity into the laws of physics at the quantum scale have run up against nonsensical infinities and deep paradoxes. The amplituhedron, or a similar geometric object, could help by removing two deeply rooted principles of physics: locality and unitarity.

“Both are hard-wired in the usual way we think about things,” said Nima Arkani-Hamed, a professor of physics at the Institute for Advanced Study in Princeton, N.J., and the lead author of the new work, which he is presenting in talks and in a forthcoming paper. “Both are suspect.”

Locality is the notion that particles can interact only from adjoining positions in space and time. And unitarity holds that the probabilities of all possible outcomes of a quantum mechanical interaction must add up to one. The concepts are the central pillars of quantum field theory in its original form, but in certain situations involving gravity, both break down, suggesting neither is a fundamental aspect of nature.

In keeping with this idea, the new geometric approach to particle interactions removes locality and unitarity from its starting assumptions. The amplituhedron is not built out of space-time and probabilities; these properties merely arise as consequences of the jewel’s geometry. The usual picture of space and time, and particles moving around in them, is a construct.

“It’s a better formulation that makes you think about everything in a completely different way,” said David Skinner, a theoretical physicist at Cambridge University.

The amplituhedron itself does not describe gravity. But Arkani-Hamed and his collaborators think there might be a related geometric object that does. Its properties would make it clear why particles appear to exist, and why they appear to move in three dimensions of space and to change over time.

Because “we know that ultimately, we need to find a theory that doesn’t have” unitarity and locality, Bourjaily said, “it’s a starting point to ultimately describing a quantum theory of gravity.”

Clunky Machinery

The amplituhedron looks like an intricate, multifaceted jewel in higher dimensions. Encoded in its volume are the most basic features of reality that can be calculated, “scattering amplitudes,” which represent the likelihood that a certain set of particles will turn into certain other particles upon colliding. These numbers are what particle physicists calculate and test to high precision at particle accelerators like the Large Hadron Collider in Switzerland.

The 60-year-old method for calculating scattering amplitudes — a major innovation at the time — was pioneered by the Nobel Prize-winning physicist Richard Feynman. He sketched line drawings of all the ways a scattering process could occur and then summed the likelihoods of the different drawings. The simplest Feynman diagrams look like trees: The particles involved in a collision come together like roots, and the particles that result shoot out like branches. More complicated diagrams have loops, where colliding particles turn into unobservable “virtual particles” that interact with each other before branching out as real final products. There are diagrams with one loop, two loops, three loops and so on — increasingly baroque iterations of the scattering process that contribute progressively less to its total amplitude. Virtual particles are never observed in nature, but they were considered mathematically necessary for unitarity — the requirement that probabilities sum to one.

“The number of Feynman diagrams is so explosively large that even computations of really simple processes weren’t done until the age of computers,” Bourjaily said. A seemingly simple event, such as two subatomic particles called gluons colliding to produce four less energetic gluons (which happens billions of times a second during collisions at the Large Hadron Collider), involves 220 diagrams, which collectively contribute thousands of terms to the calculation of the scattering amplitude.

In 1986, it became apparent that Feynman’s apparatus was a Rube Goldberg machine.

To prepare for the construction of the Superconducting Super Collider in Texas (a project that was later canceled), theorists wanted to calculate the scattering amplitudes of known particle interactions to establish a background against which interesting or exotic signals would stand out. But even 2-gluon to 4-gluon processes were so complex, a group of physicists had written two years earlier, “that they may not be evaluated in the foreseeable future.”

Stephen Parke and Tomasz Taylor, theorists at Fermi National Accelerator Laboratory in Illinois, took that statement as a challenge. Using a few mathematical tricks, they managed to simplify the 2-gluon to 4-gluon amplitude calculation from several billion terms to a 9-page-long formula, which a 1980s supercomputer could handle. Then, based on a pattern they observed in the scattering amplitudes of other gluon interactions, Parke and Taylor guessed a simple one-term expression for the amplitude. It was, the computer verified, equivalent to the 9-page formula. In other words, the traditional machinery of quantum field theory, involving hundreds of Feynman diagrams worth thousands of mathematical terms, was obfuscating something much simpler. As Bourjaily put it: “Why are you summing up millions of things when the answer is just one function?”

“We knew at the time that we had an important result,” Parke said. “We knew it instantly. But what to do with it?”

The Amplituhedron

The message of Parke and Taylor’s single-term result took decades to interpret. “That one-term, beautiful little function was like a beacon for the next 30 years,” Bourjaily said. It “really started this revolution.”

In the mid-2000s, more patterns emerged in the scattering amplitudes of particle interactions, repeatedly hinting at an underlying, coherent mathematical structure behind quantum field theory. Most important was a set of formulas called the BCFW recursion relations, named for Ruth Britto, Freddy Cachazo, Bo Feng and Edward Witten. Instead of describing scattering processes in terms of familiar variables like position and time and depicting them in thousands of Feynman diagrams, the BCFW relations are best couched in terms of strange variables called “twistors,” and particle interactions can be captured in a handful of associated twistor diagrams. The relations gained rapid adoption as tools for computing scattering amplitudes relevant to experiments, such as collisions at the Large Hadron Collider. But their simplicity was mysterious.

“The terms in these BCFW relations were coming from a different world, and we wanted to understand what that world was,” Arkani-Hamed said. “That’s what drew me into the subject five years ago.”

With the help of leading mathematicians such as Pierre Deligne, Arkani-Hamed and his collaborators discovered that the recursion relations and associated twistor diagrams corresponded to a well-known geometric object. In fact, as detailed in a paper posted to arXiv.org in December by Arkani-Hamed, Bourjaily, Cachazo, Alexander Goncharov, Alexander Postnikov and Jaroslav Trnka, the twistor diagrams gave instructions for calculating the volume of pieces of this object, called the positive Grassmannian.

 

A sketch of the amplituhedron representing an 8-gluon particle interaction. Using Feynman diagrams, the same calculation would take roughly 500 pages of algebra.
Nima Arkani-Hamed

Named for Hermann Grassmann, a 19th-century German linguist and mathematician who studied its properties, “the positive Grassmannian is the slightly more grown-up cousin of the inside of a triangle,” Arkani-Hamed explained. Just as the inside of a triangle is a region in a two-dimensional space bounded by intersecting lines, the simplest case of the positive Grassmannian is a region in an N-dimensional space bounded by intersecting planes. (N is the number of particles involved in a scattering process.)

It was a geometric representation of real particle data, such as the likelihood that two colliding gluons will turn into four gluons. But something was still missing.

The physicists hoped that the amplitude of a scattering process would emerge purely and inevitably from geometry, but locality and unitarity were dictating which pieces of the positive Grassmannian to add together to get it. They wondered whether the amplitude was “the answer to some particular mathematical question,” said Trnka, a post-doctoral researcher at the California Institute of Technology. “And it is,” he said.

Arkani-Hamed and Trnka discovered that the scattering amplitude equals the volume of a brand-new mathematical object — the amplituhedron. The details of a particular scattering process dictate the dimensionality and facets of the corresponding amplituhedron. The pieces of the positive Grassmannian that were being calculated with twistor diagrams and then added together by hand were building blocks that fit together inside this jewel, just as triangles fit together to form a polygon.

Like the twistor diagrams, the Feynman diagrams are another way of computing the volume of the amplituhedron piece by piece, but they are much less efficient. “They are local and unitary in space-time, but they are not necessarily very convenient or well-adapted to the shape of this jewel itself,” Skinner said. “Using Feynman diagrams is like taking a Ming vase and smashing it on the floor.”

Arkani-Hamed and Trnka have been able to calculate the volume of the amplituhedron directly in some cases, without using twistor diagrams to compute the volumes of its pieces. They have also found a “master amplituhedron” with an infinite number of facets, analogous to a circle in 2-D, which has an infinite number of sides. Its volume represents, in theory, the total amplitude of all physical processes. Lower-dimensional amplituhedra, which correspond to interactions between finite numbers of particles, live on the faces of this master structure.

“They are very powerful calculational techniques, but they are also incredibly suggestive,” Skinner said. “They suggest that thinking in terms of space-time was not the right way of going about this.”

Quest for Quantum Gravity

The seemingly irreconcilable conflict between gravity and quantum field theory enters crisis mode in black holes. Black holes pack a huge amount of mass into an extremely small space, making gravity a major player at the quantum scale, where it can usually be ignored. Inevitably, either locality or unitarity is the source of the conflict.

“We have indications that both ideas have got to go,” Arkani-Hamed said. “They can’t be fundamental features of the next description,” such as a theory of quantum gravity.

String theory, a framework that treats particles as invisibly small, vibrating strings, is one candidate for a theory of quantum gravity that seems to hold up in black hole situations, but its relationship to reality is unproven — or at least confusing. Recently, a strange duality has been found between string theory and quantum field theory, indicating that the former (which includes gravity) is mathematically equivalent to the latter (which does not) when the two theories describe the same event as if it is taking place in different numbers of dimensions. No one knows quite what to make of this discovery. But the new amplituhedron research suggests space-time, and therefore dimensions, may be illusory anyway.

“We can’t rely on the usual familiar quantum mechanical space-time pictures of describing physics,” Arkani-Hamed said. “We have to learn new ways of talking about it. This work is a baby step in that direction.”

Even without unitarity and locality, the amplituhedron formulation of quantum field theory does not yet incorporate gravity. But researchers are working on it. They say scattering processes that include gravity particles may be possible to describe with the amplituhedron, or with a similar geometric object. “It might be closely related but slightly different and harder to find,” Skinner said.

Physicists must also prove that the new geometric formulation applies to the exact particles that are known to exist in the universe, rather than to the idealized quantum field theory they used to develop it, called maximally supersymmetric Yang-Mills theory. This model, which includes a “superpartner” particle for every known particle and treats space-time as flat, “just happens to be the simplest test case for these new tools,” Bourjaily said. “The way to generalize these new tools to [other] theories is understood.”

Beyond making calculations easier or possibly leading the way to quantum gravity, the discovery of the amplituhedron could cause an even more profound shift, Arkani-Hamed said. That is, giving up space and time as fundamental constituents of nature and figuring out how the Big Bang and cosmological evolution of the universe arose out of pure geometry.

“In a sense, we would see that change arises from the structure of the object,” he said. “But it’s not from the object changing. The object is basically timeless.”

While more work is needed, many theoretical physicists are paying close attention to the new ideas.

The work is “very unexpected from several points of view,” said Witten, a theoretical physicist at the Institute for Advanced Study. “The field is still developing very fast, and it is difficult to guess what will happen or what the lessons will turn out to be.”

Note: This article was updated on December 10, 2013, to include a link to the first in a series of papers on the amplituhedron.

This article was reprinted on Wired.com.

About a year ago, the theoretical chemist Salvatore Torquatomet with the number theorist Matthew de Courcy-Irelandto explain that he had done something highly unorthodox with prime numbers, those positive integers that are divisible only by 1 and themselves.

A professor of chemistry at Princeton University, Torquato normally studies patterns in the structure of physical systems, such as the arrangement of particles in crystals, colloids and even, in one of his better-known results, a pack of M&Ms. In his field, a standard way to deduce structure is to diffract X-rays off things. When hit with X-rays, disorderly molecules in liquids or glass scatter them every which way, creating no discernible pattern. But the symmetrically arranged atoms in a crystal reflect light waves in sync, producing periodic bright spots where reflected waves constructively interfere. The spacing of these bright spots, known as “Bragg peaks” after the father-and-son crystallographers who pioneered diffraction in the 1910s, reveals the organization of the scattering objects.

By Natalie Wolchover  at Quanta Magazine

Torquato told de Courcy-Ireland, a final-year graduate student at Princeton who had been recommended by another mathematician, that a year before, on a hunch, he had performed diffraction on sequences of prime numbers. Hoping to highlight the elusive order in the distribution of the primes, he and his student Ge Zhanghad modeled them as a one-dimensional sequence of particles — essentially, little spheres that can scatter light. In computer experiments, they bounced light off long prime sequences, such as the million-or-so primes starting from 10,000,000,019. (They found that this “Goldilocks interval” contains enough primes to produce a strong signal without their getting too sparse to reveal an interference pattern.)

It wasn’t clear what kind of pattern would emerge or if there would be one at all. Primes, the indivisible building blocks of all natural numbers, skitter erratically up the number line like the bounces of a skipping rock, stirring up deep questions in their wake. “They are in many ways pretty hard to tell apart from a random sequence of numbers,” de Courcy-Ireland said. Although mathematicians have uncovered many rules over the centuries about the primes’ spacings, “it’s very difficult to find any clear pattern, so we just think of them as ‘something like random.’”

But in three new papers — one by Torquato, Zhang and the computational chemist Fausto Martellithat was published in the Journal of Physics Ain February, and twoothersco-authored with de Courcy-Ireland that have not yet been peer-reviewed — the researchers report that the primes, like crystals and unlike liquids, produce a diffraction pattern.

“What’s beautiful about this is it gives us a crystallographer’s view of what the primes look like,” said Henry Cohn, a mathematician at Microsoft Research New England and the Massachusetts Institute of Technology.

The resulting pattern of Bragg peaks is not quite like anything seen before, implying that the primes, as a physical system, “are a completely new category of structures,” Torquato said. The Princeton researchers have dubbed the fractal-like pattern “effective limit-periodicity.”

It consists of a periodic sequence of bright peaks, which reflect the most common spacings of primes: All of them (except 2) are at odd-integer positions on the number line, multiples of two apart. Those brightest bright peaks are interspersed at regular intervals with less bright peaks, reflecting primes that are separated by multiples of six on the number line. These have dimmer peaks between them corresponding to farther-apart pairs of primes, and so on in an infinitely dense nesting of Bragg peaks.

Dense Bragg peaks have been seen before, in the diffraction patterns of quasicrystals, those strange materials discovered in the 1980s with symmetric but nonrepeating atomic arrangements. In the primes’ case, though, distances between peaks are fractions of one another, unlike quasicrystals’ irrationally spaced Bragg peaks. “The primes are actually suggesting a completely different state of particle positions that are like quasicrystals but are not like quasicrystals,” Torquato said.

In computer experiments, theoretical chemists have diffracted light off long sequences of prime numbers to reveal the hidden order underlying their seemingly erratic distribution. The primes produce a fractal-like diffraction pattern that’s similar, yet different, to that of quasicrystals.

According to numerous number theorists interviewed, there’s no reason to expect the Princeton team’s findings to trigger advances in number theory. Most of the relevant mathematics has been seen before in other guises. Indeed, when Torquato showed his plots and formulas to de Courcy-Ireland last spring (at the suggestion of Cohn), the young mathematician quickly saw that the prime diffraction pattern “can be explained in terms of almost universally accepted conjectures in number theory.”

It was the first of many meetings between the two at the Institute for Advanced Study in Princeton, N.J., where Torquato was spending a sabbatical. The chemist told de Courcy-Ireland that he could use his formula to predict the frequency of “twin primes,” which are pairs of primes separated by two, like 17 and 19. The mathematician replied that Torquato could in fact predict all other separations as well. The formula for the Bragg peaks was mathematically equivalent to the Hardy-Littlewood k-tuple conjecture, a powerful statement made by the English mathematicians Godfrey Hardy and John Littlewood in 1923 about which “constellations” of primes can exist. One rule forbids three consecutive odd-numbered primes after {3, 5, 7}, since one in the set will always be divisible by three, as in {7, 9, 11}. This rule illustrates why the second-brightest peaks in the primes’ diffraction pattern come from pairs of primes separated by six, rather than four.

Hardy and Littlewood’s conjecture further specified how often all the allowed prime constellations will occur along the number line. Even the simplest case of Hardy-Littlewood, the “twin primes conjecture,” although it has seen a burst of modern progress, remains unproved. Because prime diffraction essentially reformulates it, experts say it’s highly unlikely to lead to a proof of Hardy-Littlewood, or for that matter the famous Riemann hypothesis, an 1859 formula linking the primes’ distribution to the “critical zeros” of the Riemann zeta function.

The findings resonate, however, in a relatively young research area called “aperiodic order,” essentially the study of nonrepeating patterns, which lies at the intersection of crystallography, dynamical systems, harmonic analysis and discrete geometry, and grew after the discovery of quasicrystals. “Techniques that were originally developed for understanding crystals … became vastly diversified with the discovery of quasicrystals,” said Marjorie Senechal, a mathematical crystallographer at Smith College. “People began to realize they suddenly had to understand much, much more than just the simple straightforward periodic diffraction,” she said, “and this has become a whole field, aperiodic order. Uniting this with number theory is just extremely exciting.”

The primes’ pattern resembles a kind of aperiodic order known since at least the 1950s called limit periodicity, “while adding a surprising twist,” Cohn said. In true limit-periodic systems, periodic spacings are nested in an infinite hierarchy, so that within any interval, the system contains parts of patterns that repeat only in a larger interval. An example is the tessellation of a strange, multipronged shape called the Taylor-Socolar tile, discovered by the Australian amateur mathematician Joan Taylor in the 1990s, and analyzed in detail with Joshua Socolarof Duke University in 2010. According to Socolar, computer experiments indicate that limit-periodic phases of matter should be able to form in nature, and calculations suggest such systems might have unusual properties. No one guessed a connection to the primes. They are “effectively” limit periodic — a new kind of order — because the synchronicities in their spacings only hold statistically across the whole system.

For his part, de Courcy-Ireland wants to better understand the “Goldilocks” scale at which effective limit-periodicity emerges in the primes. In 1976, Patrick Gallagher of Columbia University showedthat the primes’ spacings look random over short intervals; longer strips are needed for their pattern to emerge. In the new diffraction studies, de Courcy-Ireland and his chemist collaborators analyzed a quantity called an “order metric” that controls the presence of the limit-periodic pattern. “You can identify how long the interval has to be before you start seeing this quantity grow,” he said. He is intrigued that this same interval length also shows up in a different prime number rule called Maier’s theorem. But it’s too soon to tell whether this thread will lead anywhere.

The main advantage of the prime diffraction pattern, said Jonathan Keatingof the University of Bristol, is that “it is evocative” and “makes a connection with different ways of thinking.” But the esteemed number theorist Andrew Granvilleof the University of Montreal called Torquato and company’s work “pretentious” and “just a regurgitation of known ideas.”

Torquato isn’t especially concerned about how his work will be perceived by number theorists. He has found a way to glimpse the pattern of the primes. “I actually think it’s stunning,” he said. “It’s a shock.”