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空间维度与四维空间的理解 – 译学馆
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空间维度与四维空间的理解

A Beginner's Guide to the Fourth Dimension

欢迎收看《科学精灵》
We hear it all the time, that our world is three-dimensional.
我们经常听到说我们的世界是三维的
It makes sense after all,
这是有道理的
we can only define up to three directions each at a right angle to each other,
从一点出发 我们最多只能确定三个互相垂直的方向
they don’t interact with each other.
它们互不影响
No matter how far left or right you go,
无论你向左或向右走多远
you won’t move up at all.
你都不会走到上方
But the moment we try to add another dimension into the next,
但当我们尝试增加一个维度时
it all breaks down.
这个理论就不成立了
No matter how hard you try, in our three-dimensional universe,
无论你在三维宇宙里如何努力尝试
you can’t lay out four or more directions that won’t interact.
你无法找出四个或者更多互不影响的方向
Now I know what some people are thinking:
我知道有些人在想:
“Can we consider time is the dimension?”
“我们能否将时间看做一个维度?”
In simple terms, it does make sense.
简单地看 这么想也说得通
We can measure time and tell one point in time from another in a linear sense.
我们能测量时间并用线性规律描述时间变化
Four hours later is always four hours later,
四小时后永远只能是四小时后
just like how four feet to the left is always four feet to the left.
就像是向左四英尺永远只能是向左四英尺一样
Hey, you could even make time act like spacial dimension.
嘿 你甚至可以让时间像空间维度一样
Take a flipbook, each frame of the animation is a two-dimensional image,
比如一本动画书 每一帧都是一个二维画面
but by traveling through the 3D stack of these two-dimensional frames at a consistent rate,
但是以一定速度翻阅一叠二维画面
we can make time act like a spacial dimension,
我们就可以让时间成为空间维度
where height in the stack of 2D frames is the same as time between two events.
二维画面的高度就类似两个事件之间的时间
In other sense, this 2D animation just became 3D, right?
换句话说 二维动画变成了三维的 对吗?
Not necessarily.
那可不一定
Just because time can match up with the spacial dimension doesn’t mean that it is one.
就算时间可以与空间匹配也不代表它就是空间
Notice how in the animation time only moves one way,
注意在动画中 时间只向一个方向移动
from the bottom of the stack to the top.
从这堆画面的底部到顶部
But in regular 3D space,
但在平常的三维空间中
one can move either way: up or down.
物体可任意向上向下运动
This doesn’t mean it’s wrong to imagine time as a spacial dimension
这不是说不能把时间想象成空间维度
since it clearly can be done,
因为很明显这是可行的
but rather it’s wrong to assume a spacial dimension and time are exactly the same.
然而假定空间维度和时间一样则是不行的
So now that’s cleared up:
搞清楚这点之后
what would four-dimensional space be like?
那么四维空间到底是什么样?
The easiest way to answer that question is
回答这个问题的最简方法
actually to do the opposite.
就是反其道而行之
Rather than add a dimension in,
我们与其增加一个维度
let’s take a dimension out.
不如减少一个维度
Try to picture what it would be like to live on the surface of a sheet of paper.
试想生活在一张纸的表面会是什么样子
You can no longer jump up or down off the paper,
你再也不能在纸上跳上跳下
because up and down don’t exist to you anymore.
因为于你而言再也没有上下之分
You can’t look up off the paper either.
你也不能从纸面向上看
All you can do is look forward across the paper.
你所能做的只有沿着纸面向前看
When you do look in front of you, you’ll see a horizontal line.
当你看向前方 你只会看到一条水平线
And in that line, you’ll see the edges of other 3D figures.
在那条线上 你还会看到其它三维图像的边缘
Imagine a circle in a box,
想象盒子里有一个圆
clearly to us this is three-dimensional beings.
很显然这是个三维物体
We can see the circle in the box.
我们可以看到盒子里的圆
But to a two-dimensional being,
但是对于一个二维存在
the size of the box would prevent anyone from seeing inside.
盒子的边界会阻碍我们看到里面
Also consider that whenever something casts the shadow onto, say a wall,
还有 试想每当有东西在墙上投影
the shadow forms a line on the wall.
阴影在墙上形成一条线
Of course to you, all the walls will be lines anyways,
当然不管怎样 对你来说所有的墙都是一条线
so you wouldn’t really notice.
因此你不会太注意它
Let’s bring that same situation into three dimensions.
我们再把这种情况放进三维空间
As you look around, all you can see are two-dimensional objects,
当你看向四周 你只能看到二维物体
just like how even a box in two dimensions look like nothing more than a line.
就像在二维空间 一个盒子看起来不过是一条线
Since you can only view it from one side at a time,
因为你一次只能从一边看
the front of the box on three dimensions only looks like a square.
在三维空间 盒子前方看起来就像一个正方形
Also, suppose the ball was still in the box.
然后假设球仍然在盒子里
Now, sphere inside a cubic box.
现在球在一个立方盒里
We still can’t see it now because all the sides are still covered.
我们还是看不到球 因为外面都被遮住了
The shadow the box casts on the wall looks different.
盒子投射在墙上的阴影看起来不一样
Instead of just being a line like before,
不像以前那样只是一条直线
now it’s a flat 2D shape.
现在它是一个平面二维图形
If you wanted, you could even trace it on the paper
你甚至可以在纸上勾画它的轮廓
and get a 2D drawing out of it.
然后得到一个二维图像
Here’s where things get tricky.
下面就有点复杂了
Let’s try to add a new dimension that we haven’t ever seen before.
我们尝试新增一个未曾见过的维度
Let’s just dip our toe into the four dimensional water,
让我们把脚趾浸入四维的水中
and use the differences between 2D and 3D space
并且根据二维和三维空间的不同
to determine what would happen in 4D space.
来判断在四维空间会发生什么
One of the first things you would notice when you looked in a 4D space,
在四维空间你首先会注意到的是
is that you would see the world is a 3D shape.
你看到的世界是三维的
In a similar way to how you view a 3D world and can put it onto a 2D picture.
就像在三维世界 你看到的是二维图像一样
But there’s one catch: you can see all the sides of a shape.
但是有一点:你能看到这个形状的所有边
Remember how if you were to put a circle on a 2D box,
还记得你把一个圆放进二维盒子里
you could still see it looking down from above in 3D?
在三维世界从上往下看你还能看到它吗?
while now in 4D,
而在四维世界
you can see every side of the 3D box when looking down on it,
往下看你能看到三维盒子的每个面
including seeing the ball inside.
包括看到里面的球
Are you freaked out yet?
有没有被吓坏了?
After you get over the bit of wonder
克服惊奇的感觉后
you experience being able to see inside of the box,
你开始体会到看到盒子里面的感觉
you might also notice that another 4D box is casting a shadow on the wall.
你可能也注意到另一个四维盒子在墙上投影
And the shadow, it’s a 3D shape.
而这个阴影是一个三维形状
Yes, just like how 2D space has linear shadows,
是的 就像二维空间里是线性阴影
3D space has 2D shadows,
三维空间里是二维阴影
here in 4D space, we have three-dimensional shadows.
在四维空间中 我们看到的是三维阴影
But what would the shadows look like?
但阴影是什么样的呢?
Here’s an example.
举个例子
This is called the tesseract,
这个被称为超正方体
which is the shadow you would get from casting light through a 4D cube.
即光照射到四维立方体时得到的阴影
Notice how it’s made of two cubes connected at the vertices,
请注意它是由两个顶点相连的立方体组成的
just like how you can draw a cube on paper from two squares
就像你在纸上画两个正方形 连接其顶点
and connecting the corners.
形成一个立方体一样
Now the tesseract looks weird enough sitting still,
静止时这个超正方体看起来已经够奇怪了
but watch what happens the moment you start rotating it.
现在看它开始旋转时会发生什么
I’ll give you a moment to clean up you mind,
我先给你点时间来理清思路
which is most likely at this point exploded.
这可能是最让你抓狂的了
Yes, that’s what the shadow of a 4D cube looks like
没错 这就是四维正方体不停旋转时
when they keep rotates.
影子的样子
Why does it do this?
它怎么会这样?
Where the middle cube stretches out to become the outer cube?
里面的立方体怎么延伸变成外围立方体了?
Let’s take a look at the shadow of a 3D cube.
我们来看看三维立方体的阴影
First, note the two squares,
首先 看这两个正方形
one is smaller than the other because of perspective,
因为角度不同 两个正方形看起来不一样
making further objects appear smaller.
远一点的正方形似乎小一点
If you follow the small square,
如果你目光跟着小正方形走
you’ll see that it moves to the other side.
你会看到它移向另一边
before eventually moving to the front of the cube, and becoming the big square.
最终移到立方体前面 变成大正方形
Now, watch the tesseract again.
现在再来看超正方体
The little cube because of 4D perspective
由于四维角度
when rotated to be closer becomes bigger.
小立方体旋转接近时变大
where the big cube rotates to be smaller.
大立方体旋转远离时变小
So just like in a 3D cube where the big and little squares switched places,
就像三维立方体中大小正方形换位置
in a 4D cube,
在四维立方体中
the big and little cubes will switch places.
大小立方体也互换位置
All in all, four-dimensional space is not all that bad.
总之 四维空间也没那么难以理解
In fact, with the proper understanding of what will happen,
事实上 正确理解这种情况
one can devise some clever shapes
有助于设计一些精妙图形
that do things we can’t even imagine doing
而我们在三维空间甚至无法想象
in the 3D space.
这些图形能有多么神奇的特性
Ever hear of a Mobius Strip?
听说过麦比乌斯圈吗?
It’s a plane that thanks to being bent around in 3D space,
它是个平面 由于在三维空间扭曲
has only one side.
因此只有一个面
Because what would normally be the top and bottom sides are now connected,
通常情况的上下两面现在连在了一起
making them the same.
使它们变成了一个面
Things to a little 4D trickery, a figure called a Klein Bottle can exist,
另一个类似的四维精巧设计是克莱因瓶
where the inside and outside are the same.
它的里面和外面都一样
Ever notice how you can mere writing
你注意到过仅需翻转纸面
on the page by flipping it over?
就可以得到镜面文字吗?
In 4D space,
在四维空间
you can mirror a whole 3D object just by flipping it over four dimensionally.
仅仅通过翻转就可以得到三维物体的镜面
4D space shows us what is arguably the best part about mathematics,
四维空间向我们展示的大概是数学最好的方面
it works regardless of your bias.
不管你有何偏见 它始终有效
In a previous video,
之前有一期节目
I showed how we could use two digits to represent any value that 10 digits came.
我讲了如何用二进制数表示任意十进制数
And in a similar way,
类似地
just like how we can do any mathematical function
就像我们能用十个基数或者
possible in base 10, in base 36,
10个基数加26个字母做任何数学运算
we can have geometry that functions consistently,
我们可以相应地解决任何几何问题
whether it has three dimensions? Four dimensions? or possibly even more.
不管是在三维 四维 甚至更多维空间

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视频概述

对空间维度的简单介绍,以及对四维空间如何想像如何理解的方法

听录译者

www

翻译译者

ZTT

审核员

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视频来源

https://www.youtube.com/watch?v=j-ixGKZlLVc

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