Origami Patterns Solve a Major Physics Riddle
Ibrahim Rayintakath for Quanta Magazine
Introduction
The amplituhedron is a geometric shape with an almost mystical quality: Compute its volume, and you get the answer to a central calculation in physics about how particles interact.
Now, a young mathematician at Cornell University named Pavel (Pasha) Galashin has found that the amplituhedron is also mysteriously connected to another completely unrelated subject: origami, the art of paper folding. In a proof posted in October 2024, he showed that patterns that arise in origami can be translated into a set of points that together form the amplituhedron. Somehow, the way paper folds and the way particles collide produce the same geometric shape.
“Pasha has done some brilliant work related to the amplituhedron before,” said Nima Arkani-Hamed, a physicist at the Institute for Advanced Study who introduced the amplituhedron in 2013 with his graduate student at the time, Jaroslav Trnka. “But this is next-level stuff for me.”
By drawing on this new link to origami, Galashin was also able to resolve an open conjecture about the amplituhedron, one that physicists had long assumed to be true but hadn’t been able to rigorously prove: that the shape really can be cut up into simpler building blocks that correspond to the calculations physicists want to make. In other words, the pieces of the amplituhedron really do fit together the way they’re supposed to.
The result doesn’t just build a bridge between two seemingly disparate areas of study. Galashin and other mathematicians are already exploring what else that bridge can tell them. They’re using it to better understand the amplituhedron — and to answer other questions in a far broader range of settings.
Explosive Computations
Physicists want to predict what will happen when fundamental particles interact. Say two subatomic particles called gluons collide. They might bounce off each other unchanged, or transform into a set of four gluons, or do something else entirely. Each outcome occurs with a certain probability, which is represented by a mathematical expression called a scattering amplitude.
For decades, physicists calculated scattering amplitudes in one of two ways. The first used Feynman diagrams, squiggly-line drawings that describe how particles move and interact. Each diagram represents a mathematical computation; by adding together the computations corresponding to different Feynman diagrams, you can calculate a given scattering amplitude. But as the number of particles in a collision increases, the number of Feynman diagrams you need grows explosively. Things quickly get out of hand: Computing the scattering amplitudes of relatively simple events can require adding thousands or even millions of terms.
The second method, introduced in the early 2000s, is called Britto-Cachazo-Feng-Witten (BCFW) recursion. It breaks up complex particle interactions into smaller, simpler interactions that are easier to study. You can calculate amplitudes for these simpler interactions and keep track of them using collections of vertices and edges called graphs. These graphs tell you how to stitch the simpler interactions back together in order to compute the scattering amplitude of the original collision.
BCFW recursion requires less work than Feynman diagrams. Instead of adding up millions of terms, you might only need to add up hundreds. But both methods have the same problem: The final answer is often much simpler than the extensive computations it takes to get there, with many terms canceling out in the end.
Then, in 2013, Arkani-Hamed and Trnka made a surprising discovery: that the complicated math of particle collisions is actually geometry in disguise.
Saved by Geometry
In the early 2000s, Alexander Postnikov, a mathematician at the Massachusetts Institute of Technology, was studying a geometric object known as the positive Grassmannian.
The positive Grassmannian, which has been a subject of mathematical interest since the 1930s, is built in a highly abstract way. First, take an n-dimensional space and consider all the planes of some given, smaller dimension that live inside it. For example, inside the three-dimensional space we inhabit, you can find infinitely many flat two-dimensional planes that spread out in every direction.
Each plane — essentially a slice of the larger n-dimensional space — can be defined by an array of numbers called a matrix. You can compute certain values from this matrix, called minors, that tell you about properties of the plane.
Now consider only those planes in your space whose minors are all positive. The collection of all such special “positive” planes gives you a complicated geometric space — the positive Grassmannian.
To understand the positive Grassmannian’s rich internal structure, mathematicians divvy it up into different regions, so that each region consists of an assortment of planes that share certain patterns. Postnikov, hoping to make this task easier, came up with a way to keep track of the different regions and how they fit together. He invented what he called plabic (short for “planar bicolored”) graphs — networks of black and white vertices connected by edges, drawn so that no edges cross. Each plabic graph captured one region of the positive Grassmannian, giving mathematicians a visual language for what would otherwise be defined by dense algebraic formulas.
Nearly a decade after Postnikov introduced his plabic graphs, Arkani-Hamed and Trnka were trying to calculate the scattering amplitudes of various particle collisions. As they grappled with their BCFW recursion formulas, they noticed something uncanny. The graphs they were using to keep track of their calculations looked just like Postnikov’s plabic graphs. Curious, they drove up to MIT to meet him.
“At lunch we said, ‘It’s weird, we’re seeing exactly the same thing,’” Arkani-Hamed recalled.
They were right. To calculate the scattering amplitude for a collision of n particles, physicists would have to add up many BCFW terms — and each of those terms corresponded to a region of the positive Grassmannian in n dimensions.
Arkani-Hamed and Trnka realized that this geometric connection might make it easier to compute scattering amplitudes. Using data about their particle collision — the momenta of the particles, for instance — they defined a lower-dimensional shadow of the positive Grassmannian. The total volume of this shadow was equal to the scattering amplitude.
And so the amplituhedron was born.
An illustration of the amplituhedron corresponding to a particle collision involving eight gluons.
Nima Arkani-Hamed
That was only the beginning of the story. Physicists and mathematicians wanted to confirm, for instance, that the same plabic graphs that defined regions of the positive Grassmannian could also define pieces of the amplituhedron — and that those pieces would have no gaps or overlaps, perfectly fitting together to encompass the shape’s exact volume. This hope came to be known as the triangulation conjecture: Could the amplituhedron be cleanly triangulated, or subdivided, into simpler building blocks?
Proving this would cement Arkani-Hamed and Trnka’s vision: that the complicated BCFW formulas that produced a particle collision’s scattering amplitude (albeit inefficiently) could be understood as the sum of the volumes of the amplituhedron’s building blocks.
This was no easy task. For one thing, from the get-go it was clear there were really two amplituhedra. The first was defined in momentum-twistor coordinates — a clever mathematical relabeling that made the shape easier to work with because it related naturally to the positive Grassmannian and Postnikov’s plabic graphs. Mathematicians were able to prove the triangulation conjecture for this version of the amplituhedron in 2021.
The other version, known as the momentum amplituhedron, was instead defined directly in terms of the momenta of colliding particles. Physicists cared more about this second version, because it spoke the same language as real particle collisions and scattering experiments. But it was also harder to describe mathematically. As a result, the triangulation conjecture remained wide open.
If triangulation were to fail for the momentum amplituhedron, then it would mean that the amplituhedron was not the right way to make sense of BCFW formulas for computing scattering amplitudes.
For more than a decade, the uncertainty lingered — until the study of paper folds began to suggest a way forward.
Finding Bigfoot
Pavel Galashin didn’t set out to study either origami or the amplituhedron. In 2018, as one of Postnikov’s graduate students, he and a colleague had just proved an intriguing link between the positive Grassmannian and the Ising model, which is used to study the behavior of systems like ferromagnets. Galashin was now trying to understand a celebrated proof about the Ising model — in particular, about special symmetries it exhibited — in terms of the positive Grassmannian.
While working through the proof — a project he intermittently returned to over the next few years — Galashin encountered a couple of intriguing papers where researchers used other kinds of diagrams to make the geometry more tractable: origami crease patterns. These are diagrams of lines that tell you where to fold paper to make, say, a crane or frog.
This crease pattern will produce a swan.
It might seem strange for origami to crop up here. But over the years, the mathematics of origami has turned out to be surprisingly deep. Problems about origami — such as whether a given crease pattern will produce a shape that you can flatten without tearing — are computationally hard to solve. And it’s now known that origami can be used to perform all sorts of computations.
In 2023, while probing what origami was doing in papers about the Ising model, Galashin came across a question that caught his attention. Say you only have information about a crease pattern’s outer boundary — the border of the paper, which the creases divide into various line segments. In particular, say you only have information about how those line segments are situated in space before and after folding. Can you always find a complete crease pattern that both satisfies those constraints and produces an origami shape that can flatten properly? Mathematicians had conjectured that the answer was yes, but no one could prove it.
Galashin found the conjecture striking, because in his usual area of research, which deals with the positive Grassmannian, examining the boundary of an object is a common way to gain information about it.
Jaroslav Trnka (left) and Nima Arkani-Hamed introduced the amplituhedron to make it easier to perform important calculations in particle physics.
Courtesy of Jaroslav Trnka/UC Davis Dateline; Maximilien Brice Julien-Marius Ordan
But for months, he made no progress on it. Then he came to a sudden realization: The problem didn’t just have the same flavor as his own line of work. It could be rewritten in the language of the amplituhedron. The momentum amplituhedron, at that.
“It took much longer than I care to admit,” he said. “You don’t expect the connection, so you never realize it. You’re not supposed to see Bigfoot in Manhattan.”
But could he prove it?
Forget Flat
Galashin considered a collision involving some number of particles, and started with a crease pattern boundary that was divided into that number of line segments.
He described each line segment with a vector that consisted of two numbers. Next, he wrote down vectors that described what the same segments’ new positions should be after folding. These were determined based on information about the momenta of the particles in his collision of interest.
For each segment, he then combined the “before” and “after” vectors into a single four-dimensional vector. By listing the numbers in all these vectors as one set of coordinates, Galashin was able to define a point in a high-dimensional space. And this point didn’t live just anywhere in high-dimensional space — it lived in the momentum amplituhedron.
Galashin showed that the answer to the origami question about flat-folding crease patterns was indeed yes — and that whenever such a crease pattern could be found for a given boundary, the point encoded by that boundary had to reside in the amplituhedron.
It was an entirely new way to think about the shape. “That’s the most amazing thing to me about Pasha’s work, that this connection to origami just gives you this incredibly beautiful one-line definition of the momentum amplituhedron,” Arkani-Hamed said.
Galashin’s new origami-based interpretation gave him an idea for how to finally solve the momentum amplituhedron’s central riddle. He could resolve the triangulation conjecture if he could show that each origami-derived point was situated not just inside the amplituhedron, but inside a very particular region — in just such a way that the regions would lock together without gaps or overlaps.
To do that, he devised an algorithm that took a boundary pattern as its input and assigned a unique crease pattern to it. The crease pattern would always obey the rules that linked it to the geometry of the amplituhedron: Namely, when folded, the paper would still be able to flatten.
Galashin then represented the crease pattern as a plabic graph: First, he drew a point in the middle of each region of the crease pattern, coloring it white if that region would face up once the paper was folded, and black if the region would face down. He then drew an edge between vertices in regions that shared a crease.
Edges in this plabic graph connect regions that share a crease.
Finally, he showed that this graph carved out a region of the amplituhedron. The point encoded by that crease pattern’s boundary sat inside the region.
This was enough to resolve the triangulation question. If two regions in the amplituhedron overlapped — that is, if one point in the amplituhedron lived in two different regions — that would be equivalent to being able to match a boundary pattern to two different crease patterns. But Galashin had designed his algorithm to produce a unique match, so that was impossible. Similarly, the algorithm also implied that there could be no gaps: Every point in the amplituhedron could be rewritten as a boundary, and every boundary, when given as an input to the algorithm, landed neatly inside a region.
The amplituhedron fit together perfectly.
New Dreams
For mathematicians, the elegance of the argument was striking.
“To relate two seemingly unconnected ideas is always quite beautiful,” said Lauren Williams, a mathematician at Harvard University. “I hadn’t thought about origami crease patterns before, so it was a surprise to see them connected to the amplituhedron.”
Galashin shared her surprise. “I don’t have a good explanation for why boundaries of origami are points in the amplituhedron,” he said. “A priori there is no reason why one has to do with the other.” But he hopes that future investigations will uncover a deeper reason for the connection.
He is also hopeful that his result can help him with his original goal: to understand models of ferromagnetism and related systems through the lens of the positive Grassmannian. Perhaps using origami could help.
More broadly, physicists and mathematicians want to see if they can learn more about the amplituhedron — and wield it in a wider variety of theoretical calculations about particle collisions — by thinking about it in terms of origami. For instance, one goal is to be able to compute the scattering amplitude of a particle collision from the volume of the amplituhedron directly, without breaking it into pieces. Perhaps continuing to explore the link between crease patterns and particle collisions will help achieve this dream.
“As a physicist, I would not have come up with this in a million years,” Arkani-Hamed said. “But I find it a spectacular result, and I want to understand it more and see what it might tell us.”
