内容来源：https://www.quantamagazine.org/origami-patterns-solve-a-major-physics-riddle-20251006/

内容总结：

近日，一项突破性研究揭示了折纸艺术与高能物理领域之间的深刻联系。康奈尔大学数学家帕维尔·加拉申通过构建折纸折痕模型，成功证明了描述粒子碰撞的几何形状——动量放大面体（momentum amplituhedron）可被完美分割为无重叠、无空隙的几何单元，解决了困扰物理学界十余年的"三角化猜想"难题。

这一发现源于对粒子碰撞概率计算方法的长期探索。2013年，物理学家尼马·阿尔卡尼-哈米德与雅罗斯拉夫·特兰卡发现，描述粒子散射振幅的复杂计算可转化为对放大面体体积的几何求解。然而该形状能否被精确分割始终未被严格证明，直到加拉申意外发现折纸折痕的边界条件与动量放大面体的空间坐标存在数学同构。

"这项研究最令人惊叹的是，折纸艺术竟能以一行定义的形式完美描述动量放大面体。"阿尔卡尼-哈米德评价道。哈佛大学数学家劳伦·威廉姆斯也认为，两个看似无关领域的连接展现出惊人的数学美感。

该突破不仅验证了物理学家长期使用的BCFW递归计算法的几何基础，更开辟了通过折纸模型研究粒子物理的新路径。研究人员表示，这一跨学科桥梁或将助力实现物理学的终极梦想——直接通过几何体积计算粒子碰撞概率，无需依赖繁琐的分步计算。目前加拉申团队正进一步探索折纸模型在铁磁系统等更广泛物理场景中的应用前景。

中文翻译：

折纸结构破解物理学核心谜题

引言
振幅多面体——一种近乎神秘的几何结构：计算其体积，就能得到粒子相互作用这一物理学核心问题的答案。如今，康奈尔大学的青年数学家帕维尔（帕沙）·加拉申发现这个几何体与另一门毫不相干的学科存在着奇妙关联：折纸艺术。在2024年10月发布的证明中，他揭示折纸中产生的图案可以转化为构成振幅多面体的点集。纸张折叠与粒子碰撞这两种看似风马牛不相及的过程，竟能催生相同的几何形态。

"帕沙此前在振幅多面体领域已有卓越建树。"高等研究院物理学家尼马·阿尔卡尼-哈米德评价道，"但这次的研究堪称突破性进展。"这位于2013年与当时的研究生雅罗斯拉夫·特兰卡共同提出振幅多面体的学者补充说。

通过建立与折纸艺术的新关联，加拉申还解决了关于振幅多面体的悬疑猜想——这个被物理学家长期默认却无法严格证明的假设认为：该几何体确实能被分割成对应物理计算的基础单元。换言之，振幅多面体的组件确实能严丝合缝地拼合。

这项成果不仅在两个看似迥异的领域架起桥梁，加拉申与其他数学家正在探索这座桥梁蕴含的更多可能。他们借此深化对振幅多面体的认知，并试图在更广阔背景下解答其他科学问题。

爆炸式计算的困境
物理学家始终致力于预测基本粒子相互作用的结果。当两个名为胶子的亚原子粒子碰撞时，它们可能保持原状弹开，可能转化成四个胶子，也可能发生其他变化。每种结果都对应着由"散射振幅"这一数学表达式描述的概率。

数十年来，物理学家主要通过两种方法计算散射振幅。其一是运用费曼图——用蜿蜒曲线描述粒子运动与相互作用的图示。每个费曼图对应特定数学计算，通过叠加不同图示的计算结果就能求得散射振幅。但随着碰撞粒子数增加，所需费曼图数量呈爆炸式增长。即便是相对简单的碰撞过程，也需要处理数千乃至数百万项计算。

21世纪初提出的第二种方法BCFW递归法，将复杂粒子相互作用分解为更易研究的简单过程。通过顶点与边构成的图集记录这些简单相互作用的振幅，再按图索骥将各部分重新组合。虽然这种方法将计算量从百万量级降至数百量级，但两种方法存在相同缺陷：最终结果往往远简于推导过程，大量计算项在最终表达中相互抵消。

2013年，阿尔卡尼-哈米德与特兰卡的突破性发现改变了这一局面：粒子碰撞的复杂数学本质上是几何学的伪装。

几何学破局
21世纪初，麻省理工学院数学家亚历山大·波斯尼科夫正在研究名为"正格拉斯曼流形"的几何对象。这个自1930年代就引发数学界关注的抽象结构，其构建方式极具巧思：首先取n维空间及其内部所有低维平面（例如三维空间中存在无数二维平面），每个平面可由称为矩阵的数字阵列定义，通过计算矩阵的子式可获得平面特性信息。当仅考虑所有子式均为正的平面时，就构成了精妙的几何空间——正格拉斯曼流形。

为解析这个复杂空间的内部结构，数学家将其划分为不同区域，每个区域包含具有特定规律的平面集合。波斯尼科夫发明了"平面双色图"——由黑白顶点通过不交叉边线连接的网络图示，每个图对应正格拉斯曼流形的一个区域，为原本晦涩的代数公式提供了直观的视觉语言。

近十年后，当阿尔卡尼-哈米德与特兰卡钻研粒子碰撞散射振幅时，意外发现他们用于记录BCFW计算过程的图示与波斯尼科夫的平面双色图如出一辙。在麻省理工学院的午餐会上，阿尔卡尼-哈米德回忆道："我们当时惊呼'这太神奇了，我们看到的是完全相同的东西'。"

他们确实发现了本质关联：计算n粒子碰撞散射振幅需要叠加众多BCFW项，而每一项都对应n维正格拉斯曼流形的一个区域。二人意识到这种几何关联能简化散射振幅计算，通过粒子碰撞数据（如粒子动量）构建正格拉斯曼流形的低维投影，其投影体积正好等于散射振幅。

振幅多面体由此诞生。

但这仅是故事的序章。物理学家与数学家需要证实：定义正格拉斯曼流形区域的平面双色图，同样能完美分割振幅多面体，且组件间既无空隙也不重叠。这就是著名的"三角化猜想"——振幅多面体能否被精确分割成基础单元？

证明该猜想将夯实阿尔卡尼-哈米德与特兰卡的理论框架：产生粒子碰撞散射振幅的复杂BCFW公式（尽管效率不高），可理解为振幅多面体基础单元体积的总和。

然而挑战重重。从一开始就存在两种振幅多面体：第一种定义在动量扭量坐标系中，这种精妙的数学重构使其更易处理，因其与正格拉斯曼流形及平面双色图存在天然联系。2021年，数学家成功证明了该版本振幅多面体的三角化猜想。

另一种称为动量振幅多面体，直接通过碰撞粒子动量定义。物理学家更关注这个版本，因其描述语言与真实粒子碰撞及散射实验一致，但数学描述更为困难，导致三角化猜想长期悬而未决。

若动量振幅多面体无法实现三角化，将意味着振幅多面体并非理解BCFW公式的正确路径。整整十年间，这个疑云始终萦绕不散——直到折纸研究指明了前进方向。

曼哈顿遇见大脚怪
帕维尔·加拉申最初并未打算研究折纸或振幅多面体。2018年，作为波斯尼科夫的研究生，他与同事刚证明了正格拉斯曼流形与用于研究铁磁体等系统行为的伊辛模型之间存在迷人联系。随后数年，加拉申持续尝试从正格拉斯曼流形角度理解伊辛模型著名证明中展现的特殊对称性。

在研究过程中，几篇运用折纸折痕图简化几何处理的论文引起他的注意。这些指示纸张折叠成仙鹤或青蛙的线图，看似与高深数学毫不相干，但折纸数学近年来已展现出惊人深度。关于折纸的问题（如特定折痕图能否实现无损压平）在计算层面极具挑战，且已被证明能执行各类复杂计算。

2023年，在探究伊辛模型论文中折纸原理时，一个特殊问题吸引了加拉申：若仅知折痕图的外边界信息（即被折痕分割的纸张边缘线段），特别是这些线段在折叠前后的空间位置，是否能找到满足约束且可平整折叠的完整折痕图？数学家猜想答案肯定，但无人能证。

这个猜想令加拉申震撼，因为在他熟悉的正格拉斯曼流形研究中，通过物体边界获取信息是常规手段。历经数月停滞不前后，他突然灵光乍现：这个问题不仅与他的研究方向气味相投，更可用振幅多面体的语言重构——确切地说，是动量振幅多面体。

"领悟这个过程花费的时间远超预期。"他坦言，"由于意想不到这种关联，就像在曼哈顿街头撞见大脚怪般难以置信。"

但他能证明吗？

突破平面思维
加拉申设想包含若干粒子的碰撞过程，从具有相应数量边界线段的折痕图出发。他用二元向量描述每个线段，再根据目标碰撞的粒子动量信息，写出折叠后这些线段的新位置向量。将每个线段的"折叠前"与"折叠后"向量合并为四维向量后，通过将所有向量数值作为坐标集，他在高维空间中定义了一个特殊点——这个点正好落在动量振幅多面体内。

加拉申证实了折痕图问题的肯定答案，并证明只要存在满足边界条件的折痕图，其编码的边界点必位于振幅多面体内。这为理解该几何体提供了全新视角。"帕沙工作最令人惊叹之处，在于通过折纸连接给出了动量振幅多面体极其优雅的单行定义。"阿尔卡尼-哈米德赞叹道。

基于折纸的新诠释让加拉申找到了解决动量振幅多面体核心难题的钥匙。若能证明每个折纸衍生点不仅位于振幅多面体内，更处于特定区域中，且这些区域能严丝合缝地拼合，就能攻克三角化猜想。

为此他设计了一种算法：输入边界图案即可生成唯一折痕图，且该折痕图始终遵循与振幅多面体几何的关联规则——即折叠后的纸张仍可压平。随后他将折痕图转化为平面双色图：在每个区域中心绘制顶点，按折叠后朝向着色，再沿共享折痕连接顶点。

最终他证明这个图划定了振幅多面体的一个区域，而由折痕图边界编码的点正位于该区域内。这完美解决了三角化问题：若振幅多面体内两区域重叠（即某点同属两个区域），则等价于同一边界对应两个不同折痕图，但加拉申的算法确保匹配唯一性；同理，算法也排除了存在空隙的可能——振幅多面体的每个点都可改写为边界，而每个边界通过算法处理都会准确落入某个区域。

振幅多面体确实实现了完美拼合。

新梦想启航
对数学家而言，这个论证过程的优雅性令人瞩目。"将看似无关的概念建立联系总是充满美感。"哈佛大学数学家劳伦·威廉姆斯表示，"我从未涉足折纸折痕图领域，因此它们与振幅多面体的关联令人惊喜。"

加拉申同样感到意外："我尚无法解释为何折纸边界会对应振幅多面体内的点，二者本无必然联系。"但他期待后续研究能揭示更深层原理。他也希望该成果能助其实现最初目标——通过正格拉斯曼流形理解铁磁性及相关系统的模型，或许折纸将成为关键工具。

更广阔的前景在于，物理学家与数学家正尝试通过折纸视角深化对振幅多面体的认知，并将其运用于更多粒子碰撞的理论计算。例如直接通过振幅多面体体积计算散射振幅，而无需分割处理。持续探索折痕图与粒子碰撞的关联，或许终将实现这一梦想。

"作为物理学家，我永远无法构想这样的联系。"阿尔卡尼-哈米德坦言，"但这是项非凡的成就，我渴望深入理解它可能指引的方向。"

英文来源：

Origami Patterns Solve a Major Physics Riddle
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.
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.
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.
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.
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.”