What Does the Fourth Dimension Actually Look Like?

And we’re off.

我们出发了。

I’m Janna Levin.

我是詹娜·莱文。

And I’m Steve Strogatz.

我是史蒂夫·斯特罗加茨。

And this is The Joy of Why.

这就是“为什么的喜悦”。

A podcast from Quanta Magazine where we explore some of the biggest unanswered questions in math and science today.

来自《Quanta》杂志的播客，我们探讨了当今数学和科学中一些尚未解答的最大问题。

Steve, we have a really mathematical topic today.

史蒂夫，今天我们有一个真正的数学话题。

I’m glad to have a mathematician in cahoots.

我很高兴有一位数学家同谋。

Okay, at your service.

好的，竭诚为您服务。

Excellent.

出色的。

We’re speaking in this episode about a branch of mathematics known as topology, and that’s not familiar to a lot of people.

我们在这一集中讨论的是数学的一个分支，即拓扑学，很多人对此并不熟悉。

Yeah, sure.

是的，当然。

Topology, very popular subject.

拓扑学，非常热门的学科。

You know, whereas geometry is ancient, topology is only, depending where you date it, could be a few hundred years old, but it really took off in the 20th century.

你知道，虽然几何学很古​​老，但拓扑学可能只有几百年的历史（具体取决于它的年代），但它真正起飞是在 20 世纪。

No, I didn’t, but I think about the Möbius band probably an unhealthy amount.

不，我没有，但我认为莫比乌斯环的数量可能不健康。

Yeah.

是的。

It comes up in my work.

它出现在我的工作中。

Yeah.

是的。

Oh, okay.

哦好的。

Well, there was a time when one of my kids was in elementary school, and it was bring your parent to school day, and so I had the kids make Möbius strips, and then do this thing where you cut down the midline of the strip all the way around the whole ring.

嗯，曾经有一段时间，我的一个孩子在上小学，那天要带父母去上学，所以我让孩子们制作莫比乌斯带，然后做这件事，沿着整个环切掉带子的中线。

And you think you’re cutting it in half, but it doesn’t fall apart.

你认为你把它切成两半，但它并没有散开。

And so one kid started crying.

于是一个孩子开始哭了。

Oh [laughs].

哦[笑]。

Well, math is frustrating.

好吧，数学令人沮丧。

This is the thing, it was an early introduction to what all mathematicians and math teachers are trying to do, make people cry in math class.

就是这样，它是对所有数学家和数学老师都试图做的事情的早期介绍，让人们在数学课上哭泣。

It’s perfect, actually.

事实上，这很完美。

[laughs]

[笑]

Okay.

好的。

But anyway, so that’s our subject, topology.

但无论如何，这就是我们的主题：拓扑。

I mean, the Möbius strip, this property that when you cut it down the middle, it doesn’t fall into two pieces, that’s a topological property about how things are connected.

我的意思是，莫比乌斯带，当你把它从中间切开时，它不会分成两部分，这是一个关于事物如何连接的拓扑属性。

You could change a lot of geometrical things, and you wouldn’t change this property.

你可以改变很多几何形状，但你不会改变这个属性。

It’s deeper than geometry.

它比几何学更深。

Right.

正确的。

Well, we usually think about geometry, we think about the Pythagorean theorem, which is precisely telling me how to measure distances, and topology is not about that.

嗯，我们通常会想到几何学，我们会想到毕达哥拉斯定理，它准确地告诉我如何测量距离，而拓扑学与此无关。

It’s about this global connectedness.

这是关于这种全球联系。

So it has this very abstract character, and our guest today is, in fact, Maggie Miller, an assistant professor at the University of Texas at Austin, who researches knots, surfaces, and four-dimensional spaces.

所以它具有非常抽象的特征，事实上，我们今天的嘉宾是玛吉·米勒，德克萨斯大学奥斯汀分校的助理教授，研究结、曲面和四维空间。

Welcome to The Joy of Why , Maggie.

欢迎来到《为什么的喜悦》，玛吉。

It’s really a pleasure to have you on.

很高兴有您的加入。

Yeah, thanks for having me.

是的，谢谢你邀请我。

I’m excited to be here

我很高兴来到这里

I’m really fascinated by your work.

我真的对你的工作很着迷。

It intrigues me personally.

我个人对此很感兴趣。

There’s a lot of questions I wanna ask you, but let’s just start with the Maryam Mirzakhani New Frontiers Prize, which is within the Breakthrough Prize umbrella.

我想问你很多问题，但我们先从玛丽亚姆·米尔扎哈尼新领域奖开始，该奖属于突破奖范畴。

You received that a few years ago, and you mentioned that early on in college, you were split between studying math and art, which I think is a really fascinating combination.

几年前你收到了这封信，你提到在大学早期，你分别学习数学和艺术，我认为这是一个非常有趣的组合。

What in particular led you to choose math in the end?

是什么特别促使您最终选择了数学？

As a kid, even before college, I was really invested in art.

当我还是个孩子的时候，甚至在上大学之前，我就对艺术非常投入。

I took art classes for a long time.

我上了很长一段时间的艺术课。

I agree.

我同意。

This is sort of a misunderstood aspect of art.

这是艺术的一个被误解的方面。

People think art is structureless and without constraints, and that’s really not the case, especially if the artist is good.

人们认为艺术是无结构的、没有约束的，但事实并非如此，尤其是如果艺术家很优秀的话。

So I need to hear about the homeschooling.

所以我需要听听有关家庭教育的信息。

Yeah.

是的。

My parents decided to homeschool me and my twin sister, starting about halfway through third grade.

我的父母决定让我和我的双胞胎妹妹在家上学，大约从三年级一半开始。

It is very common in the state of Texas.

这在德克萨斯州很常见。

I knew a lot of other homeschooled children growing up.

我认识很多其他在家上学的孩子。

I have two brothers who are quite a bit younger than me, and so they were both homeschooled all the way, up until college, which again, it’s very common in Texas compared to some other states, and I think it’s a little easier in terms of government restrictions.

我有两个比我小很多的兄弟，所以他们一直在家上学，直到上大学，这在德克萨斯州与其他一些州相比很常见，而且我认为在政府限制方面这要容易一些。

So, Maggie, each time I ask you a question, you say something that I have to ask about.

所以，玛吉，每次我问你问题时，你都会说一些我必须问的事情。

So first it was the homeschooling, and then it was me and my twin sister.

所以首先是在家上学，然后是我和我的双胞胎妹妹。

Twin sister, an identical twin sister, and is she also a mathematician or an artist?

双胞胎姐妹，同卵双胞胎姐妹，她也是数学家还是艺术家？

No, we’re fraternal.

不，我们是兄弟。

I think we have a lot in common.

我认为我们有很多共同点。

We’re still sisters.

我们还是姐妹。

She is not particularly interested in math or art in the traditional sense.

她对传统意义上的数学或艺术并不是特别感兴趣。

She definitely has a lot of craft activities that she enjoys.

她肯定有很多她喜欢的手工艺活动。

My sister went to college and studied psychology, so I think she’s a little bit more interested in maybe social science.

我姐姐上大学学习心理学，所以我认为她可能对社会科学更感兴趣。

Mm-hmm.

嗯嗯。

Yeah.

是的。

So, I wanna get into discussing your area of mathematics, which is one of my favorite subjects: topology.

所以，我想讨论一下你的数学领域，这是我最喜欢的科目之一：拓扑学。

Can you explain what topology is?

你能解释一下什么是拓扑吗？

So topology is the study of abstract spaces, which could mean the space that we live in right now.

因此，拓扑学是对抽象空间的研究，这可能意味着我们现在居住的空间。

We live in a space where we perceive three dimensions, X, Y, and Z.

我们生活在一个可以感知三个维度（X、Y 和 Z）的空间中。

But it refers to a more general concept.

但它指的是一个更普遍的概念。

There’s what is a space is something that you start off defining in a course on topology.

空间是你在拓扑课程中开始定义的东西。

But for now, just imagine, like, a space you could live in.

但现在，想象一下，一个你可以居住的空间。

and we say that two objects or spaces are the same, equivalent if you can continuously deform one into the other.

我们说两个物体或空间是相同的，如果你可以不断地将一个物体或空间变形为另一个物体或空间，则它们是等价的。

So topology is the study of objects like stretching and twisting, but you’re never allowed to break things.

因此，拓扑学是对物体的拉伸和扭曲的研究，但你永远不允许破坏物体。

So when you say X, Y, and Z, you mean the three dimensions we’re used to occupying.

因此，当您说 X、Y 和 Z 时，您指的是我们习惯占据的三个维度。

But within topology, you also work on other dimensional spaces.

但在拓扑中，您​​还可以研究其他维度空间。

That’s right.

这是正确的。

It’s very common to study spaces of any dimension or spaces that don’t have a well-defined dimension because they can be really abstract.

研究任何维度的空间或没有明确定义维度的空间是很常见的，因为它们可能非常抽象。

So for example, a very simple example of a space is something like the circle.

例如，一个非常简单的空间例子就是像圆一样的东西。

So not the filled-in circle, but really just the edge of a disc, like if I took the edge of a plate or something.

所以不是实心圆，而实际上只是圆盘的边缘，就像我拿了盘子或其他东西的边缘一样。

So if you glue them all together at one point, it’s like a fan almost.

因此，如果你将它们全部粘在一起，它几乎就像一把扇子。

Yeah, it would be a fan, yeah.

是的，这将是一个粉丝，是的。

Or sometimes people call it an earring.

或者有时人们称其为耳环。

So some of these spaces are true to the word “space.” They represent, where we can live, or places that can be occupied, but some of them don’t have to do with that physical dimensionality that we’re used to.

因此，其中一些空间确实符合“空间”这个词。它们代表了我们可以居住的地方，或者可以占据的地方，但其中一些与我们习惯的物理维度无关。

That’s right.

这是正确的。

I think that the physical dimensionality motivates the name, and that’s a common theme in topology and other areas of math.

我认为物理维度激发了这个名字，这是拓扑学和其他数学领域的一个共同主题。

There’s one easy example which has an English name that makes sense, and then we analogize it and still use the same name even when it feels like that’s not what we mean in the English sense.

有一个简单的例子，它有一个有意义的英文名称，然后我们对其进行类比，并且仍然使用相同的名称，即使感觉这不是我们在英语意义上的意思。

Mm-hmm.

嗯嗯。

So, to gain some kind of perspective, let’s start with the idea of a three-dimensional manifold.

因此，为了获得某种视角，让我们从三维流形的概念开始。

We live presumably in a three-dimensional manifold.

我们大概生活在一个三维流形中。

Can you walk us through the sense in which we have three dimensions?

你能带我们了解一下我们拥有三个维度的意义吗？

Yeah.

是的。

So when we talk about dimension, it means how many different perpendicular directions are there so that every direction is a combination of those.

因此，当我们谈论维度时，它意味着有多少个不同的垂直方向，因此每个方向都是这些方向的组合。

So we live in a three-dimensional space because there are three perpendicular directions, forward, backward, left, right, up, down.

所以我们生活在三维空间中，因为有三个垂直的方向，前、后、左、右、上、下。

Those are all 90 degrees away from each other.

它们彼此之间都是 90 度。

So, if people aren’t used to thinking about 90 degrees and orthogonality, one way I like to think of it is I can move in one dimension if it’s orthogonal to the others without moving in the other directions at all.

所以，如果人们不习惯思考 90 度和正交性，我喜欢的一种思考方式是，如果一个维度与其他维度正交，我可以在一个维度上移动，而根本不需要在其他方向上移动。

That’s right.

这是正确的。

I mean, it’s, if I wanted to tell you exactly where I was, it would be how many different pieces of information do I need to tell you?

我的意思是，如果我想告诉你我到底在哪里，我需要告诉你多少条不同的信息？

So I need to give you my coordinates in those three directions.

所以我需要给你这三个方向的坐标。

Mm-hmm.

嗯嗯。

What you care about is if things are smoothly deformable between one space and another space.

你关心的是物体在一个空间和另一个空间之间是否可以平滑变形。

So you don’t care if it’s made of rubber and I literally stretch the distances or contract the distances.

所以你不在乎它是不是橡胶制成的，我确实会拉伸或收缩距离。

What you care about is, if I walk far enough, will I come back to where I started?

你关心的是，如果我走得足够远，我会回到起点吗？

Some kind of global property.

某种全球财产。

Would that be fair?

这样公平吗？

That’s exactly right.

完全正确。

Yeah, so we talk about local structure versus global structure.

是的，所以我们讨论局部结构与全局结构。

For a topologist, everything is made of rubber, and as long as I can stretch two objects and have them come out looking the same, then they are the same.

对于拓扑学家来说，一切都是由橡胶制成的，只要我可以拉伸两个物体并让它们看起来相同，那么它们就是相同的。

So a circle is an oval.

所以圆形是椭圆形。

But they’re definitely not the same thing as a line.

但它们绝对与线条不同。

Right.

正确的。

’Cause a line isn’t connected in that particular way.

因为线路并不是以那种特定的方式连接的。

So topology you’re saying is that study of that large, that global structure.

所以你所说的拓扑学是对那个大的、那个全局结构的研究。

Now, you’ve in particular worked in four dimensions, can you talk us through the fourth direction using the time analogy?

现在，您特别在四个维度上进行了工作，您能用时间类比来告诉我们第四个方向吗？

Because I think that helps people kind of imagine what we’re thinking about when we’re thinking about a dimension beyond the three spatial ones we’re used to occupying.

因为我认为这可以帮助人们想象当我们思考超出我们习惯占据的三个空间维度之外的维度时我们在想什么。

That’s right.

这是正确的。

I mean, having a fourth dimension is a really abstract idea.

我的意思是，拥有第四维度是一个非常抽象的想法。

People understand forward, backward, left, right, up, down.

人们理解向前、向后、向左、向右、向上、向下。

Those are three orthogonal perpendicular directions.

这是三个正交的垂直方向。

How do we move to dimension four from dimension three?

我们如何从第三维度移动到第四维度？

Well, one theme, I think, of four-dimensional topology is that you can’t easily move from three to four-dimensional topology.

嗯，我认为四维拓扑的一个主题是你不能轻易地从三维拓扑转移到四维拓扑。

So, I’ve heard you say also that we shouldn’t be so reliant on our intuition ’cause we can be incredibly surprised when we try to transfer our, our really common-sense intuition from three dimensions to four dimensions.

所以，我还听到你说我们不应该如此依赖我们的直觉，因为当我们试图将我们真正的常识性直觉从三维转移到四个维度时，我们会感到非常惊讶。

It just completely deceives us.

它完全欺骗了我们。

Yeah.

是的。

I like to tell students there are three really important classes that students take, which are real analysis, algebraic structures, and topology.

我喜欢告诉学生们，学生们要学习三门非常重要的课程，即实分析、代数结构和拓扑。

Real analysis is essentially a continuation of calculus, and most students who are taking that class probably have a lot of intuition from their experience in calculus, and probably their intuition is correct, and they’re learning how to formalize it.

实分析本质上是微积分的延续，大多数上这门课的学生可能从他们的微积分经验中获得了很多直觉，而且他们的直觉可能是正确的，他们正在学习如何将其形式化。

And when you learn about algebraic structures, you probably don’t have very much intuition at all because it’s very abstract.

当你学习代数结构时，你可能根本没有太多直觉，因为它非常抽象。

So, what are the techniques that you use to not get trapped?

那么，有哪些技巧可以让你不被困住呢？

[Laughs] Well, so if I am actually trying to understand objects through pictures, if I’m actually constructing an object visually, then I need to have a rigorous system in place for a picture to actually have meaning.

[笑]好吧，所以如果我真的想通过图片来理解物体，如果我真的在视觉上构建一个物体，那么我需要有一个严格的系统来让图片真正有意义。

It can’t just be based on my intuition for how the object behaves.

它不能仅仅基于我对对象行为的直觉。

I need to formalize if I draw a plane and a line intersecting it, what does that actually mean?

如果我画一个平面和一条与它相交的线，我需要形式化，这实际上意味着什么？

And so there’s some real analysis that goes into that.

因此，我们对此进行了一些真正的分析。

So, it’s a combination of visual representation, but formalized mathematically?

那么，它是视觉表示的组合，但以数学形式化？

That’s right, and not everything is done through a visual representation.

没错，并不是所有事情都是通过视觉表示完成的。

I like to work like that because that’s something that really draws me to mathematics, but there’s lots of excellent topologists, you know, four-dimensional topologists who really don’t make use of visual representation in their work.

我喜欢这样的工作，因为这确实吸引了我对数学的兴趣，但是有很多优秀的拓扑学家，你知道，四维拓扑学家在他们的工作中确实没有使用视觉表示。

It’s not always important.

这并不总是重要的。

With these four-dimensional spaces and your intuition faltering, what are the kinds of questions that you’re asking?

在这些四维空间和你的直觉动摇的情况下，你会问什么样的问题？

What are the hypotheses that you’re really pursuing?

你真正追求的假设是什么？

So the thing that I study most commonly is the study of knotted surfaces.

所以我最常研究的就是打结曲面的研究。

So in dimension three, the analogous object would be a knot.

所以在第三维度中，类似的物体将是一个结。

Knot — k-n-o-t.

结——k-n-o-t。

So a knot is a loop.

所以结就是一个环。

If I took a piece of string…

如果我拿一根绳子……

Wait, I actually, I have a string.

等等，实际上，我有一根绳子。

If I took a piece of string, and I actually tangled it up, but then I glued the ends together to make one closed loop, then I would call that a knot.

如果我拿一根绳子，实际上把它缠结在一起，然后我将两端粘在一起形成一个闭合环，那么我会称之为结。

Yeah, and so we, we study those objects up to continuous deformation.

是的，所以我们研究这些物体的连续变形。

So I can stretch the knot around, twist it, but I can never pass it through itself.

所以我可以把绳结拉长、扭动，但我永远无法让它自行穿过。

And this is an important object in three-dimensional topology.

这是三维拓扑中的一个重要物体。

Well, less like a ribbon and more like a balloon.

嗯，不太像丝带，而更像气球。

So I wanna have an object that doesn’t have any edges.

所以我想要一个没有任何边缘的物体。

But there’s also more complicated surfaces.

但也有更复杂的表面。

So a balloon is a sphere or a 2-sphere.

所以气球是一个球体或一个二元球体。

It’s very common to say the dimension of the space.

说空间的维度是很常见的。

I have a number of questions.

我有很多问题。

One is you said that you can understand all three-dimensional spaces in terms of these knots.

一是你说你可以通过这些结来理解所有三维空间。

That seems like a big leap.

这似乎是一个很大的飞跃。

Can you help us make that connection?

你能帮助我们建立这种联系吗？

So this is a big theorem, I think 1960s due to Lickorish and Wallace, Lickorish-Wallace theorem.

所以这是一个大定理，我认为是 20 世纪 60 年代由于 Lickorish 和 Wallace，Lickorish-Wallace 定理。

So, to be more specific, I’m studying three-dimensional manifolds, and I want to say that these three manifolds don’t have boundary, and they’re compact, meaning I don’t ever go out to infinity.

所以，更具体地说，我正在研究三维流形，我想说这三个流形没有边界，而且它们是紧的，这意味着我永远不会达到无穷大。

So I can, maybe I’ll say finite volume.

所以我可以，也许我会说有限体积。

And I also want them to be orientable.

我还希望它们能够定向。

So think about, like, a Möbius band as an example of something that’s not orientable because it has only one side, is what people often say.

因此，想象一下莫比乌斯环作为一个无法定向的例子，因为它只有一侧，这是人们常说的。

So, I put some restrictions.

所以，我做了一些限制。

I want to only study reasonable three-dimensional spaces.

我只想研究合理的三维空间。

I see.

我懂了。

So if I’m understanding correctly, the knots themselves, you were saying I can start with the simply connected finite space, which is the 3-sphere, and I can construct any other topology in three dimensions by performing these kinds of surgeries on knots.

因此，如果我理解正确的话，结本身，你是说我可以从简单连接的有限空间（即 3 球体）开始，并且我可以通过对结进行此类手术来构造三维中的任何其他拓扑。

Is that what you’re saying?

你是这么说的吗？

That’s right, and so that’s exactly right.

没错，所以这是完全正确的。

And it’s called Dehn surgery.

这就是所谓的 Dehn 手术。

And so that way you make more and more complicated spaces far beyond the simplicity of the smooth, compact, connected three-dimensional sphere.

这样你就可以创造出越来越复杂的空间，远远超出了光滑、紧凑、相连的三维球体的简单性。

You make these very complicated spaces with handles and holes and kind of an origami.

你用把手、孔和折纸来制作这些非常复杂的空间。

And you’re trying to do the same thing.

而你也在尝试做同样的事情。

You’re trying to start with the simplest space in four dimensions and find all the others on the basis of the knots?

你试图从最简单的四维空间开始，然后根据结找到所有其他空间？

Yeah.

是的。

And so it’s almost true now that I can start out with the simplest space, which is called the 4-sphere, which is harder to imagine because it’s four-dimensional.

所以现在我几乎可以从最简单的空间开始，它被称为 4 球体，这更难以想象，因为它是四维的。

But it’s some standard four-dimensional space that we all agree to start with.

但我们都同意从某种标准的四维空间开始。

And the surfaces that I’m going to take are tori, which are these surfaces of bagels or donuts sitting inside of four-dimensional space.

我要采用的表面是环面，它们是位于四维空间内的百吉饼或甜甜圈的表面。

And I’m doing surgery, which means that I’m deleting one at a time.

我正在做手术，这意味着我一次删除一个。

I’m gonna delete a torus, re-glue it back in, but I’m allowed to, like, re-glue it in a funny way, and I’ll change the four-dimensional space.

我要删除一个圆环，将其重新粘回去，但我可以以一种有趣的方式重新粘合它，并且我将更改四维空间。

I see.

我懂了。

And has that been a successful program?

这是一个成功的计划吗？

Well, so that’s a fact about dimension four.

嗯，这就是第四维的事实。

This is a cool theorem, I think it’s of Iwase, several years ago.

这是一个很酷的定理，我认为是岩濑几年前提出的。

It’s not exactly a program.

这不完全是一个程序。

It’s more a motivation for why it’s worthwhile to study surfaces.

这更多的是为什么值得研究表面的动机。

Any question about four-dimensional spaces, in principle, I could rephrase as a question about surfaces, and that’s just one example.

原则上，任何关于四维空间的问题，我都可以改写为关于曲面的问题，这只是一个例子。

There’s other aspects of four-dimensional spaces that we tend to study using surfaces.

我们倾向于使用曲面来研究四维空间的其他方面。

So when I listen to that with my topologist hat on, I’m thinking that there’s a gambit that Maggie is using here that we’re trained to use in topology all the time.

因此，当我戴着拓扑学家的帽子听时，我在想玛吉在这里使用的一个策略，我们一直受过拓扑学训练。

To use, as she puts it, these smaller dimensional objects to probe higher dimensional objects.

正如她所说，使用这些较小维度的物体来探测更高维度的物体。

And since it’s not something we all do every day, I thought rather than talking about the four-dimensional usage, what if we just talked about something like the kitchen table?

由于这不是我们每天都会做的事情，我想与其谈论四维的用法，不如我们只谈论厨房桌子之类的东西呢？

You know, so like if I put a piece of string on the table and just make it into the shape of a circle, there’s a very deep topological thing that has happened as soon as I do that, which is that the loop of string separates the table into two parts, one of which we call the inside and one of which we call the outside.

你知道，就像我把一根绳子放在桌子上，然后把它做成一个圆形，一旦我这样做，就会发生一个非常深刻的拓扑学的事情，那就是绳子的循环将桌子分成两部分，其中一个我们称为内部，另一个我们称为外部。

Yeah.

是的。

Well, look, we have, skin, two-dimensional surface, and that separates in three dimensions our inside from our outside.

好吧，看，我们有皮肤，二维表面，它在三个维度上将我们的内部与外部分开。

But it wouldn’t if we lived in 4D.

但如果我们生活在 4D 环境中就不会了。

Our skin would not be sufficient to completely enclose our internal organs.

我们的皮肤不足以完全包围我们的内脏。

Very convenient then that we don’t live in 4D because we’d be spilling.

这非常方便，因为我们不用生活在 4D 环境中，因为我们会溢出。

Right.

正确的。

We’d, we’d just- You’d be able to just, like, go in and pluck out, you know, without surgery.

我们，我们只是 - 你可以，就像，进去然后拔出来，你知道，无需手术。

Surgery would be a lot easier.

手术会容易很多。

Yeah.

是的。

Well, she also talks about something that’s similar to your classroom experience, which is in some sense, the connectedness of spaces can be categorized by thinking about how you can divide things, how many cuts you have to make, essentially, to really cut the thing in half.

嗯，她还谈到了一些与你的课堂体验类似的东西，从某种意义上说，空间的连通性可以通过思考如何划分事物、你必须做多少次切割来分类，本质上，就是把东西真正切成两半。

Right.

正确的。

The other thing that hit me in her discussion was the difficulty of saying what you mean by a hole.

在她的讨论中让我印象深刻的另一件事是很难说出洞的含义。

So, you know, where this really became a live discussion on the internet was if I look at a pair of pants, so just ordinary pants, how many holes are in there?

所以，你知道，这真正成为互联网上的热门讨论的是，如果我看一条裤子，那么只是普通的裤子，里面有多少个洞？

Okay, so there’s one, two.

好的，所以有一个，两个。

But some people wanna say three because, you know, like there’s a loop at the bottom of your leg and then on your other leg and then there’s your belt.

但有些人想说三个，因为，你知道，就像你的腿底部有一个环，然后是你的另一条腿，然后是你的腰带。

It seems like there’s three.

好像有三个。

Well, okay.

好吧，好吧。

Wait.

等待。

So, incontractible loops, I have one, two…

所以，不可收缩的循环，我有一个、两个……

I guess…

我猜……

three.

三.

I’m gonna let the audience ponder this one.

我会让观众思考这个问题。

Where are you taking us next?

接下来你要带我们去哪里？

Okay.

好的。

Well, there is something, Steve, that I wanted your help with.

好吧，史蒂夫，我有件事需要你的帮助。

There’s a topic that comes up after the break, and that is Seifert surfaces.

休息后出现了一个话题，那就是塞弗特的表面。

I think you’ve thought about this.

我想你已经考虑过这一点。

I have.

我有。

They’ve actually come up in some of my applied math work.

它们实际上出现在我的一些应用数学工作中。

And probably an easy way to picture them is to imagine a game you might have played as a kid where you take a loop of wire to make bubbles outside.

想象它们的一个简单方法可能是想象一个你小时候玩过的游戏，你用一圈金属丝在外面制造泡泡。

And then if you dunk that in the soapy water, you get a kind of soap film that spans that loop.

然后，如果你把它浸入肥皂水中，你就会得到一种跨越该环的肥皂膜。

Well, that disc is an example of a Seifert surface because it’s a surface whose boundary is a given curve, in this case, a loop.

那么，该圆盘是 Seifert 曲面的一个示例，因为它的边界是给定曲线（在本例中为环）。

That’s actually a helpful visualization, and we’re gonna get into that and more just after the break.

这实际上是一个有用的可视化，我们将在休息后讨论这一点以及更多内容。

Welcome back to The Joy of Why .

欢迎回到“为什么的喜悦”。

We’re joined today by Maggie Miller, a topologist at the University of Texas at Austin.

今天，德克萨斯大学奥斯汀分校的拓扑学家玛吉·米勒 (Maggie Miller) 也加入了我们的行列。

Yeah, that’s right.

是的，没错。

And I’ll just say that was with Kyle Hayden, uh, Sungwan Kim, JungHwan Park, and Isaac Sundberg.

我只想说那是与凯尔·海登、呃、Sungwan Kim、JungHwan Park 和 Isaac Sundberg 一起的。

A Seifert surface is a surface where it’s not just an abstract surface, it’s a surface that actually lives inside of three-dimensional space.

Seifert 曲面不仅仅是一个抽象曲面，它还是一个实际存在于三维空间内的曲面。

So I really wanna make this distinction, even though I know a donut lives inside of our three-dimensional space.

所以我真的很想做出这种区分，尽管我知道甜甜圈生活在我们的三维空间内。

I’m treating it as its own abstract object, like, outside of space.

我将它视为自己的抽象对象，就像空间之外的一样。

It’s in some weird place.

它在某个奇怪的地方。

But a Seifert surface is really here.

但塞弗特曲面确实存在。

So the idea is you’re imagining, at least with some of the knots, that if you increase the number of dimensions in which they live, that you might be able to unknot them one into the other without cutting or breaking or gluing.

所以你的想法是，至少对于一些结来说，如果你增加它们生存的维度数量，你可能能够将它们从一个结解开到另一个结，而无需切割、断裂或粘合。

That’s right.

这是正确的。

I mean, that happens very often.

我的意思是，这种情况经常发生。

So in the case of knots, like a knotted loop, it’s only knotted because we’re in three-dimensional space.

因此，就结而言，就像打结的环一样，它之所以打结只是因为我们处于三维空间中。

If I had a fourth dimension of freedom, if I moved into a four-dimensional space, then my knot would actually become trivial.

如果我有第四维的自由，如果我进入到一个四维的空间，那么我的结其实就变得微不足道了。

It’s so small in dimension compared to the ambient space that it can’t be complicated anymore.

与周围空间相比，它的尺寸是如此之小，以至于不能再复杂了。

So it would untie effectively?

那么这样就能有效解开吗？

It would unknot effectively?

它会有效地解开吗？

Yeah.

是的。

That’s right.

这是正确的。

It would just untie.

它只会解开。

So in, in four dimensions, in the standard four-dimensional space, there is exactly one knotted loop.

那么在四个维度中,在标准的四维空间中,恰好存在一个有结的环。

They’re all the same.

他们都是一样的。

Ah, I see.

啊，我明白了。

But that’s not the case if they’re not …

但如果他们不是的话，情况就不是这样了……

Wait, you’re saying if you- if there are knots in three dimensions and you up the dimension to four, then they’re all the same?

等等，你是说如果你-如果在三个维度上有结并且你把维度增加到四个，那么它们都是一样的？

Then they’re all the same.

那么他们都是一样的。

But if you have a knot that lives in four dimensions, so it’s a surface knot, and you go up a dimension, are they also then all the same?

但是如果你有一个存在于四个维度的结，那么它是一个表面结，并且你上升到一个维度，那么它们也是一样的吗？

That’s not so clear, but if I go up two dimensions to dimension six, then they’re all the same.

这还不太清楚，但如果我从二维上升到六维，那么它们都是一样的。

If I increase the dimension enough, eventually they’ll all be the same.

如果我将维度增加得足够多，最终它们都会是一样的。

So when I learned a little bit about this, you can take kind of a two-dimensional example, just which might be easier for people to visualize.

所以当我了解了一些关于这一点的内容时，你可以举一个二维的例子，这可能更容易让人们形象化。

If you imagine one small rubber band inside a bigger rubber band just on a table.

如果你想象桌子上有一根小橡皮筋套在一根大橡皮筋里面。

You can’t get them past each other, right?

你无法让他们互相超越，对吗？

That’s right.

这是正确的。

That’s exactly the right idea.

这是完全正确的想法。

That’s the same reason as why knots in dimension three become trivial or all the same in dimension four.

这与为什么第三维度中的结在第四维度中变得微不足道或完全相同的原因相同。

Like, somehow I just need that one extra dimension to pull the rubber band.

就像，不知怎的，我只需要一个额外的维度来拉橡皮筋。

So let’s go back to the Seifert surfaces.

让我们回到 Seifert 曲面。

What was the question that you addressed about their behavior in four dimensions?

关于他们的行为，您在四个维度上提出了什么问题？

Yeah.

是的。

So we were really interested in this paper by Charles Livingston, Chuck Livingston, where he was studying Seifert surfaces for the unknot and the unlink, where the unlink is a bunch of unknots that are all unlinked from each other.

因此，我们对 Charles Livingston、Chuck Livingston 的这篇论文非常感兴趣，他正在研究 Seifert 曲面的 unknot 和 unlink，其中 unlink 是一堆彼此不链接的 unknot。

So just a bunch of round circles that have nothing to do with each other.

所以只是一堆彼此无关的圆圈。

Yeah.

是的。

So even though the unlink is somehow not very interesting, I could still find complicated Seifert surfaces for the unlink that are different from each other.

因此，即使取消链接在某种程度上不是很有趣，我仍然可以找到彼此不同的复杂 Seifert 曲面。

So the surface boundary, trivial, not interesting.

所以表面的边界，琐碎，无趣。

But the surface itself is complicated.

但表面本身很复杂。

Yeah.

是的。

I think we could do a little more justice to the idea of these are the same.

我认为我们可以更公正地对待这些相同的想法。

That is really one of the core principles of topology is, what do you mean by the same?

这确实是拓扑的核心原则之一，“相同”是什么意思？

Can you clarify that?

你能澄清一下吗？

Yeah, it’s a core principle what does it mean for two, uh, objects to be the same.

是的，这是一个核心原则，两个对象相同意味着什么。

It’s also a core principle that there are lots of different notions of the sameness.

同一性有很多不同的概念，这也是一个核心原则。

So this is always whenever you define an object, you have to say, what is it going to mean for two to be the same.

所以每当你定义一个对象时，你总是必须说，两个相同的对象意味着什么。

So far we’ve talked about manifolds, which included surfaces and knots.

到目前为止，我们已经讨论了流形，其中包括曲面和结。

So for manifolds, we typically use the definition homeomorphic or homeomorphism.

因此，对于流形，我们通常使用同胚或同态的定义。

So we say that two manifolds are homeomorphic if there is a homeomorphism from one to the other.

因此，如果一个流形与另一个流形之间存在同胚，我们就说两个流形是同胚的。

And what that means, if I have my two manifolds, let’s call them X and Y, they’re homeomorphic if I can find a function, F, that goes from X to Y.

这意味着，如果我有两个流形，我们称它们为 X 和 Y，如果我能找到一个从 X 到 Y 的函数 F，它们就是同胚的。

And the function needs to be continuous, and it also needs to be a bijection, which means that it’s one-to-one and on two.

并且该函数需要是连续的，并且还需要是双射，这意味着它是一对一和对二的。

If I were to give this formalized mathematical description to a child with some Play-Doh, I would say, “The game is, can you make this shape into that shape without breaking it, without puncturing or re-gluing?” As long as you’re just smoothly mushing dough, then topologically we’re gonna consider those homeomorphic.

如果我要给一个拿着培乐多的孩子这个形式化的数学描述，我会说，“游戏是，你能把这个形状变成那个形状而不打破它，不刺穿或重新粘合吗？”只要你能顺利地搅碎面团，那么从拓扑上讲，我们就会认为它们是同胚的。

Right.

正确的。

Yeah.

是的。

So it’s okay to stretch things.

所以拉伸事情是可以的。

It’s okay to twist things.

扭曲事情是可以的。

If I have two knots, if I have one knot and I’m trying to turn it into another knot, I can make the string longer and I can add twists, but I can never pass the string through itself, which would break it.

如果我有两个结，如果我有一个结，并且我想把它变成另一个结，我可以使绳子更长，我可以添加扭曲，但我永远不能让绳子穿过它自己，这会破坏它。

Now, I think the question of games is not totally irrelevant, because for people outside of mathematics, when they’re listening to the problems that mathematicians are addressing, there’s an attempt to assess, is this just a game that’s being played?

现在，我认为游戏的问题并不是完全无关紧要的，因为对于数学之外的人来说，当他们听到数学家正在解决的问题时，他们会试图评估，这只是一个正在玩的游戏吗？

Oh, that’s interesting.

哦，这很有趣。

I think, hmm, I think the fundamental or the most important results are things that are discovered, but the way that we get to them is often by winning a game.

我认为，嗯，我认为最根本或最重要的结果是被发现的东西，但我们获得这些结果的方式通常是通过赢得一场比赛。

I think that’s exactly right.

我认为这是完全正确的。

If I were trying to solve a problem, let’s say I have a conjecture that I think is true, and I need to prove that it’s true for all knots.

如果我试图解决一个问题，假设我有一个猜想，我认为它是正确的，并且我需要证明它对于所有结都是正确的。

Then I might phrase it as a game, well, if I, if I give you a picture of a knot, and I tell you, “Okay, you can start off with this picture, but you’re allowed to change the picture in the following ways,” and I tell you exactly what’s allowed, is it possible to do only those things and eventually get to whatever it is that I’ve conjectured is true?

然后我可以把它说成是一个游戏，好吧，如果我给你一张结的图片，然后我告诉你，“好吧，你可以从这张图片开始，但你可以用以下方式改变图片，”我确切地告诉你什么是允许的，是否有可能只做这些事情，并最终得到我猜想的真实的东西？

For people who are outside of mathematics, who are trying to understand what’s at stake, why are these different games important and compelling?

对于那些不懂数学、试图理解其中的利害关系的人来说，为什么这些不同的游戏如此重要且引人注目？

And I know that’s an existential question for mathematics.

我知道这是数学的一个存在主义问题。

The games that we were mentioning before are always going into, well, if I could win this game, then I prove this conjecture.

我们之前提到的比赛总是会进入，嗯，如果我能赢得这场比赛，那么我就证明了这个猜想。

So at the heart of it, it’s really why is pure mathematics important?

那么，从本质上讲，这就是为什么纯数学如此重要？

The one issue is that it’s not always clear to anybody working in the field why it’s important for the real world.

一个问题是，该领域的工作人员并不总是清楚为什么它对现实世界如此重要。

Yeah.

是的。

And I understand that there’s a sense in which it’s an unfair question.

我知道从某种意义上来说这是一个不公平的问题。

It’s more sort of giving mathematicians an opportunity to address it, because all the time physicists face similar questions.

这更像是给数学家一个解决这个问题的机会，因为物理学家一直面临着类似的问题。

What is the significance of your work to my daily life?

你的工作对我的日常生活有何意义？

And the real answer is that’s not what we care about, and that’s okay.

真正的答案是，这不是我们关心的，没关系。

Well yeah, the real answer is that’s not the goal.

是的，真正的答案是这不是目标。

In fact, I think most topologists, if you ask, they’d probably try to link it back to physics and then hope you don’t ask why the physics part matters.

事实上，我认为大多数拓扑学家，如果你问的话，他们可能会尝试将其与物理学联系起来，然后希望你不要问为什么物理学部分很重要。

Right.

正确的。

Well, let’s, you know what?

好吧，让我们，你知道吗？

Let’s play that game because that’s interesting and fun.

让我们玩这个游戏吧，因为它很有趣。

There, there’s a really valuable sense in which topology has huge application to theoretical physics.

在那里，拓扑学在理论物理中有着巨大的应用，这是非常有价值的。

There’s lots of complex ways in which topology applies, but there’s one that’s kind of intuitive, which is that the universe is probably a three-dimensional manifold.

拓扑的应用有很多复杂的方式，但有一种是直观的，那就是宇宙可能是一个三维流形。

It might be four-dimensional manifold if we include time in the real sense, right?

如果我们包括真正意义上的时间，它可能是四维流形，对吗？

Not just another spatial direction, but time in which measures are different on that manifold for time than they are for space.

不仅仅是另一个空间方向，还有时间，其中时间流形上的测量与空间上的测量不同。

Do you think about those implications for physics?

您考虑过这些对物理学的影响吗？

Well, so actually what’s interesting is that I think most topologists would go the opposite direction and think, “Well, I never think about what physicists need, sorry.” But I’m very interested in what physicists bring to topology.

好吧，实际上有趣的是，我认为大多数拓扑学家会走相反的方向并思考，“好吧，我从来没有考虑过物理学家需要什么，对不起。”但我对物理学家给拓扑学带来的影响非常感兴趣。

I love that.

我喜欢那个。

I love that it goes in both directions.

我喜欢它是双向的。

I mean, of course, physics also has to grapple with the question of why three dimensions.

我的意思是，当然，物理学也必须解决为什么是三维的问题。

And so a lot of physicists are interested in higher-dimensional spaces because we think the universe may well be higher dimensional.

因此，许多物理学家对高维空间感兴趣，因为我们认为宇宙很可能是更高维的。

Yeah.

是的。

That totally makes sense.

这完全有道理。

And of course, there’s lots of mathematicians who study higher dimensional spaces.

当然，有很多数学家研究高维空间。

A lot of people would say, “Oh, I’m a 3-manifold topologist.

很多人会说：“哦，我是一个三流形拓扑学家。

I’m a 4-manifold topologist.” It’s not so common to have a specific higher number.

我是一个四流形拓扑学家。”具有特定的更高数字并不常见。

Like, oh, nobody would say they study 8-dimensional topology…

就像，哦，没有人会说他们研究 8 维拓扑......

No?

没有？

How come?

怎么会？

’Cause it’s too generalizable after eight, it’s just N?

因为八之后就太笼统了，只是 N？

They tend to be techniques that are special to three, special to four, and then higher.

它们往往是三级特有的技术，四级特有的技术，然后是更高级别的技术。

But there are lots of people who work in higher dimension, especially algebraic topology tends to be higher arbitrary dimension.

但有很多人在更高维度上工作，特别是代数拓扑往往是更高的任意维度。

Just a curiosity, is there something special about three dimensions and four dimensions?

只是好奇，三维和四维有什么特别之处吗？

Yeah, there is something special, which I think in dimension three, in some ways it’s similar to dimension two.

是的，有一些特别的东西，我认为在第三维度中，在某些方面它与第二维度相似。

We all know that it’s impossible even for the most accomplished mathematician to literally visualize higher dimensional spaces.

我们都知道，即​​使是最有成就的数学家也不可能真正想象出更高维的空间。

Is there a sense in which you feel you have a kind of a mental theater?

您是否觉得自己有一种心理剧场？

Yeah, I think in dimension four, I really literally imagine and will draw what I described before as a movie.

是的，我认为在第四维度中，我真的想象并将画出我之前描述的电影。

I really like to just draw a sequence of three-dimensional spaces, and so okay, here’s three-dimensional space at time zero, and a little bit to the right of it, I’ll draw another three-dimensional space, be like, this is time one, and a little bit to the left, I’ll draw the same three-dimensional space, be like, this is time negative one.

我真的很喜欢绘制一系列三维空间，所以好吧，这是时间为零的三维空间，在它的右侧一点，我将绘制另一个三维空间，就像，这是时间一，在左侧一点，我将绘制相同的三维空间，就像，这是时间负空间。

And in some sense, that’s a literal picture of a four-dimensional space.

从某种意义上说，这是四维空间的字面图片。

There’s some interpolation of, it’s not really time straight from zero straight to one.

有一些插值，这并不是真正从零直接到一的时间。

There has to be like, there’s one half in the middle, We can’t draw all of them.

必须有这样的情况，中间有一半，我们不能把它们全部画出来。

You could go up one more.

还可以再上去一层。

I have this, like, sort of secret goal of I would really like to draw like a six or maybe seven-dimensional picture where the, one of the dimensions is you turn the page.

我有一个秘密目标，我真的很想画一张六维或七维的图画，其中一个维度是你翻页的地方。

Right.

正确的。

Exactly.

确切地。

Or you have to build it in a room.

或者你必须把它建在一个房间里。

Oh, that’d be interesting too.

哦，那也很有趣。

I wanted to ask you a question that we like to ask here at The Joy of Why , which is what brings you the most joy in your work?

我想问你一个我们喜欢在“为什么的快乐”中问的问题，即什么给你带来工作中最大的快乐？

I mean, I think that I have probably the same answer that a lot of mathematicians, where we tend to work on problems that are really hard and abstract, and maybe you think about something for several years before you understand the answer.

我的意思是，我认为我的答案可能与许多数学家相同，我们倾向于研究非常困难和抽象的问题，也许你会思考一些事情几年才能理解答案。

I was working on more than one thing at a time.

我一次只做一件事。

I know very few people who are stuck on something for several years, and just that’s the only thing they’re doing.

我知道很少有人会在某件事上坚持好几年，而这就是他们唯一做的事情。

But every once in a while, you actually solve a problem.

但每隔一段时间，你就会真正解决一个问题。

It feels really good.

感觉真的很好。

I mean, I think everybody can relate to that feeling of you’re working on a hard puzzle of some kind, like a crossword, and just this aha moment where you finally get an answer.

我的意思是，我认为每个人都能体会到你正在解决某种难题的感觉，比如填字游戏，而就在这个顿悟时刻，你终于得到了答案。

It’s a big relief.

这是一个很大的解脱。

Yeah.

是的。

I think it’s interesting to be in a subject where you’re always working at the very edge of your own abilities and maybe what’s even possible in the field.

我认为在一个学科中很有趣，你总是在自己能力的最边缘工作，甚至是在该领域的可能性。

And it can be very frustrating.

这可能会非常令人沮丧。

It’s so frustrating.

太令人沮丧了。

All the time I just like, “Why can’t I have a hint button?” You know?

我总是喜欢问“为什么我不能有提示按钮？”你知道？

[Laughs] Right?

[笑]对吧？

Well, Maggie, thank you so much.

嗯，玛吉，非常感谢你。

It’s such an incredible field and I love this connection with the visual.

这是一个令人难以置信的领域，我喜欢这种与视觉的联系。

Thank you so much for taking time with us.

非常感谢您抽出时间与我们在一起。

Yeah, thank you for having me.

是的，谢谢你邀请我。

It was interesting to talk about math.

谈论数学很有趣。

[Laughs] It’s very charming to listen to the two of you talking about what is really a data visualization problem, and I’m reminded of the great book by Edward Tufte, Visual Display of Quantitative Information .

[笑]听你们两个谈论什么是真正的数据可视化问题非常迷人，我想起了爱德华·塔夫特（Edward Tufte）的伟大著作《定量信息的可视化显示》。

Oh, well, okay.

哦，好吧，好吧。

Do you know the book?

你知道这本书吗？

Oh, come on.

哦，来吧。

You’re teasing me.

你在逗我。

Okay, fine.

好吧，好吧。

The title may not be the selling point for the book, but he has one diagram of Napoleon’s march into Russia.

书名可能不是这本书的卖点，但他有一张拿破仑进军俄罗斯的图表。

So you see the thickness of a line that represent how many soldiers Napoleon has at a given time.

所以你可以看到一条线的粗细，它代表拿破仑在给定时间有多少士兵。

And as this line starts moving across the page, this black band starts getting thinner and thinner as Napoleon’s army is getting depleted.

当这条线开始在页面上移动时，随着拿破仑的军队逐渐耗尽，这条黑色带子开始变得越来越细。

Meanwhile, there’s other data showing the temperature, how cold it was, and what the date was when they attacked this or that place.

与此同时，还有其他数据显示温度、有多冷，以及他们袭击这个或那个地方的日期。

Yeah, visualizations in mathematics is fascinating.

是的，数学可视化很有趣。

I personally really appreciate visualizations, and yet I understand it’s a limitation.

我个人非常欣赏可视化，但我知道这是一个限制。

But what she described is actually a really fun way to imagine a higher dimensional space, is to unpack it in some sense.

但她所描述的实际上是想象高维空间的一种非常有趣的方式，就是在某种意义上解开它。

Just draw it here, then draw it there.

只需将其绘制到此处，然后将其绘制到那里即可。

Well, we have to unfortunately leave this abstract realm behind.

好吧，不幸的是我们不得不把这个抽象领域抛在脑后。

Back to reality.

回到现实。

But Steve, thank you so much for being a partner in crime

但是史蒂夫，非常感谢你成为犯罪同伙

Very good.

非常好。

See you next time.

下次见。

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