Two Twisty Shapes Resolve a Centuries-Old Topology Puzzle

For the first time, mathematicians have found an example of a compact doughnutlike surface (as seen above) that shares its local geometric information with another surface, despite having a completely different global structure.

数学家第一次发现了一个紧凑的甜甜圈状表面的例子（如上所示），尽管具有完全不同的全局结构，但它与另一个表面共享其局部几何信息。

Mark Belan/ Quanta Magazine ; source: Publications mathématiques de l’IHÉS 142, 241–293 (2025)

马克·贝兰/广达杂志；资料来源：Publications mathématiques de l’IHÉS 142, 241–293 (2025)

Imagine if our skies were always filled with a thick layer of opaque clouds.

想象一下，如果我们的天空总是充满厚厚的不透明云层。

With no way to see the stars, or to view our planet from above, would we have ever discovered that the Earth is round?

如果没有办法看到星星，也没有办法从上面观察我们的星球，我们会发现地球是圆的吗？

The answer is yes.

答案是肯定的。

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.

通过测量地面上的特定距离和角度，即使没有卫星图片，我们也可以确定地球是一个球体，而不是平面或甜甜圈形状。

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.

数学家发现，对于更普遍的二维表面来说，这通常是正确的：关于表面的相对少量的局部信息就足以弄清楚其整体形式。

The part uniquely defines the whole.

部分唯一地定义了整体。

But in some exceptional cases, this limited local information might describe more than one surface.

但在某些特殊情况下，这种有限的局部信息可能描述多个表面。

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.

在过去的 150 年里，数学家们一直在对这些例外情况进行分类：通常只定义一个表面的局部测量实际上描述了多个表面。

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.

但他们找到的唯一例外并不是球体或甜甜圈等漂亮的封闭表面，而是它们在某个方向上永远延伸，或者有可能掉落的边缘。

Nobody could find a closed-up surface that broke the rule.

没有人能找到一个违反规则的封闭表面。

It began to seem as though there simply weren’t any.

看起来好像根本就没有。

Perhaps such surfaces could always be uniquely defined by the usual local information.

也许这样的表面总是可以由通常的本地信息唯一地定义。

Now, mathematicians have finally uncovered one of those long-sought exceptions.

现在，数学家终于发现了长期寻找的例外之一。

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.

在 10 月份发表的一篇论文中，三名研究人员——柏林工业大学的 Alexander Bobenko、慕尼黑工业大学的 Tim Hoffmann 和北卡罗来纳州立大学的 Andrew Sageman-Furnas——描述了一对非常扭曲、封闭的表面，尽管它们具有相同的局部信息，但具有完全不同的全局结构。

Finding them took years of toil, a few very overheated laptops, and an unexpected clue from a seemingly unrelated corner of geometry.

找到它们需要多年的辛劳，几台过热的笔记本电脑，以及来自看似不相关的几何角落的意想不到的线索。

Mathematicians have all sorts of ways to describe a surface locally, but two are especially useful.

数学家有各种各样的方法来局部描述表面，但其中有两种特别有用。

One captures information about the surface’s “extrinsic” curvature.

一种捕获有关表面“外在”曲率的信息。

Choose a point on your surface.

选择表面上的一个点。

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.

此时，您可以在无限多个方向上计算表面在空间中弯曲的速度，即所谓的曲率。

Focus only on the directions where you get the biggest and smallest curvature values, then take the average of the two.

仅关注获得最大和最小曲率值的方向，然后取两者的平均值。

The number you get is called the mean curvature.

您得到的数字称为平均曲率。

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.

您可以计算表面上任何给定点的平均曲率，以更好地了解它在周围空间中的位置。

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.

另一种测量捕获有关表面“固有”曲率的信息——一种不依赖于表面所在空间的几何属性。

Consider a flat sheet of paper.

考虑一张平坦的纸。

You can wrap it into a cylindrical tube without stretching or tearing it.

您可以将其包裹成圆柱形管，而无需拉伸或撕裂它。

If two points are connected by a curve on the sheet of paper, that curve will have the same length on the cylinder.

如果用纸上的曲线连接两个点，则该曲线在圆柱体上的长度相同。

This means that the sheet of paper and the cylinder have the same “metric,” or notion of distance.

这意味着纸张和圆柱体具有相同的“公制”或距离概念。

But try to wrap the sheet of paper around a sphere, and that’s no longer the case.

但尝试将这张纸包裹在一个球体上，情况就不再是这样了。

You’ll have to stretch, cut, or crinkle the paper, and the lengths of curves between points will change.

你必须拉伸、剪切或弄皱纸张，点之间的曲线长度也会改变。

The two surfaces therefore have different metrics.

因此，这两个表面具有不同的度量。

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.

1867 年，法国数学家 Pierre Ossian Bonnet 表明，如果您知道曲面上每个点的公制曲率和平均曲率，就足以告诉您该曲面是什么。

Most of the time.

大多数时候。

But most of the time is not all the time, and that’s the kind of caveat that makes mathematicians itch.

但大多数时候并不总是如此，这就是让数学家们感到痒痒的警告。

In the 150 years since Bonnet’s proof, mathematicians have discovered various kinds of surfaces that defy his rule of thumb.

自博内特证明以来的 150 年间，数学家们发现了各种违背他经验法则的曲面。

These surfaces have the same metric and mean curvature, yet they don’t have the same global structure.

这些曲面具有相同的度量和平均曲率，但它们不具有相同的全局结构。

Pierre Ossian Bonnet asked when it was possible for two different shapes to be defined by the same local data.

Pierre Ossian Bonnet 询问何时可以用相同的本地数据定义两种不同的形状。

But all these surfaces are what mathematicians call non-compact.

但所有这些曲面都是数学家所说的非紧曲面。

They don’t wrap up nicely the way spheres, doughnuts, and other “compact” surfaces do.

它们不像球体、甜甜圈和其他“紧凑”表面那样很好地包裹起来。

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).

相反，非紧凑表面可能会在某个方向上无限延伸（例如平面或圆柱体），或者具有突然终止的边缘（例如从较大形状中切出的一块）。

Compact surfaces are more restricted.

紧凑的表面受到更多限制。

They have to satisfy various constraints to twist back on themselves and close up perfectly.

他们必须满足各种限制才能扭转自己并完美闭合。

So it seemed reasonable to think they might be uniquely defined by their metric and mean curvature.

因此，认为它们可能由其度量和平均曲率唯一定义似乎是合理的。

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.

1981 年，数学家布莱恩·劳森 (Blaine Lawson) 和雷纳托·德·阿泽维多·特里布齐 (Renato de Azevedo Tribuzy) 证明，对于球体以及任何拓扑上与其等效的表面（即任何没有孔的紧凑表面）来说，这都是正确的。

When it came to compact surfaces with a hole (topological doughnuts called tori), there was a bit more wiggle room.

当涉及到带有孔的紧凑表面（称为圆环的拓扑甜甜圈）时，就有更多的回旋余地。

The mathematicians showed that a given metric and mean curvature could correspond to at most two different tori.

数学家表明，给定的度量和平均曲率最多可以对应两个不同的环面。

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.

然而，没有人能找到这种“紧凑的阀盖对”的例子，因此几十年来，普遍的观点是环面就像球体，给定的度量和平均曲率将定义一个环面。

“People believed that for a long time,” said Robert Bryant of Duke University, “because they couldn’t construct any examples.”

“人们很长一段时间都相信这一点，”杜克大学的罗伯特·布莱恩特说，“因为他们无法构造任何例子。”

Alexander Bobenko has spent the past 20 years chewing on mathematical doughnuts.

过去 20 年来，亚历山大·博本科 (Alexander Bobenko) 一直在咀嚼数学甜甜圈。

In the 2000s, he tried to prove that compact Bonnet pairs do indeed exist.

2000年代，他试图证明紧致邦尼对确实存在。

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.

但在意识到这个问题需要几个月的时间才能解决后，他把它放在一边，专注于他认为可以取得更快进展的问题。

He turned to an area of mathematics that seemed unrelated to the Bonnet problem.

他转向了一个似乎与邦尼特问题无关的数学领域。

But that area would end up being the key to solving it.

但该领域最终将成为解决该问题的关键。

Bobenko started to think about “discrete” surfaces, which are a bit like pixelated low-resolution versions of smooth surfaces.

博本科开始考虑“离散”表面，这有点像光滑表面的像素化低分辨率版本。

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.

数学家研究离散曲面，因为它们本身具有重要的几何特性，以及在计算机科学、物理、工程等领域的实际应用。

To get a discrete surface, take a finite collection of points and connect them by lines to form a shape with flat faces.

要获得离散表面，请采用有限的点集合并用线连接它们以形成具有平面的形状。

By choosing different points, you can represent a given smooth surface in different ways.

通过选择不同的点，您可以用不同的方式表示给定的光滑表面。

Here are some examples of how you might represent a sphere, for instance:

以下是如何表示球体的一些示例，例如：

Some discrete surfaces are better representations than others.

一些离散曲面比其他曲面能更好地表示。

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.

博本科和他的经常合作者蒂姆·霍夫曼花了近二十年的时间来开发一种理论，研究如何使用离散表面来保留光滑表面最显着的几何特征。

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.

2010 年代，当时还是哥廷根大学博士生的安德鲁·萨吉曼-弗纳斯 (Andrew Sageman-Furnas) 加入了这一努力，并将邦尼特问题重新纳入考虑范围。

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.

萨吉曼-弗纳斯因对渔网等机织织物（本质上是离散表面）的力学的兴趣而被离散数学所吸引。

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?

一路上，他提出了邦尼特问题的离散版本：局部信息何时会唯一地定义离散表面，何时不会？

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.

通过采用一种已知的方法来生成邦内规则的例外情况，萨奇曼-弗纳斯与他的顾问马克斯·沃德茨基和霍夫曼一起，找到了在离散情况下炮制例外情况的方法。

As in the smooth case, these exceptions were always non-compact.

与平滑情况一样，这些例外总是非紧凑的。

But because discrete surfaces don’t contain infinitely many points, it’s possible to study them using computers.

但由于离散曲面不包含无限多个点，因此可以使用计算机来研究它们。

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?

Sageman-Furnas 想知道，是否有可能使用计算暴力方法在离散几何世界中找到紧致的 Bonnet 对？

If so, then perhaps the discrete case could lead the way to a smooth solution as well.

如果是这样，那么离散情况也许也能带来顺利的解决方案。

And so he joined Bobenko and Hoffmann in Berlin as a postdoctoral researcher in Bobenko’s group and got to work.

因此，他加入了柏林的博本科和霍夫曼，在博本科的小组中担任博士后研究员并开始工作。

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.

2018 年春天，Sageman-Furnas 开始通过计算机搜索一种特殊类型的表面——一种可以转化为 Bonnet 对的表面，类似于酵母发酵剂作为搅打不同种类面包的基础的方式。

This “starter” surface would be like the ones he had used to make discrete Bonnet pairs as a graduate student.

这个“入门”表面就像他在研究生时用来制作离散阀帽对的表面一样。

Except this time, he required it to be a torus.

只是这一次，他要求它是一个环面。

That is, it had to be compact with one or more holes.

也就是说，它必须是紧凑的，有一个或多个孔。

He disappeared for weeks, if not months, Hoffmann recalled.

霍夫曼回忆说，他失踪了几周甚至几个月。

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.

当这位年轻的数学家最终重新出现时，他找到了他一直在寻找的东西：一个非常尖的形状，看起来更像是折纸犀牛而不是圆环。

Mark Belan/ Quanta Magazine ; source: Publications mathématiques de l’IHÉS 142, 241–293 (2025)

马克·贝兰/广达杂志；资料来源：Publications mathématiques de l’IHÉS 142, 241–293 (2025)

But a torus it was.

但它是一个环面。

And according to Sageman-Furnas’ computer program, it had all the other properties required of a starter surface that would generate Bonnet pairs.

根据 Sageman-Furnas 的计算机程序，它具有生成 Bonnet 对的起始表面所需的所有其他属性。

Even more important, when Sageman-Furnas generated those pairs on his computer, they were also tori.

更重要的是，当萨奇曼-弗纳斯在他的计算机上生成这些对时，它们也是环面。

The transformations from the rhino to the Bonnet pair didn’t seem to twist the rhino open into non-compact surfaces.

从犀牛到帽子对的转变似乎并没有将犀牛扭曲成非紧凑的表面。

The surfaces stayed compact.

表面保持紧凑。

“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.”

“当你开始进行计算探索和设计时，”萨奇曼-弗纳斯说，“你可以获得远远超出你认为可能的新例子。”

But was it too good to be true?

但这是不是好得令人难以置信？

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.

计算机程序会产生舍入错误：萨奇曼-弗纳斯的犀牛可能看起来符合所需的标准，而它生成的邦尼特对可能看起来是圆环，但这可能都是海市蜃楼，是小计算错误的产物。

Without a rigorous proof, the mathematicians couldn’t be sure.

如果没有严格的证明，数学家就无法确定。

“He showed up, and he showed us some weird-looking geometric object that really looked like it could have been numerical crap,” Hoffmann said.

“他出现了，他向我们展示了一些看起来很奇怪的几何物体，看起来真的像是数字垃圾，”霍夫曼说。

“Tongue in cheek, probably my most precious contribution to the whole project was that at the time I said, ‘I’ve seen worse.’”

“说实话，我对整个项目最宝贵的贡献可能就是当时我说，‘我见过更糟糕的情况。’”

Andrew Sageman-Furnas (left), Tim Hoffmann (center), and Alexander Bobenko constructed a pair of novel shapes that settled a long-standing conjecture.

安德鲁·萨奇曼-弗纳斯（左）、蒂姆·霍夫曼（中）和亚历山大·博本科构建了一对新颖的形状，解决了一个长期存在的猜想。

From left: Courtesy of Andrew Sageman-Furnas; N.

从左至右：由安德鲁·萨吉曼-弗纳斯 (Andrew Sageman-Furnas) 提供； N。

Kutz; Courtesy of Alexander Bobenko

库茨；由亚历山大·博本科提供

It took some time, but Hoffmann and Sageman-Furnas were eventually able to convince themselves that the rhino was worth taking seriously.

虽然花了一些时间，但霍夫曼和萨吉曼-弗纳斯最终说服自己，犀牛值得认真对待。

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.

如果有可能找到这样一个可能的离散阀帽对的例子，也许光滑的情况并不是那么绝望。

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.

霍夫曼和萨吉曼-弗纳斯在那个闷热的夏天里在犀牛身上寻找线索，有时一次花八到十二小时进行视频聊天，寻找不寻常的特性和几何约束，这可能有助于他们缩小寻找光滑帽环的范围。

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.

随着九月的到来，他们终于找到了一个新的线索，这个线索让人感觉很有希望，以至于博本科恩科又回到了他几十年前放弃的问题上。

The clue had to do with particular lines that loop around the rhino along its edges.

这条线索与沿着犀牛边缘环绕的特定线条有关。

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.

众所周知，这些线条可以提供有关犀牛曲率的重要信息——追踪犀牛弯曲和折叠最多和最少的方向。

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.

由于犀牛是一个生活在三维空间中的二维表面，数学家们预计这些线也能在整个 3D 空间中开辟出路径。

But instead, they always lay either in a plane or on a sphere.

但相反，它们总是位于平面或球体上。

It was exceedingly unlikely that these alignments had happened by chance.

这些排列不可能是偶然发生的。

“That suggested to us that there was really something special happening,” Sageman-Furnas said.

“这向我们表明，确实发生了一些特别的事情，”萨吉曼-弗纳斯说。

It was “spectacular.”

这是“壮观的”。

Unlike discrete surfaces, smooth surfaces don’t have edges.

与离散表面不同，光滑表面没有边缘。

But you can still draw “curvature lines” that trace out the paths of maximum and minimum bending.

但您仍然可以绘制“曲率线”来描绘最大和最小弯曲的路径。

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.

萨格曼-弗纳斯、博本科和霍夫曼决定寻找犀牛的光滑类似物，其曲率线同样仅限于生活在平面或球体上。

Perhaps a starter surface with those properties could give rise to smooth Bonnet tori.

也许具有这些特性的起始表面可以产生光滑的阀盖环面。

But it wasn’t clear if such a surface even existed.

但尚不清楚这样的表面是否存在。

Jean Gaston Darboux came up with formulas that, more than a century later, turned out to be the missing link in work on the Bonnet problem.

让·加斯顿·达布 (Jean Gaston Darboux) 提出的公式在一个多世纪后被证明是邦尼问题研究中缺失的一环。

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.

然后博本科意识到，一个多世纪前，法国数学家让·加斯顿·达布 (Jean Gaston Darboux) 几乎准确地列出了数学家现在所需要的东西。

Darboux had come up with formulas for generating surfaces that had the right kinds of curvature lines.

达布提出了生成具有正确曲率线的曲面的公式。

The problem was that his formulas wouldn’t produce curvature lines that looped back on themselves.

问题是他的公式不会产生自行循环的曲率线。

Instead, they “look like spirals and go to infinity,” Bobenko said.

相反，它们“看起来像螺旋并且趋向无穷大”，博本科说。

“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.

“没有机会让他们关闭。”这意味着虽然曲率线可能存在于平面和球体上，但整个表面不会是环面。

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.

经过多年的努力，数学家们结合纸笔技术和计算实验，找到了如何调整达布公式以使曲率线闭合的方法。

They’d finally found their smooth analogue of the rhino (although the two didn’t look much alike).

他们终于找到了犀牛的光滑相似物（尽管两者看起来不太相似）。

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.

此外，正如他们所希望的那样，这头光滑的犀牛可以生成一对新的环面，它们具有相同的平均曲率和公制数据，但整体结构不同。

The team finally had their answer to the original Bonnet problem: Some tori can’t be uniquely defined by their local features after all.

团队终于找到了最初的 Bonnet 问题的答案：有些花环毕竟不能通过其局部特征来唯一定义。

But when they worked out what this Bonnet pair actually looked like, they found that the two tori were mirror images of each other.

但当他们弄清楚这对邦尼特的实际样子时，他们发现这两个花托是彼此的镜像。

“Technically, this wasn’t an issue,” Sageman-Furnas said.

“从技术上讲，这不是问题，”萨奇曼-弗纳斯说。

“Formally, it solved the problem.” But, he added, it was still unsatisfying.

“从形式上来说，它解决了问题。”但他补充说，这仍然令人不满意。

And so over the next year, they tried to tweak their smooth rhino in various ways.

因此，在接下来的一年里，他们尝试以各种方式调整光滑的犀牛。

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.

最终，他们意识到，如果放弃一组曲率线必须位于球体上的要求，他们就可以构造出一头新的光滑犀牛来完成他们想要的任务。

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.

然后，他们使用这个曲面生成了一个新的 Bonnet 对，这一次是两个非常扭曲的环面，它们是明显不同的曲面，但仍然具有相同的公制和平均曲率。

The team’s final compact Bonnet pair.

该团队的最后一对紧凑型发动机罩。

Mark Belan/ Quanta Magazine ; source: Publications mathématiques de l’IHÉS 142, 241–293 (2025)

马克·贝兰/广达杂志；资料来源：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.

这一结果令马萨诸塞大学阿默斯特分校的数学家 Rob Kusner 感到惊讶。

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.

据他说，这表明即使是托里——一些最漂亮、研究得最好的表面——也不能总是用它们的当地特征来完美描述。

“It’s an example of something where our intuition wasn’t good enough,” said Bryant, the Duke mathematician.

“这是我们的直觉不够好的一个例子，”杜克大学数学家布莱恩特说。

Still, the two tori that the mathematicians found are a bit strange: They pass through themselves like figure eights.

尽管如此，数学家发现的两个环面还是有点奇怪：它们像八字形一样穿过自身。

Bobenko now hopes to prove that there are Bonnet tori that don’t intersect themselves.

博本科现在希望证明存在不相交的帽环。

The Bonnet tori are a welcome validation of Bobenko and Hoffmann’s decades of work on discrete surfaces.

Bonnet tori 是对 Bobenko 和 Hoffmann 数十年在离散曲面方面工作的一个值得欢迎的验证。

Traditionally, the geometry of smooth shapes has advanced much faster, dragging the less developed theory of discrete geometry along behind it.

传统上，光滑形状的几何学发展得更快，拖累了不太发达的离散几何理论。

But in this work, the discrete theory charged ahead and was ultimately what made progress on the smooth side possible.

但在这项工作中，离散理论向前推进，并最终使顺利的进展成为可能。

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.

霍夫曼认为，这凸显了这样一个事实：虽然离散曲面可能看起来不像光滑曲面的复杂模型，但它们有自己的数学寿命。

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.

离散世界可以与平滑世界一样丰富，甚至更丰富，揭示出额外的对称性和联系，否则这些对称性和联系可能会丢失。

“People sort of forgot about this discrete aspect,” Hoffmann said.

“人们有点忘记了这个离散的方面，”霍夫曼说。

But “there are still things to gain from it.”

但“仍然可以从中获益。”

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