What Does the Fourth Dimension Actually Look Like?

但是……

But...

所以至少有一组非常简单的四维空间，我可以根据想要得到的目标空间来选择起始空间。

So at least there is a very simple family of four dimensional spaces that I start out with depending on what space I wanna get.

我知道该从哪个开始，只需要对环面进行手术，就能得到我想要的任何空间。

I know which one to start with and all I have to do is do surgery on tori and I can get to whatever space I wanted.

明白了。

I see.

那这个方案进展如何？

And how's that been a successful program?

嗯，这是关于四维空间的一个事实。

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

如果这是个了不起的定理，我想那是几年前由Iwasse提出的。

If this is a cool theorem, I think it's of Iwasse several years ago.

这并不完全算是一个方案。

It's not exactly a program.

这更像是研究曲面的动机，说明为什么曲面值得研究。

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

任何关于四维空间的问题，原则上我都可以重新表述为关于曲面的问题。

Any question about four dimensional spaces, in principle, I could rephrase as a question about surfaces.

而这只是一个例子。

And that's just one example.

四维空间还有其他方面，我们倾向于用曲面来研究。

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

所以举个例子，让我们回到更低维的空间。

So for example, let's move back down to a lower dimensional space.

让我们再看看曲面。

Let's actually look at surfaces again.

我之前举过两个曲面的例子，就是二维球面和环面。

I gave two examples of surfaces before, which were the two sphere and the torus.

第一次学这些对象时，真的很难用语言说清楚它们之间的区别。

It's actually pretty hard the first time you learn about these objects to really verbalize what is the difference between them.

当我在面向更普通听众的演讲中，我会说，哦，环面有个洞，而球面没有洞。

When I give a talk that's a little bit more general audience, I'll say something like, oh, well, the torus has a hole in it and the sphere doesn't have a hole.

但一般来说，如果我问一个学生，好吧。

But it's pretty hard on average if I asked a student, okay.

现在告诉我什么是洞。

Now tell me what a hole is.

比如，环面有洞是什么意思？

Like, what does it mean that the torus is a hole?

这是个棘手的问题。

That's a tricky question.

所以我喜欢这样想：如果我在二维球面上画一个环，比如地球的赤道，它总是会把球面分成两部分。

And so the way that I like to think about it is that, well, if I draw a loop on a two sphere, so think like the equator of the Earth, it will always divide the two sphere into two pieces.

赤道把地球分成北半球和南半球，不穿过赤道就无法在两者之间移动。

The equator divides the Earth into the northern hemisphere and the southern hemisphere, and you can't go between them without crossing the equator.

环面没有这个性质，因为我可以取一个环绕洞的环。

The torus doesn't have that property because I could take a loop on the torus that goes around the hole.

如果我沿着它切开，实际上并不会把环面切成两半。

And if I cut, it actually doesn't cut the torus into two pieces.

但这样的环只能放一个。

But I can only fit one loop like that.

如果试图沿两条环切开，那最终会把环面切成两半。

If I try to cut along two loops, then I'm gonna end up cutting the torus into two pieces.

所以这告诉我环面有一个洞，我可以把这些子空间，也就是嵌入在曲面里的低维圆圈，形式化地表达出来。

So that tells me that the torus has one hole, and I can formalize it in terms of these subspaces, these circles that are smaller in dimension that sit inside of the surface.

同样，我也可以在更高维度上这样做。

So I can do that in higher dimensions too.

如果我想区分两个四维流形，有时我可以把这个问题转化为：什么样的曲面嵌入在这些四维空间里？

If I'm trying to distinguish two four manifolds from each other, sometimes I can rephrase that as a question about, well, what kind of surfaces sit inside of these four dimensional spaces?

当我以拓扑学家的身份听这些时，我在想，玛吉在这里用了一个我们拓扑学中经常训练使用的策略。

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.

就是像她说的，用这些低维对象来探测高维对象。

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

而且因为这不是我们每天都会做的事情，我想与其讨论四维空间的应用，不如我们聊聊厨房桌子之类的东西？

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?

你懂我的意思吧？

You know?

所以，比如说，如果我在桌子上放一根绳子，把它围成一个圆圈，那么一旦我这样做，就发生了一件非常深刻的拓扑学事情，那就是这个绳圈把桌子分成了两个部分，一个我们称之为内部，另一个我们称之为外部。

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.

这其实挺有意思的。

And that's kind of interesting.

你可能会说，不。

I mean, might say, no.

这没什么。

It's not.

这太明显了。

That's so totally obvious.

但这正是数学家会思考的那种事情。

But this is the kind of thing a mathematician ponders.

好的。

Like, okay.

让我说得更抽象一点。

Let me be more abstract.

与其说桌面，不如说无限平面。

Rather than a tabletop, how about the infinite plane?

就是那个向两个方向无限延伸的XY平面。

You know, the x y plane going out to infinity in both directions.

如果你在XY平面上画一个圆，它会把平面分成两部分，一部分在里面，对。

If you put a circle in the x y plane, it separates the plane into an inside part Yeah.

那部分是有界的，而外面那部分包含无穷远。

That's bounded and an outside part that includes infinity.

即使你把这个圆变形，也总是有一个里面和一个外面。

And that's true even if I deform the loop, there's always one inside and one outside.

嗯，你看。

Well, look.

我们有皮肤，这是一个二维表面，它把我们的内部和外部在三维空间中分隔开来。

We have skin, two dimensional surface, and that separates in three dimensions our inside from our outside.

如果我们生活在四维空间，那就不会这样了。

It wouldn't if we lived in four d.

我们的皮肤就不足以完全包裹住我们的内部器官。

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

那幸好我们不是生活在四维空间，不然我们就要漏出来了。

Very convenient then that we don't live in four d because we'd be spilling.

我们就会……

We'd we'd just...

你就能直接伸手进去把东西拿出来，不用手术。

You'd be able to just like go in and pluck out, you know, without surgery.

手术会容易得多。

Surgery would be a lot easier.

她还提到了一些和你课堂经历相似的东西，那就是在某种意义上，空间的连通性可以通过思考如何分割事物来分类。

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.

基本上需要切几刀才能真正把东西切成两半。

How many cuts you have essentially to really cut the thing in half.

她的讨论中另一个让我触动的地方是，很难说清楚你说的“整体”是什么意思。

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

所以这在互联网上真正引发热议的地方是，如果我拿一条裤子来看，就是普通的裤子，它有几个洞？

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?

好。

Okay.

所以有一个，两个。

So there's one, two.

但有些人会说三个，因为裤腿底部有一个圈，

But some people wanna say three because, you know, like there's a loop at the bottom of your leg

然后另一条腿，

and then on

然后还有你的腰带。

your other leg and then there's your belt.

看起来有三个。

It seems like there's three.

嗯，好吧。

Well, okay.

等等。

Wait.

所以在可收缩的环里，我有一个、两个，我猜是三个。

So in contractible loops, I have one, two, I guess three.

我要

I'm going to

让听众思考一下这个问题。

let the audience ponder this one.

你要带我们去哪里？

Where are you taking Well, us

史蒂夫，这正是我想请你帮忙的事情。

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

休息之后要讨论的一个话题是密码曲面。

There's a topic that comes up after the break, and that is ciphered surfaces.

我觉得你对这个有过思考。

I think you've thought about this.

确实。

I have.

它们其实在我的一些应用数学工作中出现过。

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

可能一个简单的理解方式就是想象小时候玩过的游戏，拿一个铁丝环在外面吹泡泡。

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.

然后把它浸到肥皂水里，就会得到一层横跨这个环的肥皂薄膜。

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

那个薄膜就是一个密码曲面的例子，因为它的边界是一条给定的曲线，这里就是那个环。

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

但密码曲面也可以用于纽结曲线，这时曲面的边界就是那个纽结。

But you could also have ciphered surfaces for knotted curves where the boundary of that surface is the knot.

这在纽结理论中是一个非常实用的构造。

It's a really useful construction in knot theory.

这个可视化确实很有帮助，我们稍后会更深入地讨论这个话题。

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

欢迎回到《为什么的乐趣》。

Welcome back to The Joy of Why.

今天我们的嘉宾是奥斯汀得克萨斯大学的拓扑学家玛吉·米勒。

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

我们来谈谈你与你的合作者解决的一个长期悬而未决的问题，这个问题由查尔斯·利文斯通在1982年提出，引起了广泛关注，与塞弗特曲面有关。

Let's talk about this work you did to answer a long standing question that was posed by Charles Livingston in 1982 with you and your collaborators that received a lot of attention, and that had to do with the Seifert surfaces.

这些曲面有什么特别之处？

What's special about those surfaces?

没错。

That's right.

我要提一下，这项工作是和凯尔·海登、金承元、朴正韩和艾萨克·桑德伯格合作的。

And I'll just say that was with Kyle Hayden, Seung Won Kim, Jung Han Park, and Isaac Sundberg.

塞弗特曲面不是那种抽象的曲面。

The Seifert surface is a surface where it's not just an abstract surface.

它是真正存在于三维空间里的曲面。

It's a surface that actually lives inside of three-dimensional space.

所以我想特别强调这个区别，虽然我知道甜甜圈是存在于我们的三维空间里的。

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

但我是把它当作一个独立的抽象物体来看待的，就像在空间之外。

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

它在某个奇怪的地方。

It's in some weird place.

但塞弗特曲面是真实存在于这里的。

But a seiphert surface is really here.

它们必须存在于三维空间里，这一点很重要，因为当我研究这个曲面时，我可以连续地变形它，但它必须保持在三维空间内。

It's important that they live inside of three-dimensional space because when I study the surface, I'm allowed to continuously deform it, but it has to stay inside of three-dimensional space.

所以我可以有两个不同的环面，它们都是环面，但在三维空间里，我实际上无法把一个变成另一个。

So I could have two different tori that are both tori, but I can't actually turn one into the other inside of three-dimensional space.

更简单的例子就是结，也就是打结的环。

A simpler thing is just even knots, knotted loops.

如果我拿一个真正的圆环，就是一个漂亮的圆形，我们称之为平凡结。

If I take a loop that's really a circle, just a nice round circle, we call that the unknot.

如果我拿一个环，把它打成某种复杂的结，那它就不是平凡结。

If I take a loop and actually tie it into some complicated thing, it won't be the unknot.

它们虽然都是圆，但并不是同一种结。

And they're still both circles, but they're not the same knot.

如果不让它穿过自身，我就无法把一个变成另一个。

I can't actually turn one into the other without passing it through itself.

赛弗特曲面也是同样的道理。

Seifert surfaces are the same thing.

但对于赛弗特曲面，我希望它们有边界。

But for a seifert surfaces, I do want them to have boundary.

它们应该有一个边缘，而这个边缘就是一个结，一个打结的环。

They should have an edge, and the edge will be a knot, a knotted loop.

所以这个想法是，你想象一下，至少对于某些纽结来说，如果你增加它们所在空间的维度，也许就能在不切割、不破坏、不粘合的情况下，把它们互相解开。

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.

没错。

That's right.

这种情况经常发生。

I mean, that happens very often.

所以对于纽结，比如一个打结的环，它之所以是打结的，仅仅是因为我们处在三维空间中。

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

如果我有第四维度的自由度，如果我进入四维空间，那么我的纽结实际上就会变得平凡。

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

它相对于周围空间来说太小了，所以它不能再复杂了。

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

所以它实际上会解开？

So it would untie effectively?

它会有效地解开？

It would unknot effectively?

对。

That's right.

它就会直接解开。

It would just untie.

所以在四维空间里，在标准的四维空间中，恰好只有一种纽结环。

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

它们全都是同一种。

They're all the same.

哦，我明白了。

Ah, I see.

但如果它们是纽结的话，情况就不是这样了……

But that's not the case if they're knots...

等等。

Wait.

你是说如果你……

You're saying if you...

如果在三维空间里有结，把维度提升到四维，它们就都一样了。

If there are knots in three dimensions and you up the dimension to four, then they're all the same.

对，它们就都一样了。

Then they're all the same.

但如果有一个结存在于四维空间，那它就是一个曲面。

But if you have a knot that lives in four dimensions, so it's a surface

结，然后你再升一个维度，它们是不是也都一样了？

Knot, and you go up a dimension, are they also then all the same?

这个就不太确定了。

That's not so clear.

如果升两个维度到六维，那它们就都一样了。

If I go up two dimensions to dimension six, then they're all the same.

只要维度升得足够高，最终它们都会变得一样。

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

我稍微了解过一点这个，你可以拿一个二维的例子来……

So when I learned a little bit about this, you can take kind of a two dimensional example just...

这个可能大家比较容易想象。

Which might be easy for people to visualize.

想象一下，桌上有一个小橡皮筋套在一个大橡皮筋里面，你没法让它们互相穿过。

If you imagine one small rubber band inside a bigger rubber band just on a table, you can't get them past each other.

但如果有三维空间，那就容易多了。

But if you have three dimensions, it's real easy.

你把其中一个抬起来，拉大，然后就能把它放到另一个外面。

You lift one up, you stretch it out, and then you can lay it outside the other.

所以，维度数量让你能够解开结或者在不切割、不粘贴的情况下分离物体，这个想法挺直观的。

So this idea of the number of dimensions allowing you to unknot things or to separate things without cutting and pasting is kind of intuitive.

我们经常用到这个。

We use it all the time.

对，没错。

That's right.

这正是正确的思路。

That's exactly the right idea.

这和为什么三维中的纽结在四维中变得平凡或全都一样是同一个道理。

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

就像，不知怎么的，我只需要多一个维度就能把橡皮筋拉出来。

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

那我们回到那些密码曲面吧。

So let's go back to the seiphered surfaces.

你研究的问题是关于它们在四维中的行为，具体是什么？

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

我们当时对查尔斯·利文斯顿，也就是查克·利文斯顿的那篇论文很感兴趣，他在研究平凡纽结和平凡链环的密码曲面，平凡链环就是一堆互不缠绕的平凡纽结。

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

就是一堆互不相干的圆圈。

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

所以即使平凡链环本身没什么有趣的，我还是能找到复杂的密码曲面，而且它们彼此不同。

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

所以这个曲面的边界……

So the surface boundary...

平凡，不有趣，但曲面本身很复杂。

Trivial, not interesting, but the surface itself is complicated.

利文斯顿证明了，如果我有两个，比如说，连通的密码曲面，它们对应同一个非链环，而且我知道这两个曲面有相同的亏格，也就是我们用来衡量曲面有多少个洞的指标，即使这些曲面在三维空间里可能不一样，但一旦我加上第四维的自由度，它们就变成一样的了。

And Livingston proved that if I have two, let's say, connected ciphered surfaces for the same unlink, and I know that those two surfaces have the same genus, which is our measure of how many holes the surface has, Even though those surfaces are probably not the same inside of three-dimensional space, if I add a fourth dimension of freedom, then they become the same.

我有这个额外的方向可以把它们拉过去。

I have this extra direction that I can pull them in.

在那篇论文里，他特别强调了边界是非链环。

In the paper, he was very specific about the boundary being the unlink.

他还指出，如果边界不是非链环，那这个结论肯定不成立，但他没有给出证明，主要是因为要构造高维对象的奇怪例子真的很难。

And he pointed out that it's surely not true if the boundary is not the unlink, but he didn't have a proof of that primarily because it's really hard to come up with weird examples of higher dimensional objects.

你得先想出一些疯狂的、有相同边界的曲面，然后才能去证明它们彼此不同。

You have to come up with some crazy pair of surfaces with the same boundary in the first place before you can even then try to prove that they aren't the same as each other.

这在拓扑学里一直是个难题。

That's always the problem in topology.

我觉得我们可以更深入地解释一下“它们是相同的”这个概念。

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

这确实是拓扑学的核心原则之一，就是你说的“相同”是什么意思？

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

你能解释一下吗？

Can you clarify that?

这是一个核心原则。

It's a core principle.

两个物体相同意味着什么？

What does it mean for two objects to be the same?

同样重要的是，相同有很多不同的定义。

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

所以这总是……

So this is always...

每当你定义一个对象时，你必须说明两个对象相同是什么意思。

Whenever you define an object, you have to say, what is it going to mean for two objects to be the same?

到目前为止，我们讨论了流形，包括曲面和纽结。

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

对于流形，我们通常用同胚或同胚映射这个定义。

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

也就是说，如果存在一个从流形A到流形B的同胚映射，我们就说这两个流形是同胚的。

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

这意味着，如果我有两个流形，叫它们X和Y，它们同胚，当且仅当我能找到一个从X到Y的函数f，而且这个函数必须是连续的。

And what that means, if I have my two manifolds, call them x and y, they're homeomorphic if I can find a function f that goes from x to y and the function needs to be continuous.

同时它还得是双射，也就是说它既是一一对应的，又是满射的。

And it also needs to be a bijection, which means that it's one to one and onto.

所以X中的任意两个不同的点，在Y中必须对应到两个不同的点。

So any two points in x have to go to two different points in y.

它们不能对应到同一个点。

They can't go to the same point.

而且我还得覆盖Y中的每一个点。

And I also have to hit every single point in y.

X中的某个点必须映射到它。

Something from x has to map to it.

最后一点是，我希望 f 也有一个连续的逆映射。

And then the last thing is that I want f to have a continuous inverse as well.

所以这其实就是一组从 x 到 y 的连续指令，但这三个要点让它成为一个严谨的定义。

So it's just a set of continuous instructions to go from x to y, but these three bullet points make it a rigorous definition.

所以在纽结的背景下，我其实可以用同样的定义。

So in the context of knots, I can use the same definition actually.

我可以说，如果存在一个同胚映射，也就是这种从三维空间到自身的受限函数，那么两个纽结就是等价的。

I could say that two knots are equivalent if there's a homeomorphism, so this kind of restricted function from three space to itself.

当我输入第一个纽结时，它输出第二个纽结。

But when I plug in the first it outputs the second knot.

如果我要把这个形式化的数学描述讲给一个玩橡皮泥的孩子听，我会说，游戏规则是，你能不能在不弄断、不穿孔、也不重新粘合的情况下，把这个形状变成那个形状？

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 regluing?

只要你只是平滑地揉捏面团，那么在拓扑学上，我们就认为它们是同胚的。

As long as you're just smoothly mushing dough, then topologically, we're gonna consider those homeomorphic.

所以拉伸是可以的。

So it's okay to stretch things.

可以扭转东西。

It's okay to twist things.

如果我有两个结……

If I have two knots...

如果我有两个结，想把它变成另一个结，我可以把绳子拉长，也可以加一些扭转，但绝不能让绳子穿过自身，那样就断了。

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.

对。

No.

我觉得游戏这个问题并不是完全不相关。

I think the question of games is not totally irrelevant.

因为对于数学圈外的人来说，听到数学家研究的问题时，他们会想，这到底是不是只是在玩一个游戏？

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?

我觉得从哲学角度来问这个问题就是，你认为数学是被发现的，在某种程度上存在于外部世界，还是说……

And I think the sort of philosophical way to phrase the question is, do you consider mathematics to be something that is discovered, that it exists in some sense out there, or is this...

还是说数学其实是一系列由人类发明的不同游戏？

Or is mathematics sort of a series of different games that are invented by human beings?

哦，这很有意思。

Oh, that's interesting.

我觉得，最根本或最重要的成果是那些被发现的，但我们达到这些成果的方式往往是通过赢得一场游戏。

I think 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.

我觉得这完全正确。

I think that's exactly right.

如果我要解决一个问题，比如说我有一个猜想，我认为它是正确的，我需要证明它对所有纽结都成立，那么我可能会把它表述成一个游戏。

If I were trying to solve a problem, let's say I I have a conjecture that I think is true and I need to prove that it's true for all knots, then I might phrase it as a game.

嗯，如果我给你一张纽结的图片，我告诉你，好的，你可以从这张图片开始，但你可以用以下方式改变图片。

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 picture in the following ways.

我告诉你具体允许做什么。

And I tell you exactly what's allowed.

是否有可能只做那些事情，最终达到我所猜想的那个结果？

Is it possible to do only those things and eventually get to whatever it is that I've conjectured is true?

比如，我认为发现的部分在于首先想出这个游戏。

Like, the part that it's the discovery I think is coming up with the game in the first place.

提出猜想真的很难，因为我们并不总是有很好的直觉。

It's really hard to make a conjecture because we don't always have good intuition.

然后有很多论文就是，嗯，这里有一个我们都感兴趣的大问题。

And then there's lots of papers that are just, well, here's a big problem that we're all interested in.

而我的定理是，我实际上无法回答它，但我可以证明，如果你总能赢得这个游戏，那么这个定理就是真的。

And my theorem is that I can't actually answer it, but I can prove that if you can always win this game, then the theorem is true.

对于数学之外的人来说，他们想理解这些不同的游戏为什么重要、为什么吸引人？

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

我知道这对数学来说是一个根本性的问题。

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

我们之前提到的那些游戏，总是归结为，如果我能赢得这个游戏，那么我就证明了这个猜想。

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

所以归根结底，这其实是纯数学为什么重要的问题。

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

一个问题是，对于在这个领域工作的任何人来说，它为什么对现实世界重要，并不总是很明确。

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

大家喜欢开玩笑说，纯数学从被发明或发现（随便你怎么说）到真正应用到现实生活中，大概有五十年左右的时间差。

People like to joke that there's like a fifty year time gap between pure math being invented or discovered or whichever verb you want to use versus actually being applied in real life.

比如说，在拓扑学里，有一个相对较新的领域叫拓扑数据分析。

So for example, in topology, there's a relatively recent field called topological data analysis.

从名字就能看出来，它跟现实生活的联系更紧密。

You can tell from the name immediately it's much more applicable to real life.

它的核心思想是，如何利用拓扑学来高效地理解非常庞大的数据集，或者从中找出一些规律。

It's the idea of how can we understand very large data sets efficiently or understand some patterns in them using topology.

假设我有一个数据集，它太大了，没法直接搜索，但我可以理解它的整体结构。

Maybe I have some dataset which is way too massive for me to do a search through, but I can understand global patterns.

比如，这个数据集可以近似成某个空间，这一点本身就很有意义，因为它是这个空间，就告诉了我关于数据的一些信息。

Like, it's meaningful that somehow this dataset I can approximate as this space, and the fact that it's this space tells me something about the data.

但拓扑数据分析里用到的那些拓扑学，也就是纯数学的部分，其实已经相对比较古老了，就纯数学本身而言。

But the actual topology, the pure math part that goes into topological data analysis, is at this point relatively old in terms of just literally the pure math component of it.

我们可能用到的是曲面的分类，或者拓扑学里一些经典的拓扑不变量。

Maybe we're using things like classification of surfaces or some classical invariants in topology.

所以它还是在追赶的过程中。

And so it's still somehow catching up.

这些东西很重要，但我们这周做的拓扑学内容还没有被应用到应用数学中。

It's important that we have those things, but the stuff that we're doing in topology this week isn't being implemented in applied math yet.

我理解这个问题在某种意义上有失公平。

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

这更像是给数学家一个机会来回应，因为物理学家也经常面临类似的问题。

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

你的工作对我的日常生活有什么意义？

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

真正的答案是，这不是我们关心的。

And the real answer is that's not what we care about.

这没关系。

And that's okay.

真正的答案是，这不是目标。

The real answer is that's not the goal.

对。

The Yeah.

事实上，我觉得大多数拓扑学家，如果你问他们，他们可能会试图把它和物理学联系起来，然后希望你别再问为什么物理学部分重要。

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.

好吧，我们……

Well, let's...

你知道吗？

You know what?

我们来玩这个游戏吧，因为那很有趣也很好玩。

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

有……

There...

有一种非常有价值的意义，拓扑学对理论物理学有巨大的应用。

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

拓扑学有很多复杂的应用方式，但有一种比较直观，那就是宇宙很可能是一个三维流形。

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.

如果从真正的意义上把时间算进去，宇宙可能是一个四维流形，对吧？不只是另一个空间方向，而是时间，在那个流形上，时间的度量跟空间的度量是不同的。

It might be four dimensional manifold if we include time in the real sense, right, not just another spatial direction, but time in which measures are different on that manifold for time than they are for space.

你觉得这对物理学有什么影响吗？

Do you think about those implications for physics?

嗯，其实有趣的是，我觉得大多数拓扑学家会反过来说，他们根本不会去想物理学家需要什么。

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.

抱歉。

Sorry.

但我很感兴趣的是物理学家给拓扑学带来了什么。

But I'm very interested in what physicists bring to topology.

所以当我读研究生的时候，我选了一门课，那个教授开了个玩笑，我到现在还经常想起。

So when I was in grad school, a professor I was taking a course with made this joke, which I still think about all the time.

他说得完全对。

He's totally right.

每当拓扑学家想找一种新的方法来区分两个空间时，他们就会穿过走廊，走到物理系，物理系肯定就在同一栋楼里，对吧。

Whenever topologists want a new way of distinguishing two spaces from each other, what they do is they walk across the hall to the physics department, which is surely in the same building Mhmm.

然后去问他们，哦，能不能给我一个新的偏微分方程，能告诉我一些关于空间的信息？

And ask them, oh, can you please give me a new partial differential equation that tells me something about spaces?

然后你把它带回来，完全忘了它跟物理学有什么关系，那就是你的新方法。

And then you bring it back and you totally forget that it had anything to do with physics and that's just your new way.

所以我们喜欢说，哦，物理学家告诉我有个描述热流的偏微分方程，我可以数数它有多少个稳定解之类的。

So we we like to say, oh, well, physicists told me that there's this partial differential equation that describes heat flow and I can count how many stable solutions there are or whatever.

我以后就记住那个数字就行了。

And I'm just gonna remember that number from now on.

我不会再去想热流的事。

I'm not gonna think about heat flow.

但这两个空间在这个数字上不一样，所以它们肯定不同。

But it's different for these two spaces, so they must be different.

我喜欢这个。

I love that.

我喜欢这种双向的交流。

I love that it goes in both directions.

当然，物理学也得面对为什么是三维这个问题。

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

所以很多物理学家对高维空间感兴趣，因为我们觉得宇宙很可能就是高维的。

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

所以当我们试图理解可能的宇宙是什么样的，可能的宇宙的几何结构是什么时，我们就得求助于数学家，说，如果我们生活在四维空间，这些宇宙在某种意义上是独一无二的。

And so when when we're trying to understand what are the possible universes, what are the geometries of the possible universe, we have to look to the mathematicians and say, well, if we're in four dimensions, these are unique universes in some sense.

而这些，虽然看起来可能不同，但实际上是一样的。

And these, although they might seem different, are actually the same.

这当然和我们正在做的事情非常相关。

And this is, of course, very relevant for what we're trying to do.

这完全说得通。

That totally makes sense.

而且，当然有很多数学家研究高维空间。

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

很多人会说，哦，我是研究三维流形的，我是研究四维流形的。

Lot of people would say, oh, I'm a three manifold topologist, I'm a four manifold topologist.

不太常见有人专门研究某个更高的数字，比如，没人会说他们研究八维拓扑学，不会的。

It's not so common to have a specific higher number like, oh, nobody would say they study eight dimensional topology No.

我觉得。

I think.

欢迎。

Welcome.

但是……

But...

因为八维以上就太一般化了。

Because it's too generalizable after eight.

就是n维了。

It's just n.

他们往往有一些专门针对三维、专门针对四维的技巧，然后更高的维度就不同了。

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

但是有很多人研究高维空间，尤其是代数拓扑学，往往是在任意高维上研究。

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

只是好奇一下，三维和四维有什么特别之处吗？

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

确实有特别之处，我觉得三维在某些方面和二维很像。

There is something special which I think in dimension three, in some ways, it's similar to dimension two.

在二维曲面方面，我可以列举出来。

In dimension two surfaces, I can enumerate.

我可以完全列出所有不同的二维物体。

I can totally list out all of the different two dimensional objects.

而且你不需要给我很多数据，我就能完全确定是哪种二维流形。

And there's really not very many pieces of data that you would have to give me to totally determine which two dimensional manifold I have.

在三维方面，说起来稍微复杂一点，但确实有对三维物体的完整分类。

In dimension three, it's a little bit more complicated to state, but there is a complete classification of three-dimensional objects.

这个分类要现代得多。

And this is much more modern.

这大概是从七十年代到两千年代，是瑟斯顿和佩雷尔曼的工作。

This is going from maybe seventies to two thousands, work of Thurston and Perelman.

我们完全……

We totally...

嗯，也许“完全”这个词我得小心点，但我想说我们确实完全理解三维流形。

Well, maybe I wanna be careful with the word totally, but I would say that we totally understand three manifolds.

至少我们有一个完整的刻画。

At least we have a complete characterization.

仍然有很多有趣的问题，但通常不是问这两个空间是否不同。

There's still a lot of interesting questions, but it's generally not, are these two spaces different from each other?

在四维中，这个问题更难，但有很多兴趣不一定在于那种分类。

In dimension four, that's a harder question, but there is a lot of interest not necessarily in that classification.

那是不可能的。

That's not possible.

但四维之所以特别，是因为高维或低维的定理在四维中失效，所以动机上有点不同。

But it's special because of the failure of theorems from higher or lower dimension in dimension four, so it's a little bit different in motivation.

但在更高维中，我们不再有那个，也没有分类。

But in higher dimension, we don't have that anymore and we also don't have the classification.

而且还有一点心理上的因素，想象四维空间已经够难的了，但至少借助时间的类比，你还能勉强理解。

And then also, there's this kind of psychological thing of it's hard enough to imagine a four dimensional space, but at least with the time analogy, you can kind of get there.

但要想象一个十四维的空间，就有点难了。

But it's a little bit hard to conceptualize, you know, as a 14 dimensional space.

我觉得那种动力就没那么强了。

I don't think there's that same level of motivation.

我们都知道，即使是最杰出的数学家，也不可能真正在字面上可视化高维空间。

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

你有没有一种感觉，就是你在脑海里有一个类似舞台的东西？

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

我觉得在四维的情况下，我真的会字面地去想象，并且会画出我之前描述的那种电影。

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

我真的很喜欢画一系列的三维空间。

I really like to just draw a sequence of three-dimensional spaces.

然后，好的，这是时间零点的三维空间，在它右边一点，我再画一个三维空间，就像，这是时间一。

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.

稍微靠左一点，我会画出同一个三维空间，然后说这是时间负一。

And a little bit to the left, I'll draw the same three-dimensional space, be like this is time negative one.

从某种意义上说，这就是四维空间的直观图。

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

中间有一些插值……

There's some interpolation of...

并不是直接从零跳到一。

It's not really time straight from zero, straight to one.

中间得有过渡，对吧。

There has to be like Mhmm.

中间还有一半。

There's one half in the middle.

我们不可能把所有的都画出来。

We can't draw all of them.

原则上，在更高维度上你也可以这样做。

You can, in principle, do that in higher dimensions too.

我有篇论文，或者可能几篇论文，里面画了五维空间。

I I have a paper or maybe a few papers where I draw five dimensional spaces as well.

这个就是……

This is...

我画四维空间的方法是画一排三维的图。

I draw four dimensions by drawing a row of three-dimensional pictures.

所以如果我把这些行堆叠起来，形成一个网格，那就是五维了。

So if I stack rows and have a grid, now it's five dimensional.

你可以再往上加一维。

You could go up one more.

我有一个秘密目标，真的很想画一个六维或者七维的图，其中一维是翻页。

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

没错。

Exactly.

或者你得在房间里把它搭出来。

Or you have to build it in a room.

哦，那也会很有趣。

Oh, that'd be interesting too.

我想问你一个问题，这是我们在《为什么的乐趣》中常问的，那就是在你的工作中，什么给你带来最大的快乐？

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?

我想我的答案可能和很多数学家一样，我们往往研究那些非常困难且抽象的问题。

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.

也许你会花上好几年思考某个问题，然后才明白答案。

And maybe you think about something for several years before you understand the answer.

当然，我们同时会处理不止一件事。

Of course, we're working on more than one thing at a time.

我知道很少有人会只专注于一个难题好几年，而不做其他事情。

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

但偶尔，你真的会解决一个问题。

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

那种感觉真的很好。

It feels really good.

我想每个人都能体会到那种感觉，就是你在解一个很难的谜题，哪怕是填字游戏，终于得到答案的那一刻。

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

就像是一种巨大的解脱。

It's like a big relief.

我觉得有意思的是，你所在的领域总是让你在自己的能力极限上工作，甚至可能是在这个领域可能性的极限上。

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.

而且这可能会非常令人沮丧。

And it can be very frustrating.

确实非常令人沮丧。

It's so frustrating.

我一直都在想，为什么我就不能有个提示按钮呢？

All the time, I'm just like, why can't I have a hint button?

你懂吗？

You know?

好的，Maggie，非常感谢你。

Well, Maggie, thank you so much.

这是一个非常了不起的领域，我很喜欢它和视觉之间的联系。

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

非常感谢你抽出时间和我们交流。

Thank you so much for taking time with us.

谢谢你们，能聊数学真是件有趣的事。

Thank you for having interesting to talk about math.

听你们俩讨论这个数据可视化的问题，真的很有意思。

It's very charming to listen to the two of you talking about what is really a data visualization problem.

这让我想起了爱德华·塔夫特的那本经典著作《定量信息的视觉展示》。

And I'm reminded of the great book by Edward Tufte, a visual display of quantitative information.

书名很吸引人。

Gripping title.

哦，好吧。

Oh, well, okay.

你知道这本书吗？

Do you know the book?

我没读过。

I don't.

哦，别这样。

Oh, come on.

你在逗我玩吧。

You're teasing me.

好吧。

Okay.

行吧。

Fine.

书名可能不是这本书的卖点，但书里有一张拿破仑进军俄国的图表。

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

你可以看到一条线的粗细，代表拿破仑在某个时间点有多少士兵。

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

当这条线在页面上移动时，这条黑色带子变得越来越细，因为拿破仑的军队在逐渐减少。

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

同时，还有数据显示温度，当时有多冷，以及他们进攻某个地方时的日期。

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

所以你不只得到地理上的二维信息，你还得到军队的规模，这是第三个维度。

So you're actually getting not only geographical two d information, you're getting the size of the army, that's a third dimension.

你还能得到日期，这是第四个维度，等等。

You're getting a fourth dimension for the date and so on.

所以人们长期以来一直在思考如何在二维图片上可视化多个维度，这真的很聪明。

So it's really clever that people have been thinking about how to visualize multiple dimensions even on a two dimensional picture for a long time.

数学中的可视化非常吸引人。

Visualizations in mathematics is fascinating.

我个人非常欣赏可视化。

I personally really appreciate visualizations.

不过，我明白它也有局限性。

Yet, I understand it's a limitation.

但她描述的那种方式，实际上是一种很有趣的想象高维空间的方法，就是在某种意义上把它展开。

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

在这里画一下，然后在那里画一下。

Just draw it here, then draw it there.

想象一个球体穿过，比如说，平面国，一个所有生物都只生活在二维世界的世界，你会看到什么？

You know, this idea of a sphere passing through, let's say, flatland, a world in which all creatures only live in two dimensions, and what do you see?

你可以把三维球体理解为一组同心圆，随着球体进入而扩大，随着球体离开而缩小。

You can understand the three-dimensional sphere as a series of concentric circles that widen as the sphere enters and then shrink as the sphere leaves.

这真的有助于你理解，嗯，我或许可以用一个四维球体穿过三维空间来做到这一点。

And, it really helps you say, well, I could probably do that with a four sphere passing through three dimensions.

确实可以。

And indeed, you can.

这真的有助于你与这些面的物理性建立联系。

It really sort of helps connect with the physicality of these faces.

哇。

Wow.

好吧，我们不得不离开这个抽象领域，回到现实。

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

但是史蒂夫，非常感谢你。谢谢你，詹娜。

But Steve, thank you so much for Thank you, Janna.

犯罪。

Crime.

非常好。

Very good.

下次再见。

Till next time.

下次见。

You next time.

如果你喜欢《为什么的乐趣》并且还没有订阅，请在你收听的地方点击订阅或关注按钮。

You're enjoying the Joy of Why and you're not already subscribed, hit the subscribe or follow button where you're listening.

你也可以给节目留下评论。

You can also leave a review for the show.

这有助于人们找到这个播客。

It helps people find this podcast.

在quantamagazine.org上可以找到文章、通讯、视频等更多内容。

Find articles, newsletters, videos, and more at quantamagazine.org.

《探索的乐趣》是《Quanta杂志》的一档播客，这是一份由西蒙斯基金会支持的编辑独立出版物。

The Joy of Why is a podcast from Quanta Magazine, an editorially independent publication supported by the Simons Foundation.

西蒙斯基金会的资金决策不会影响本播客或《Quanta杂志》中主题、嘉宾或其他编辑决策的选择。

Funding decisions by the Simons Foundation have no influence on the selection of topics, guests, or other editorial decisions in this podcast or in Quanta Magazine.

《探索的乐趣》由PRX Productions制作。

The Joy of Why is produced by PRX Productions.

制作团队包括凯特琳·福尔斯、贾德·阿卜杜勒·马利克、吉纳维芙·斯邦斯勒和梅里特·雅各布。

The production team is Caitlin Falls, Jade Abdul Malik, Genevieve Sponsler, and Merritt Jacob.

PRX Productions的执行制片人是乔斯琳·冈萨雷斯。

The executive producer of PRX Productions is Jocelyn Gonzales.

埃德温·埃乔阿是我们的项目经理。

Edwin Echoa is our project manager.

来自《Quanta杂志》的西蒙·弗朗斯和萨米尔·帕特尔提供了编辑指导，并得到了塞缪尔·贝拉斯科、基特·苏达尔、西蒙·巴尔和迈克尔·坎扬戈洛的支持。

From Quanta Magazine, Simon France and Samir Patel provided editorial guidance with support from Samuel Velasco, Kit Soudal, Simone Barr, and Michael Kanyangolo.

萨米尔·帕特尔是《量子杂志》的主编。

Samir Patel is Quanta's editor in chief.

本期节目的封面艺术由香奈儿·尼贝林克创作，我们的标志由杰基·金和克里斯蒂娜·阿米蒂奇设计。

The episode art is by Chanel Nibelink, and our logo is by Jackie King and Christina Armitage.

特别感谢康奈尔广播工作室的加思·埃弗里。

Special thanks to Garth Avery at the Cornell Broadcast Studio.

我是你们的主持人，詹娜·莱文。

I'm your host, Janna Levin.

如果你有任何问题或评论，请发送邮件至Quanta@SimonsFoundation.org。

If you have any questions or comments, please email us at Quanta@SimonsFoundation.org.

感谢收听。

Thanks for listening.

来自PRX。

From PRX.