内容来源：https://www.quantamagazine.org/two-twisty-shapes-resolve-a-centuries-old-topology-puzzle-20260120/

内容总结：

经过长达一个半世纪的探索，数学家们终于破解了一个经典几何谜题：他们首次发现了一对特殊的“紧致”曲面，即使局部几何信息完全相同，其整体结构却截然不同。这一发现颠覆了数学界长期以来的认知。

自1867年法国数学家博内提出相关定理以来，数学界普遍认为，若已知曲面上每一点的度量（内蕴性质）和平均曲率（外蕴性质），通常就能唯一确定该曲面的整体形状。尽管过去也发现过例外情况，但那些都是非紧致曲面（如无限延伸的平面或带有边界的曲面）。对于像球体、环面这类封闭的“紧致曲面”，尤其是拓扑环面（形如甜甜圈），学界曾猜测其可由局部信息唯一确定。

由柏林工业大学、慕尼黑工业大学和北卡罗来纳州立大学的三位数学家组成的研究团队，通过结合离散几何理论与计算机搜索，成功构造出了一对自交的紧致环面。它们拥有完全相同的局部度量与平均曲率，但整体形状却无法通过刚性变换相互转换。这项研究的关键突破源于对“离散曲面”（一种由有限点与平面多边形构成的近似模型）的深入计算分析。研究团队首先在离散模型中找到了一个形状奇特的“启动曲面”，进而通过调整19世纪数学家达布留下的公式，最终在光滑曲面范畴中构造出了这对反例。

该成果不仅解决了博内问题在紧致曲面情形下的长期悬念，也彰显了离散几何方法的强大力量——它不再是光滑几何的简单近似，而能独立揭示更深层的数学结构。正如研究者所言，这一发现提醒我们，即使对于环面这样被深入研究的经典对象，直觉也可能存在局限，而离散与光滑几何的对话将继续推动数学前沿的发展。

中文翻译：

两个扭曲形状破解百年拓扑谜题

引言

想象一下，如果我们的天空永远被厚重的云层笼罩。既看不见星辰，也无法从高空俯瞰地球，我们还能发现地球是球体吗？

答案是肯定的。即使没有卫星图像，通过测量地面上的特定距离和角度，我们也能确定地球是一个球体，而非平面或甜甜圈形状。

数学家们发现，这一规律普遍适用于二维曲面：通常只需获取相对少量的局部曲面信息，就能推断出其整体形态。局部足以定义整体。

但在某些特殊情况下，有限的局部信息可能对应多种曲面形态。过去150年间，数学家们一直在系统归类这些例外情况——即通常能唯一确定曲面的局部测量数据，实际上却对应多个不同曲面的情形。然而他们找到的所有例外都不是球体或甜甜圈这类光滑封闭曲面，而是要么朝某个方向无限延伸，要么存在可能"坠落"的边缘。

始终无人能找到违反这一规律的封闭曲面。这似乎暗示此类曲面根本不存在，或许常规的局部信息总能唯一确定这类曲面。

如今，数学家终于发现了这种长期寻觅的例外。在十月发表的论文中，柏林工业大学的亚历山大·博本科、慕尼黑工业大学的蒂姆·霍夫曼与北卡罗来纳州立大学的安德鲁·塞奇曼-弗纳斯三位研究者，描述了一对极度扭曲的封闭曲面——尽管具有相同的局部信息，却拥有完全不同的整体结构。

寻找这对曲面耗费了数年心血，烧坏了几台笔记本电脑，还意外获得了来自几何学看似无关领域的线索。

几何异类

数学家拥有多种描述曲面局部特征的方法，其中两种尤为常用。

第一种捕捉曲面的"外在"曲率信息。在曲面上任选一点，该点存在无数个方向可以计算曲面在空间中的弯曲速率（即曲率）。仅关注最大与最小曲率对应的方向，取二者的平均值，所得数值称为平均曲率。通过计算曲面上任意点的平均曲率，可以更好地理解曲面在空间中的姿态。

第二种测量则捕捉曲面的"内在"曲率信息——这种几何特性不依赖于曲面所处的空间。以平整纸张为例：无需拉伸或撕裂即可将其卷成圆柱筒。若纸面上两点间存在曲线，该曲线在圆柱面上的长度保持不变。这意味着纸张与圆柱面具有相同的"度量"（即距离概念）。但若尝试将纸张包裹球体，情况就截然不同——必须拉伸、裁剪或揉皱纸张，点间曲线长度也会改变，因此两种曲面具有不同的度量。

1867年，法国数学家皮埃尔·奥西安·博内证明：若已知曲面上每一点的度量与平均曲率，通常就足以确定曲面的形态。

但"通常"不等于"永远"，正是这种例外情况让数学家们心痒难耐。

博内定理问世后的150年间，数学家发现了多种违反其经验法则的曲面。这些曲面具有相同的度量与平均曲率，却拥有不同的整体结构。

但所有这些曲面都是数学家所称的"非紧致"曲面。它们不像球体、甜甜圈等"紧致"曲面那样完美闭合，而是可能朝某个方向无限延伸（如平面或圆柱面），或存在突然终止的边缘（如从更大形状中切割出的碎片）。

紧致曲面限制更严格。它们必须满足各种约束条件才能自我扭曲并完美闭合。因此人们有理由认为，这类曲面或许能由其度量与平均曲率唯一确定。1981年，数学家布莱恩·劳森与雷纳托·德阿泽维多·特里布齐证明：对于球体及任何拓扑等价的曲面（即无洞的紧致曲面），这一猜想成立。

对于带有一个洞的紧致曲面（称为环面的拓扑甜甜圈），情况则更灵活。数学家证明：给定的度量与平均曲率最多可能对应两个不同的环面。

然而始终无人找到此类"紧致博内对"的实例，因此数十年来主流观点认为环面与球体类似，给定度量与平均曲率将唯一确定一个环面。"人们长期相信这一点，"杜克大学的罗伯特·布莱恩特说，"因为他们构造不出任何反例。"

但他们错了。

像素化的世界

过去二十年间，亚历山大·博本科一直在钻研数学意义上的甜甜圈。本世纪初，他曾试图证明紧致博内对确实存在。但在意识到解决该问题需要远超数月的时间后，他暂时搁置了这项研究，转而攻克那些他认为能更快取得进展的课题。

他转向了一个看似与博内问题无关的数学领域。但这个领域最终成为解决难题的关键。

博本科开始研究"离散"曲面——这种曲面类似于光滑曲面的像素化低分辨率版本。数学家研究离散曲面不仅因其本身具有重要的几何性质，更在计算机科学、物理学、工程学等领域具有实际应用。

要构建离散曲面，需选取有限点集并用线段连接，形成具有平坦面的形状。通过选择不同的点集，可以用多种方式表现给定的光滑曲面。以下是以离散形式表现球体的几种示例：

某些离散曲面比其他形式更具表现力。博本科与其长期合作者蒂姆·霍夫曼投入近二十年时间，发展了一套关于如何用离散曲面保留光滑曲面最显著几何特征的理论。

2010年代，当时在哥廷根大学攻读博士的安德鲁·塞奇曼-弗纳斯加入研究，并将博内问题重新纳入研究视野。

塞奇曼-弗纳斯因对渔网等编织物力学特性产生兴趣而涉足离散数学领域，这类织物本质就是离散曲面。在研究过程中，他提出了博内问题的离散版本：局部信息何时能唯一确定离散曲面？何时不能？通过改造已知的博内法则例外生成方法，塞奇曼-弗纳斯与导师马克斯·瓦德茨基及霍夫曼合作，找到了在离散情形下构造例外曲面的方法。

与光滑情形类似，这些例外总是非紧致的。但由于离散曲面不包含无限多个点，可以用计算机进行研究。塞奇曼-弗纳斯思考：能否在离散几何世界中，通过计算暴力搜索找到紧致博内对？如果可行，离散案例或许也能为光滑情形的解决指明方向。

于是他作为博士后研究员加入柏林博本科的研究小组，与霍夫曼共同展开了工作。

曲面狩猎

2018年春天，塞奇曼-弗纳斯开始用计算机搜索一种特殊曲面——这种曲面能转化为博内对，类似于用酵头发酵不同面包的原理。这种"起始"曲面与他研究生时期用于制作离散博内对的曲面类似，但这次他要求必须是环面，即具有一个或多个洞的紧致曲面。

霍夫曼回忆，这位年轻数学家消失了数周甚至数月。当他最终重现时，找到了梦寐以求的目标：一个布满尖刺、更像折纸犀牛而非传统环面的形状。

但这确实是环面。根据塞奇曼-弗纳斯的计算机程序，它具有起始曲面生成博内对所需的所有特性。更重要的是，当他在计算机上生成博内对时，得到的也是环面。从犀牛曲面到博内对的变换似乎没有将其扭曲成非紧致曲面，所有曲面都保持紧致。

"当你开始进行计算探索与设计时，"塞奇曼-弗纳斯说，"可能会获得远超想象的新案例。"

但这会不会好得令人难以置信？计算机程序存在舍入误差：塞奇曼-弗纳斯的犀牛曲面可能看似满足条件，其生成的博内对也可能看似环面，但这都可能是微小计算误差造成的幻象。没有严格证明，数学家们无法确信。

"他带着这个看起来怪异的几何对象出现时，"霍夫曼说，"确实像是数值计算产生的垃圾。说句玩笑话，我对整个项目最宝贵的贡献可能就是当时说了句：'我见过更糟的。'"

经过一段时间，霍夫曼和塞奇曼-弗纳斯最终确信犀牛曲面值得深入研究。既然有可能找到离散博内对的潜在案例，或许光滑情形也并非毫无希望。在那个炎热的夏天，两人仔细剖析犀牛曲面寻找线索，有时连续进行8到12小时的视频讨论，搜寻非常规特性与几何约束，以缩小光滑博内环面的搜索范围。

九月来临之际，他们终于发现了充满希望的新线索，这个线索将博本科重新吸引回他数十年前放弃的难题。

闭合环路

这个线索与犀牛曲面边缘环绕的特定线条有关。

已知这些线条能提供关于曲面曲率的重要信息——标示出弯曲折叠程度最大与最小的方向。由于犀牛曲面是存在于三维空间的二维曲面，数学家原本预期这些线条也会在三维空间中勾勒出路径。但事实上，它们始终位于平面或球面上。这种排列几乎不可能是偶然发生的。

"这暗示确实存在特殊规律，"塞奇曼-弗纳斯说，这种现象"非常壮观"。

与离散曲面不同，光滑曲面没有边缘。但仍可绘制标示最大最小弯曲路径的"曲率线"。塞奇曼-弗纳斯、博本科和霍夫曼决定寻找犀牛曲面的光滑版本，要求其曲率线同样被限制在平面或球面上。或许具有这些特性的起始曲面能催生光滑的博内环面。

但当时尚不清楚这种曲面是否存在。

此时博本科意识到，早在一个多世纪前，法国数学家让·加斯东·达布就已几乎完整阐述了数学家现在所需的内容。

达布提出了生成具有特定曲率线曲面的公式。问题在于他的公式产生的曲率线无法自我闭合。"它们看起来像螺旋线并趋向无穷远，"博本科说，"根本没有闭合的可能。"这意味着虽然曲率线可能位于平面和球面上，但整体曲面不会是环面。

经过数年努力，三位数学家结合纸笔演算与计算实验，最终找到了调整达布公式使曲率线闭合的方法。他们终于发现了犀牛曲面的光滑版本（尽管两者外观差异很大）。

更重要的是，正如他们所期望的，这个光滑犀牛曲面能生成一对具有相同平均曲率与度量数据、但整体结构不同的新环面。研究团队终于回答了最初的博内问题：某些环面确实无法由其局部特征唯一确定。

但当他们计算出这对博内环面的实际形态时，发现两者互为镜像。"技术上这不成问题，"塞奇曼-弗纳斯说，"形式上已经解决了难题。"但他补充道，这仍令人不够满意。

于是在接下来的一年里，他们尝试用各种方法调整光滑犀牛曲面。最终意识到，如果放弃一组曲率线必须位于球面上的要求，就能构造出符合预期的新光滑犀牛曲面。他们利用这个曲面生成了新的博内对——这次得到了两个明显不同、极度扭曲的环面，它们仍具有相同的度量与平均曲率。

这一结果令马萨诸塞大学阿默斯特分校的数学家罗布·库斯纳感到惊讶。在他看来，这证明即使是环面这种最典型、研究最深入的曲面，也并非总能被局部特征完美描述。

"这个例子说明我们的直觉还不够敏锐，"杜克大学的数学家布莱恩特评论道。

不过数学家发现的这对环面仍有些奇特：它们像八字形一样自我交叉。博本科现在希望证明存在不自交的博内环面。

博内环面的发现验证了博本科与霍夫曼数十年来在离散曲面研究上的努力。传统上，光滑形状的几何学发展更快，相对滞后的离散几何理论往往跟随其后。但在这项工作中，离散理论率先突破，最终推动了光滑情形的研究进展。

霍夫曼指出，这凸显了一个事实：虽然离散曲面看似是光滑曲面的简化模型，但它们具有独立的数学生命力。离散世界可能与光滑世界同样丰富，甚至更为丰富，能揭示那些原本可能被忽略的对称性与关联。

"人们某种程度上遗忘了离散几何的这个层面，"霍夫曼说，"但它仍能带来新的收获。"

英文来源：

Two Twisty Shapes Resolve a Centuries-Old Topology Puzzle
Introduction
Imagine if our skies were always filled with a thick layer of opaque clouds. With no way to see the stars, or to view our planet from above, would we have ever discovered that the Earth is round?
The answer is yes. By measuring particular distances and angles on the ground, we can determine that the Earth is a sphere and not, say, flat or doughnut-shaped — even without a satellite picture.
Mathematicians have found that this is often true of two-dimensional surfaces more generally: A relatively small amount of local information about the surface is all you need to figure out its overall form. The part uniquely defines the whole.
But in some exceptional cases, this limited local information might describe more than one surface. Mathematicians have spent the past 150 years cataloging these exceptions: instances in which local measurements that usually define just one surface in fact describe more than one. But the only exceptions they managed to find weren’t nice, closed-up surfaces like orbs or doughnuts — instead, they stretched on forever in some direction, or had edges you could fall off of.
Nobody could find a closed-up surface that broke the rule. It began to seem as though there simply weren’t any. Perhaps such surfaces could always be uniquely defined by the usual local information.
Now, mathematicians have finally uncovered one of those long-sought exceptions. In a paper published in October, three researchers — Alexander Bobenko of the Technical University of Berlin, Tim Hoffmann of the Technical University of Munich, and Andrew Sageman-Furnas of North Carolina State University — describe a pair of very twisty, closed-up surfaces that, despite having the same local information, have completely different global structures.
Finding them took years of toil, a few very overheated laptops, and an unexpected clue from a seemingly unrelated corner of geometry.
Geometric Misfits
Mathematicians have all sorts of ways to describe a surface locally, but two are especially useful.
One captures information about the surface’s “extrinsic” curvature. Choose a point on your surface. At that point, there are infinitely many directions in which you can calculate how quickly the surface bends in space — what’s known as its curvature. Focus only on the directions where you get the biggest and smallest curvature values, then take the average of the two. The number you get is called the mean curvature. You can compute the mean curvature for any given point on the surface to gain a better understanding of how it’s situated in the space surrounding it.
Another kind of measurement captures information about the surface’s “intrinsic” curvature — a geometric property that doesn’t depend on the space that the surface lives in. Consider a flat sheet of paper. You can wrap it into a cylindrical tube without stretching or tearing it. If two points are connected by a curve on the sheet of paper, that curve will have the same length on the cylinder. This means that the sheet of paper and the cylinder have the same “metric,” or notion of distance. But try to wrap the sheet of paper around a sphere, and that’s no longer the case. You’ll have to stretch, cut, or crinkle the paper, and the lengths of curves between points will change. The two surfaces therefore have different metrics.
Mark Belan/Quanta Magazine
In 1867, the French mathematician Pierre Ossian Bonnet showed that if you know both the metric and the mean curvature at every point on a surface, that’s enough to tell you what the surface is. Most of the time.
But most of the time is not all the time, and that’s the kind of caveat that makes mathematicians itch.
In the 150 years since Bonnet’s proof, mathematicians have discovered various kinds of surfaces that defy his rule of thumb. These surfaces have the same metric and mean curvature, yet they don’t have the same global structure.
But all these surfaces are what mathematicians call non-compact. They don’t wrap up nicely the way spheres, doughnuts, and other “compact” surfaces do. Rather, a non-compact surface might stretch out infinitely in some direction (like a plane or cylinder), or have edges where it suddenly ends (like a piece cut out from a larger shape).
Compact surfaces are more restricted. They have to satisfy various constraints to twist back on themselves and close up perfectly. So it seemed reasonable to think they might be uniquely defined by their metric and mean curvature. In 1981, the mathematicians Blaine Lawson and Renato de Azevedo Tribuzy proved that this is true for the sphere and any surfaces topologically equivalent to it — that is, any compact surfaces that have no holes.
When it came to compact surfaces with a hole (topological doughnuts called tori), there was a bit more wiggle room. The mathematicians showed that a given metric and mean curvature could correspond to at most two different tori.
No one could find examples of such “compact Bonnet pairs,” however, and so for decades, the prevailing view was that tori were like spheres, and that a given metric and mean curvature would define a single torus. “People believed that for a long time,” said Robert Bryant of Duke University, “because they couldn’t construct any examples.”
But they were wrong.
A Pixelated World
Alexander Bobenko has spent the past 20 years chewing on mathematical doughnuts. In the 2000s, he tried to prove that compact Bonnet pairs do indeed exist. But after realizing that the problem would take him more than a few months to solve, he set it aside to focus on questions he thought he could make faster progress on.
He turned to an area of mathematics that seemed unrelated to the Bonnet problem. But that area would end up being the key to solving it.
Bobenko started to think about “discrete” surfaces, which are a bit like pixelated low-resolution versions of smooth surfaces. Mathematicians study discrete surfaces because they have important geometric properties in their own right, as well as practical applications in computer science, physics, engineering, and more.
To get a discrete surface, take a finite collection of points and connect them by lines to form a shape with flat faces. By choosing different points, you can represent a given smooth surface in different ways. Here are some examples of how you might represent a sphere, for instance:
Mark Belan/Quanta Magazine
Some discrete surfaces are better representations than others. Bobenko and his frequent collaborator Tim Hoffmann have dedicated nearly two decades to developing a theory for how to preserve the most salient geometric features of smooth surfaces using discrete ones.
In the 2010s, Andrew Sageman-Furnas, then a doctoral student at the University of Göttingen, joined the effort — and brought the Bonnet problem back into the mix.
Sageman-Furnas had been drawn into discrete mathematics through his interest in the mechanics of woven fabrics like fishing nets, which are essentially discrete surfaces. Along the way, he’d asked a discrete version of the Bonnet question: When will local information uniquely define a discrete surface, and when won’t it? By adapting a known method for generating exceptions to Bonnet’s rule, Sageman-Furnas, along with his adviser Max Wardetzky and Hoffmann, found a recipe for concocting exceptions in the discrete case.
As in the smooth case, these exceptions were always non-compact. But because discrete surfaces don’t contain infinitely many points, it’s possible to study them using computers. Might it be possible, Sageman-Furnas wondered, to use computational brute-force methods to find a compact Bonnet pair in the world of discrete geometry? If so, then perhaps the discrete case could lead the way to a smooth solution as well.
And so he joined Bobenko and Hoffmann in Berlin as a postdoctoral researcher in Bobenko’s group and got to work.
Surface Safari
In the spring of 2018, Sageman-Furnas began a computer search for a special type of surface — one that could be transformed into a Bonnet pair, akin to how a sourdough starter acts as a base for whipping up different kinds of bread. This “starter” surface would be like the ones he had used to make discrete Bonnet pairs as a graduate student. Except this time, he required it to be a torus. That is, it had to be compact with one or more holes.
He disappeared for weeks, if not months, Hoffmann recalled. When the younger mathematician finally reemerged, he had found what he’d been looking for: a very spiky shape that looked more like an origami rhino than a torus.
Mark Belan/Quanta Magazine; source: Publications mathématiques de l’IHÉS 142, 241–293 (2025)
But a torus it was. And according to Sageman-Furnas’ computer program, it had all the other properties required of a starter surface that would generate Bonnet pairs. Even more important, when Sageman-Furnas generated those pairs on his computer, they were also tori. The transformations from the rhino to the Bonnet pair didn’t seem to twist the rhino open into non-compact surfaces. The surfaces stayed compact.
“When you start to do computational exploration and design,” Sageman-Furnas said, “you can get new examples that are far outside of what you thought was possible.”
But was it too good to be true? Computer programs make rounding errors: Sageman-Furnas’ rhino might appear to meet the desired criteria, and the Bonnet pair it generated might appear to be tori, but that could all be a mirage, an artifact of small computational errors. Without a rigorous proof, the mathematicians couldn’t be sure.
“He showed up, and he showed us some weird-looking geometric object that really looked like it could have been numerical crap,” Hoffmann said. “Tongue in cheek, probably my most precious contribution to the whole project was that at the time I said, ‘I’ve seen worse.’”
From left: Courtesy of Andrew Sageman-Furnas; N. Kutz; Courtesy of Alexander Bobenko
It took some time, but Hoffmann and Sageman-Furnas were eventually able to convince themselves that the rhino was worth taking seriously. And if it was possible to find such a likely example of a discrete Bonnet pair, maybe the smooth case wasn’t so hopeless after all. Hoffmann and Sageman-Furnas spent that sweltering summer scouring the rhino for clues, sometimes sitting in video chats for eight to 12 hours at a time, searching for unusual properties and geometric constraints that might help them narrow down where to look for smooth Bonnet tori.
As September rolled around, they finally found a new lead that felt so promising that it drew Bobenko back into the problem he’d abandoned decades earlier.
Closed Loops
The clue had to do with particular lines that loop around the rhino along its edges.
These lines were already known to provide important information about the rhino’s curvature — tracing out the directions in which it bent and folded the most and least. Since the rhino is a two-dimensional surface that lives in three-dimensional space, the mathematicians had expected these lines to carve out paths throughout 3D space as well. But instead, they always lay either in a plane or on a sphere. It was exceedingly unlikely that these alignments had happened by chance.
“That suggested to us that there was really something special happening,” Sageman-Furnas said. It was “spectacular.”
Unlike discrete surfaces, smooth surfaces don’t have edges. But you can still draw “curvature lines” that trace out the paths of maximum and minimum bending. Sageman-Furnas, Bobenko, and Hoffmann decided to look for a smooth analogue of the rhino whose curvature lines were similarly restricted to living in planes or on spheres. Perhaps a starter surface with those properties could give rise to smooth Bonnet tori.
But it wasn’t clear if such a surface even existed.
Then Bobenko realized that more than a century ago, the French mathematician Jean Gaston Darboux had laid out almost exactly what the mathematicians now needed.
Darboux had come up with formulas for generating surfaces that had the right kinds of curvature lines. The problem was that his formulas wouldn’t produce curvature lines that looped back on themselves. Instead, they “look like spirals and go to infinity,” Bobenko said. “No chance to get them closed.” Which meant that while the curvature lines might live on planes and spheres, the overall surface wouldn’t be a torus.
After years of toil, the mathematicians — using a combination of pen-and-paper techniques and computational experiments — figured out how to adjust Darboux’s formulas so that the curvature lines would close up. They’d finally found their smooth analogue of the rhino (although the two didn’t look much alike).
Moreover, as they’d hoped, this smooth rhino could generate a pair of new tori that had the same mean curvature and metric data but different overall structures. The team finally had their answer to the original Bonnet problem: Some tori can’t be uniquely defined by their local features after all.
But when they worked out what this Bonnet pair actually looked like, they found that the two tori were mirror images of each other. “Technically, this wasn’t an issue,” Sageman-Furnas said. “Formally, it solved the problem.” But, he added, it was still unsatisfying.
And so over the next year, they tried to tweak their smooth rhino in various ways. Ultimately, they realized that if they dropped the requirement that one set of curvature lines had to sit on spheres, they could construct a new smooth rhino that did what they wanted. They then used this surface to generate a new Bonnet pair — this time, two very twisty tori that were much more obviously different surfaces but still had the same metric and mean curvature.
Mark Belan/Quanta Magazine; source: Publications mathématiques de l’IHÉS 142, 241–293 (2025)
The result came as a surprise to Rob Kusner, a mathematician at the University of Massachusetts, Amherst. According to him, it demonstrates that even tori — some of the nicest, best-studied surfaces — can’t always be perfectly described by their local characteristics.
“It’s an example of something where our intuition wasn’t good enough,” said Bryant, the Duke mathematician.
Still, the two tori that the mathematicians found are a bit strange: They pass through themselves like figure eights. Bobenko now hopes to prove that there are Bonnet tori that don’t intersect themselves.
The Bonnet tori are a welcome validation of Bobenko and Hoffmann’s decades of work on discrete surfaces. Traditionally, the geometry of smooth shapes has advanced much faster, dragging the less developed theory of discrete geometry along behind it. But in this work, the discrete theory charged ahead and was ultimately what made progress on the smooth side possible.
According to Hoffmann, this highlights the fact that while discrete surfaces might seem like less sophisticated models of their smooth counterparts, they have a mathematical life of their own. The discrete world can be just as rich as the smooth one, if not richer, revealing extra symmetries and connections that might otherwise get lost.
“People sort of forgot about this discrete aspect,” Hoffmann said. But “there are still things to gain from it.”