So something just arrived in the mail. I'm very excited about it.
有些东西刚刚寄到了。这个东西令我很兴奋。
It's this huge humongous thing here some of you may be familiar with it -
这是一个巨大的庞大的事情，你们中的一些人可能熟悉它 -
It's a Klein bottle.
这是一个克莱因壺(Klein Bottle)。
It's a humongous Klein bottle cost me a couple of thousand dollars
这是一个巨大的克莱因壺，花了我几千美元
Definitely worth it as you can see I'm pretty nuts about these things I've got like a pretty huge collection of Klein bottles
绝对值得。如你所见，我对这些东西相当疯狂。我有一个非常庞大的克莱因壺收藏
I'll just put a couple here on the table. You know as many as could fit in this frame!
我只把其中几个放在桌子上。你懂的，基本上是画框的范围内尽量放！
Basically here, and so I want to talk a little bit about Klein bottles. Why I'm nuts about them?
那…我是想说一点关于克莱因壺的事，为什么我为之而疯狂啦
but also
还有就是
We want to do something. You know do something related to what we've been doing before so
我们也想做些什么。就是说，和之前我们一直在做的事情有关的事情
就是说…
I've put Rubik's cubes in bottles and today the final thing I'm going to do is I'm going to put some Rubik's cubes in Klein bottles.
我先前把魔方放进了瓶子里
而今天我要做的最后一件事就是我要把一些魔方放在克莱因壺里。
It's going to be something all right, so I think before we start talking about these these various bottles,
那是一件大事。好了，我想在开始谈论这些不同的壺之前，
I better give you an explanation of what it's all about mathematically.
我应该用解释一下这东西的来龙去脉，在数学的意义上
Let me tell you a little bit about what makes these Klein bottle interesting,
让我稍为告诉你为什么克莱因壺这么有趣
Mathematicians are really nuts about them,
（数学家真的很为之疯狂）
so what we do to build one,
我们来造一个
We just take a sheet of paper well not paper, some flexible material and do this, okay?
我们拿一张纸——其实不是纸而是有弹性的材料——来造吧，好吗？
So we'll bring the red edges together make them into a tube
因此，我们将红色边缘组合在一起，使它们成为一个管子
And then we connect up the yellow circles in a special way, okay?
然后我们以特殊方式连接黄色圆圈，好吗？
If you just bring them together like this we get a like a donut shape
如果你像这样将它们组合在一起，我们会得到一个甜甜圈形状
But we can do this in a second way
但我们可以用第二种方式接合
让它成为克莱因壺。让我们来看看，所以我们像这样拉长再拉长
and that gives the Klein bottle. Let's have a look, so we extend out like this and then extend and
together comes the Klein bottle okay?
再接合就变成克莱因壺了，对不？
Alright, so that's one way of building it the other way actually focuses on the yellow edges first
就那样。那这是第一种做法。另一个做法是首先关注黄色边缘
Okay, so let's just go straight to that
好的，让我们直截了当地说
So what I'll do here is I'll take a rectangle
这次我将一个矩形
that's a bit thinner to start with okay, and then the way the yellow edges are brought together is
开始时窄一点，然后将黄色边缘放在一起的方式是
Well as there's just two different ways
嗯，有两种不同的方式
and so you could just do like this and you get a nice ring
你可以就这样接，那你会得到一个很好的圆环
Or you could give it a twist and bring it together like this
或者呢，把它转一下再将它们这样接合
And that's of course something that pretty much everybody who watches this I guess knows
这当然就是，我猜几乎所有在看的人都知道，
This is a Mobius strip all right, and what's special about a Mobius strip?
一个莫比乌斯带，莫比乌斯带有什么特別之处？
I mean you start out with two sides,
你一开始是有两个面，对不？
and when you connect like this
如果你像这样连接
When you connect like this, white gets connected to white and yellow is connected to yellow and you get two sides again
当你这样连接时，白色连接到白色，黄色连接到黄色，你再次获得两面
But when you kind of switch over like that you actually connect the two sides
但是当你像那样反转时，你其实连接了两个面
So you connect the white side to the yellow side the yellow side to the white side and actually now what you get here is
所以你将白色侧面连接到黄色侧面，黄色侧面连接到白色侧面，实际上现在你得到的是
a one-sided surface okay, so if you take two different points like one point here and
只有一个面的表面，所以如果你在这里选两个不同的点。像是这个点
还有对面的一个点，你可以沿着表面从这里走到那里
Like on the opposite side you can actually travel from here to there by walking along the surface, okay?
So just walking along the surface
只是沿着表面走
So that's that's something very special a one-sided surface one sided surface
这是非常特殊的东西，只有一个面的表面
Okay, now the Klein bottle is actually just a mobius strip,
好的，克莱因壺实际上只是一个莫比乌斯带，
extended in such a way that eventually the
以这样的方式延伸，最终令
the two edges come together and merge together
两个边缘合在一起并接合
Okay, let's just have a look at how that works so first we connect this up into a Mobius strip in a special way
好的，让我们看看它是如何工作的，所以首先我们以一种特殊的方式将它连接到一个Mobius带
So what we have to do is
所以我们要做的是
We have to connect
我们必须联系
The yellow point to the green point and the green point to the yellow point, and we do this in a special way here
黄点到绿点，绿点到黄点，我们以特殊方式来做
So with the twist is kind of hidden a little bit okay, so goes like that just extend it out
这个扭转有点儿不明显，就这样延长它
Bring it around and now you can see if I just extend this out
把它带过来，现在你看到，我只要将这边廷伸
those two points are going to get
这两点将会
Merged those two points going to get merge and you get a Mobius strip okay,
合并。这两点将合并。而你得到一个莫比乌斯带，
and now let's just go into our real starting position
现在让我们进入我们真正的起始位置
Which is basically this so there's your Mobius strip, okay?
——基本上就是这样子。一个莫比乌斯带
And now we kind of extend things out,
现在我们将这个延伸，
So we extend things out we extend further we extend further we extend further and if you just extend a little bit further
这样地延伸，我们进一步延伸我们进一步延伸我们进一步延伸，只要你再进一步延伸一点点
It's all going to come together and you're going to get your Klein bottle
这一切都将聚集在一起，你将获得你的克莱因壺
What's nice about the Klein bottle is that doesn't have any edges whatsoever? It's like a sphere basically
克莱因壺的美妙之处在于它没有任何边沿。它基本上就像一个球体
It's like a sphere doesn't have any edges It's a closed surface as opposed to one with an edge the mobius strip. So that makes it somewhat special, okay?
这就像一个球体没有任何边缘
它是一个封闭的表面，而不是具有边缘的莫比乌斯带。所以这有点特別，好吗？
Okay, now
好的，现在
Klein bottles they actually don't live in our world.
克莱因瓶这东西呢，实际上不存在于我们的世界。
They really live in four dimensions, and they're really I mean these sorts of guys they really feel uncomfortable here
他们其实存在于四维空间，他们真的是——我的意思是这些家伙在这里真的感到不舒服
They're really kind of crammed into three dimensions, and you can see it here kind of they're kind of clash into it themselves
他们被挤进了三维空间。你可以看到它会在这里撞到自已［自我交集］
They don't want to do this, they want to be like like donut shapes. Donut shapes, they don't clash into themselves
他们不想这样做，他们想要成为甜甜圈的形状。甜甜圈的形状，他们不会撞到自己
They want to be like that and in four dimensions. They can be like this. I just want to show you
他们希望成为那样。而在四维空间他们可以是那样的
我是想告诉你要如何把第四维放进这里
How you actually get the fourth dimension in here? So if you've got a figure eight living in a plane, all right?
如果有个８字活在一个平面上
现在这里就会相撞
Now there's got a clash here
If you want to get rid of this clash what you do is you kind of just take a little bit of this
如果你想避免这个相撞，你要做的是
把这数字８这一点点提高
This figure eight here and kind of raise
让它慢慢地进入第三维然后再回来
It slowly up into the third dimension and bring it back in like
像这样
我们可以说是造了一座桥，这一下子就不会相撞了
we've kind of built a bridge and all of a sudden the whole thing doesn't clash into itself you can do the same [sort] of
对克莱因壺也可以做同样的事
你要做的是在这个位置
Thing here with the Klein bottle so what you do is at this stage here
你慢慢地把东西推向第四维度
You kind of push things off slowly into the fourth dimension
有点像在第四维中建造一座桥那样走进第四维
And you kind of just go across like in the fourth dimension kind of building a bridge in the fourth dimension
and then
那样的话
well from our perspective it just seems to vanish here right
从我们的观点，它似乎在这里消失了
From the perspective of somebody who lives in here the curves just seems to vanish and then to reappear, but it's still there
从居住在这里（平面）的人的观点，曲线似乎消失然后重新出现，但它仍然存在
It's still there in three-dimensional space same sort of thing is possible here for a Klein bottle
它仍然存在于三维空间中
相同的事情也可以在克莱因壺上发生
所以这里它就像是开始消失
So here it just kind of starts vanishing
But it's actually still there in the fourth dimension goes across and you don't get this running into itself
但它实际上依然存在。在第四维度中跨过去，并且不会撞到自己
So this is really a four dimensional creature, and it's a one sided surface
所以这其实是一个四维生物，它是一个单面的表面
And there's a couple of other things that are really amazing about it. So one of them
还有其他一些令人惊讶的东西。其中一个呢
Which not just mathematicians find amazing, but just about anybody finds amazing is that they provide
不仅仅让数学家感到惊讶，而且几乎任何人都觉得惊人的东西是
他们提供了
一个提示
You know a hint that the universe we live in might actually have a very strange property and I just want to show you this
我们生活的宇宙实际上可能有一个非常奇怪的属性，我只是想告诉你这个
By looking at a mobius strip real quick. So here's a really really big mobius strip okay, and
我们再回看一下莫比乌斯带。这是一个非常大的莫比乌斯带
Let's say there is a creature living in here
假设有一个生物住在这里
So there's like our two-dimensional man lives in here, and that mobius strip that it doesn't have any
想像我们有一个二维人住在这里，而那个莫比乌斯带
third dimension it's just that it's there.
是没有任何三维的。它就只是，只是。。。
But it doesn't have any third dimension. So what we can do is this man
它是存在的但它没有任何第三维度。
我们可以对这个人做的呢，就是让他在莫比乌斯带上行走
我们来让他绕着莫比乌斯带走一圈
Here is we can let him travel around the Mobius strip, okay?
so let's just let him travel around the Mobius strip
and now he kind of just turns on the spot and
这下子呢他只要在这里转个身
Now something amazing has happened
现在发生了惊人的事情
只是走在莫比乌斯带上走一圈，这个男人就变成了自已的镜像
but just walking around a mobius strip this man has actually turned into his mirror image, so
所以呢
在我们的宇宙中，类似的事情在理论上是可行的，我们可以在宇宙中旅行
Something similar is theoretically possible in our universe that we kind of go off for a trip through the universe
We come back and we come back mirror reversed
我们回来了，我们回来的时候是反转的
It could theoretically
理论上可以发生
Happen, and it could happen in a kind of a three-dimensional counterpart of of the Klein bottle so that's what makes Klein bottles interesting
它可能在克莱因壺的三维对应物中发生，这就是克莱因壺有趣的理由
For new Mathematicians and also you know I hope
至少对于数学家。而且，你知道，我希望
对于在看的各位大家也是一样。它基本上给了你一个空间
Everybody watches here. It gives you basically a space a
Flat analog of of the three-dimensional universe we live in
一个对应的平面，反映我们存在的三维宇宙
That has this special property that if you kind of just walk around in the right way
有一个特殊的属性，只要你以正确的方式走动
You kind of turn into your mirror image, and you know you won't feel it
你就会变成了你的镜像。你不会对此有感觉
You won't feel it that you know you'll feel perfectly fine all the way throughout the trip, but you know I come back it's strange
你在整个旅程中都会感觉完全正常，但是，你懂的，回来后郤很奇怪
所以呢，我们来认真看看
So let's get serious about putting some
把魔方放进克莱因壺这件事
Rubik's Cube in Klein bottles and
talking just in general a little bit about about my collection here, so
还有关于我的收藏品的概况
What have I got here, I've got
我有什么呢，我有
Different sizes, so this is kind of just straightforward shape basically four different sizes here
不同的尺寸，所以这里只是简单的形状，基本上有四种不同的尺寸
Well really five if I've got all of these really small ones here, so there's another one
其实是五种 因为我在这里有所有这些非常小的，这是一个
So five different sizes you can use them for in earrings or pendants or whatever now. They're all made from glass by real glass blowers
因此，是五种不同尺寸 你可以用来做耳环或吊坠或其他什么。它们都是玻璃匠用真正的玻璃制成的
Pretty much all of these come from a place in California
几乎所有这些都来自加利福尼亚州的一个地方
I'm going to put a link somewhere here
我打算在这里放一个链接
So they're all coming from California. I've got one here this one came from the UK so,
所以他们都来自加利福尼亚州。我这里有一个这个来自英国
They actually attached something here, so they can use the Klein bottle as a
他们实际上在这里附上了一些东西，因此他们可以使用克莱因壺作为杯具
Drinking vessel and then I've got something very strange here
然后我在这里有一些非常奇怪的东西
three Klein bottles kind of merge together like this very strange
像这样的三个克莱因壺合并在一起
非常奇怪
Then I've got a Klein bottle opener so you can actually open beer bottles like this
然后我有一个克莱因壺开瓶器 你可以像这样打开啤酒瓶
Which is actually 3D printed.
这实际上是3D打印的。
And it's in metal and there's another 3D printed version just from plastic, and this is something that I use on a daily basis
它是金属的，还有另一个3D打印版本，只是塑料，这是我每天使用的东西
So this is my Klein bottle mug
所以这是我的克莱因壺—杯
所以，如果你真的很近很近地看
So if you actually have a really really close look
This is actually a real mug, and you can see that water's kind of sitting in here in a strange way And that's basically you know the talos space.
这实际上是一个真正的马克杯，你可以看到水以一种奇怪的方式留在这里
基本上呢。。你懂的，就像这里的空洞
每隔一段时间喝一些水，好吧
Get some water in every once in a while, okay
now how do we get a
现在我们怎么
Rubik's Cube in a Klein bottle
把一个魔方放到克莱因壺中呢
Okay, so we want to get maybe this one here into this Klein bottle
好的，所以我们想把这个放到这个克莱因壺里
I mean from the movie you maybe remember well this thing actually only has one side, right?
我的意思是从电影中 你可能记得很清楚这件事实际上只有一面，对吗？
So it actually doesn't have an inside and an outside
所以它实际上没有内部和外部
Mathematically speaking, so pretty much you could just say well,
从数学的角度来说。所以，你差不多可以说它已经在"里面"了
不过没有人会觉得厉利。再说那也不是很正确。不是很正确，因为
It but nobody will be impressed also. It's not really true. It's not really true because
while we're talking about a Klein bottle kind of squeezed into our 3D space
我们谈论的是挤进我们3D空间的克莱因壺
It would be true if we had what kind of you know doing this video in four dimensions
如果我们在四维空间中拍这个视频，那就会是正确的
那么对这个东西做任何事都没有意义
Then I mean there would really be no point to do anything with this thing
You know just be inside whichever way you look at it, right?
无论你怎么看它都只能在"里面"，对吧？
But because it's kind of squeezed into three dimensions. You can actually see well. There is really an inside here somehow, right?
但因为它被挤压进了三维空间。你看到这里其实真的有一个内部，对吗？
There's really an inside so how do we get it into this space inside it looks hard and actually it would be completely impossible
真的有内部的话，那我们怎么把它放到这个空间内呢
看起来很难，实际上它会是完全不可能的
Were it not for a strange flaw
如果没有一个奇怪的缺陷
Okay, let's let's be kind and call it feature of all these glass Klein bottles, okay?
好吧，让我们好心的，把它称为所有这些玻璃克莱因壺的特征，好吗？
So when you look at them
所以当你看着他们
And when you remember so there's basically a tube going through to the inside here
如是你记得，基本上这根管子进入里面
And then there's a surface kind of cutting across here, so I mean here we've got this kind of circular
然后在这里是有一个表面横着穿过这里，所以在这里我们有这种圆形的交滙
intersection so a circular intersection and really
一个圆形的交滙
What should be there it should two surfaces in two intersecting right, so this surface
那里应该有两个相交的表面，所以这个表面
And that surface here there should be intersecting and actually that hole should be closed
和那个表面应该是交叉的，实际上这个洞应该是封闭的
But it isn't you can actually get in through the tunnel here and move things to the inside
但实际上并不是
你可以通过这里的隧道进入并将东西移到里面
It's a floor. I guess. It's just very hard for for glassblower to get any sheet of of glass in here
这是一个缺陷。我猜对于玻璃吹制者来说，在这里加一块玻璃是非常困难的
So that's actually the case for all of these
这里的所有东西都有相同情况
all of these
全部这些
Handmade glass bottles and strangely enough it's also the case for this plastic one. I mean for this plastic one There's actually no excuse for not putting anything
手工制作的玻璃瓶，奇怪的是，这种塑料瓶也是如此。我的意思是这个塑料的
实际上没有借口不放任何东西
But I guess whoever made this he just looked at the glass Klein bottle copied it and didn't realize that
但我想造这个出来的那个什么人，他只是看着玻璃克莱因壺然后复制它，并没有意识到这一点
He was actually making a mistake there now the person who made this one here
他实际上在那里犯了一个错误
但是呢，造了这边的这个出来的人呢
very famous 3d printing artist, Bathsheba Grossman
非常著名的3D打印艺术家，Bathsheba Grossman
She got it right, so if you actually look in the hole here So there's the surface cutting across so among all those things that I have here. It's really this is really the only one
她做对了，所以如果你真看看这里的洞里面
是有一个横跨的面。在我放在这里的全部东西里呢，这真的是唯一一个
That's you know perfect right in that sense, but I mean we're not really worried about this
从那个意义上说，是完美的，但我的意思是我们并不是真的太在意这一点
We're actually pretty happy about this because at least there's a way in you know, so otherwise yeah, there's no hope
我们其实对此感到非常高兴，因为至少有一种方法可以进入。否则就是，没有希望
Okay, so how do we get something in?
好的，那么我们如何放东西进去呢
Well was this large one
对于这个大的
it's actually pretty easy, so I got to just kind of take it like this and I get a
实际上非常简单，所以我只要这样拿，然后我有这个
small cube, maybe my favorite cube the 1x1x1, and I just kind of pop it in
小魔方—也许是我最喜欢的魔方1x1x1—就这样放进去
alright, and it's in
好吧，它在里面了
Nobody will be impressed, well I think you'd still be impressed, by me holding this thing
没有人会有什么感觉，也许你仍然会觉得厉害，对于我抱着的这个东西
So anyway, so you can get a cube in like this, but of course you want to want a real 3x3x3
无论如何，所以你可以像这样放一个魔方进去，但当然你想要一个真正的3x3x3
okay, well that's that there's still a really really tricky way of doing it because I've got this really really small 3x3x3
好吧，那就是说还有一个非常取巧的方法，因为我有一个真的很小很小的3x3x3
So that's actually you know that's actually the smallest mass-produced
实际上你知道这实际上是最小的批量生产
3x3x3 cube. That's just as big as
3x3x3魔方。它的大小就像是
One of the little cubes that make up an ordinary one and you can really you know
一个构成普通魔方的小立方体，你真的可以
twist the sides of this thing and can you scramble it up
扭动这个东西的两侧，你可以把弄乱它
And you can solve it and do all sorts of things. It's called a Nano Cube
你可以还原它并做各种各样的事情。它叫做纳米魔方(Nano cube)
So this one let's put it in probably a world first
所以这个让我们放进去。大概是世界首次
There, a 3x3x3 in a Klein bottle just like that
就像那样，克莱因壺中的3x3x3
Okay, but again
好的，但又一次
I mean you've never seen this before and I'm sure it's going to be very memorable you'll never forget this
我的意思是你以前从未见过这个，我相信它会让你非常难忘，你永远不会忘记这一点
Image of me throwing this thing in but still you know it's too easy. I mean, I could have done it
我扔这个东西进去的影象，但你懂的。这太容易了。我的意思是，我本可以做到的
I mean this this one here
我的意思是这个
You only really impressed because I got this thing in that doesn't really fit through the hole alright
你真的会觉得厉害的原因是，我放进去的东西不可能好好地通过这个洞。对不？
因此，我们必须以某种方式将这样的立方体装入其中一个克莱因壺中
So we have to somehow get this a cube like this into one of those Klein bottles
Now I'm not going to use this one here because I mean if you've actually watched the video where I put this one in you
现在我不打算在这里使用这个，因为我的意思是，如果你真的看过我把这个放进去的视频
know It's dangerous, sometimes I destroy these things in the process right, so I'm not going to use the several thousand dollar
那你就知道
这很危险，有时我会在这个过程中破坏这些东西，所以我不打算使用几千美元
Sample here of which only like two exists in the whole wide world
的这个样本。只有两个存在于整个广濶世界之中
So I'm going to use one of the other ones right, so I've got quite a few of the other ones and so
所以我将使用其他的，我有很多其他的，所以
What I thought I'd try is, I'm going to try and put
我以为我会尝试的是，我会尝试放某些东西到
Something into this one here
这个里面
Okay, so this one here and well
好的，所以是这个和
I mean this this 3x3x3, definitely doesn't fit in all right. So can't can't fit that one inside
我的意思是这个3x3x3，绝对是放不进。所以把这不能放到里面
and even larger one would be this 2x2x2 right so that that would I think just just just fit inside and
这个2x2x2更大，所以我认为这只是刚刚仅仅放得进内部和
Well, I guess we could do another world first since I'm here we could put a Klein bottle inside a Klein bottle since we've got
好吧，我想我们可以先做另一个世界首次，因为在这里我们可以把一个克莱因壺放入克莱因壺中，因为我们已经有这个
It here. We might as well do it I've got this little Klein bottle here
在这里。我们不妨这样做，我在这里有这个小小的克莱因壺
Okay, so we're going to put a Klein bottle inside a Klein bottle, there it is, never seen before, world-first
好的，所以我们要把一个Klein瓶装在一个Klein瓶子里面，就这样，前所未见的世界首次
All right, okay. Let's leave this in there. I think I might just leave this in there permanently and
好的，好的。我们把它放在那里。我想我可能永远把它留在那里
Now I'm going to go away and then
现在我要离开然后
And then I'm going to see what I can get this thing inside, okay?
然后我会看看里面能放进什么东西，好吗？
alright. I'm back. I'm wearing my Klein bottle hat
好的。我回来了。我戴着我的克莱因壺帽
Which you can get also for the place where we get these glass Klein bottles from so it's a very neat hat
你也可以买到它，就在这些克莱因壺的来源地，它就是一个非常好的帽子
So if you go inside, you pull
所以，如果你从里面拉
You can actually see the two kind of sliding inside so very nice anyway, so I succeeded
你实际上可以看到里面的管在滑动，所以非常好，所以我成功了
as you can see
如你看到的
The Rubik's Cube is in alright. The Rubik's Cube is in the Klein bottle and
魔方在里面。 魔方在克莱因壺中
let me just show you a little bit, so
让我展示给你看，所以
Here it is right. I just kind of move it around a little bit here
就在这里。我稍微移动它
Wow, okay, anyway
哇，好吧，无论如何
That's as far as we'll go. It's actually pretty tight fit. This way, so you know great stuff?
我就只达成了这么多。实在只是刚刚好放得进。这样就，你懂的，很棒