Exploring other dimensions - Alex Rosenthal and George Zaidan

翻译人员: Gabriella Hu 校对人员: Geoff Chen

We live in a three-dimensional world

我们生活在一个三维世界里

where everything has length,

每件物体都有长度

width,

宽度

and height.

和高度

But what if our world were two-dimensional?

但是如果我们的世界是二维的呢？

We would be squashed down

我们会被压扁

to occupy a single plane of existence,

只存在于一个平面，

geometrically speaking, of course.

当然，只是从几何的角度来看

And what would that world look and feel like?

那那个世界看上去 和感觉上又是怎样的呢？

This is the premise

这就是埃德温·艾勃特

of Edwin Abbott's 1884 novella, Flatland.

1884年的中篇小说《平面国》中的假设

Flatland is a fun, mathematical thought experiment

平面国是一个有趣的数学思想实验

that follows the trials and tribulations of a square

通过一个正方形在三维世界里的

考验和磨难

exposed to the third dimension.

But what is a dimension, anyway?

但，什么是一个维度呢？

For our purposes, a dimension is a direction,

简单的来说，一个维度是一个方向

which we can picture as a line.

我们可以把它想象成一条直线

For our direction to be a dimension,

一个方向能成为一个维度

it has to be at right angles to all other dimensions.

它必须和其它的维度形成直角

So, a one-dimensional space is just a line.

所以，一维空间只是一条直线

A two-dimensional space is defined

一个二维空间是由

by two perpendicular lines,

两条垂直的线组成

which describe a flat plane

形成一个平面

like a piece of paper.

像一张纸一样

And a three-dimensional space

一个三维空间

adds a third perpendicular line,

再添加了第三条垂直的线

which gives us height

给了我们高度

and the world we're familiar with.

和我们所熟悉的世界

So, what about four dimensions?

那四维世界呢？

And five?

五维？

And eleven?

十一维？

Where do we put these new perpendicular lines?

我们在哪里放置这些新的垂直线条呢？

This is where Flatland can help us.

这就是《平面国》可以 帮助我们找到的答案

Let's look at our square protagonist's world.

让我们看看书中的主角 - 正方形 - 的世界

Flatland is populated by geometric shapes,

平面国中充满不同的几何形状

ranging from isosceles trianges

从等腰三角形

to equilateral triangles

到等边三角形

to squares,

pentagons,

到正方形

五角形

hexagons,

六边形

all the way up to circles.

一直到圆形

These shapes are all scurrying around a flat world,

这些图形在一个平面世界中四处游荡

living their flat lives.

过着它们的平面生活

They have a single eye on the front of their faces,

它们的脸上有一只眼睛

and let's see what the world looks like

让我们看看从它们的角度

from their perspective.

这个世界是什么样子的

What they see is essentially one dimension,

它们所看到的是一维的

a line.

一条直线

But in Abbott's Flatland,

但是在艾勃特的《平面国》中

closer objects are brighter,

距离更近的物体更加明亮

and that's how they see depth.

这是它们如何察觉深度的

So a triangle looks different from a square,

所以，一个三角形看上去 就和一个正方形不同

looks different a circle,

和圆形不一样

and so on.

等等

Their brains cannot comprehend the third dimension.

它们的大脑无法理解第三维度

In fact, they vehemently deny its existence

其实，它们强烈地否认它的存在

because it's simply not part of their world

因为那不是它们的世界的一部分

or experience.

它们也没有经历过

But all they need,

但是它们所需要的

as it turns out,

事实证明

is a little boost.

是一点小鼓励

One day a sphere shows up in Flatland

有一天，一个球体出现在平面国中

to visit our square hero.

去拜访那个正方形

Here's what it looks like

当一个球体从平面国中通过时，

when the sphere passes through Flatland

从一个正方形的角度

from the square's perspective,

它看到的画面是这样的

and this blows his little square mind.

这彻底颠覆了它的二维观念

Then the sphere lifts the square

然后，那个球体把正方形举起来

into the third dimension,

进入了第三维度

the height direction where no Flatlander has gone before

高度上升了，到了平面国中的形状们 从来没去过的地方

and shows him his world.

给正方形看看它的世界

From up here, the square can see everything:

从上空中，正方形看到了所有的东西：

the shapes of buildings,

高楼的形状

all the precious gems hidden in the Earth,

地球中埋藏着的珍贵的宝石

and even the insides of his friends,

甚至它的朋友们的内侧

which is probably pretty awkward.

这应该挺尴尬的

Once the hapless square

一旦这个正方形

comes to terms with the third dimension,

适应了第三个维度

he begs his host to help him

它央求球体帮助它

visit the fourth and higher dimensions,

看到第四个，更高的维度

but the sphere bristles at the mere suggestion

但是听见之后十分生气

of dimensions higher than three

居然还有比三维更高的维度

and exiles the square back to Flatland.

然后把正方形放逐回平面国

Now, the sphere's indignation is understandable.

我们可以理解球体的愤怒

A fourth dimension is very difficult

第四个维度的想法很难

to reconcile with our experience of the world.

和我们在世界中的经历并存

Short of being lifted into the fourth dimension

我们不可能被一个超立方体举起

by visiting hypercube,

被带到第四个维度

we can't experience it,

我们无法体验它

but we can get close.

但是我们能接近它

You'll recall that when the sphere

回想一下，当球体

first visited the second dimension,

第一次来到二维世界时

he looked like a series of circles

它看上去像一系列的圆形

that started as a point

当它在平面国落地时

when he touched Flatland,

它看上去像一个点

grew bigger until he was halfway through,

一直变大，直到它一半的体积陷进地面

and then shrank smaller again.

然后又开始变小

We can think of this visit

我们可以把这个看做

as a series of 2D cross-sections of a 3D object.

一个三维物体的一系列 二维横截面

Well, we can do the same thing

我们可以用同样的方法

in the third dimension with a four-dimensional object.

从一个三维世界看一个四维物体

Let's say that a hypersphere

比如，一个超球是一个

is the 4D equivalent of a 3D sphere.

相当于三维球体的四维物体

When the 4D object passes through the third dimension,

当四维物体通过一个三维世界时

it'll look something like this.

它看上去是这样的

Let's look at one more way

我们用另一个角度来看

of representing a four-dimensional object.

四维物体

Let's say we have a point,

我们画一个点

a zero-dimensional shape.

这是一个零维形状

Now we extend it out one inch

现在我们把它扩展到一英寸

and we have a one-dimensional line segment.

我们就有一个一维直线

Extend the whole line segment by an inch,

再把直线扩展一英寸

and we get a 2D square.

我们就有了一个二维正方形

Take the whole square and extend it out one inch,

把整个正方形再扩大一英寸

and we get a 3D cube.

就有了一个三维正方体

You can see where we're going with this.

这就是大概的思路

Take the whole cube

把整个正方体

and extend it out one inch,

再扩充一英寸

this time perpendicular to all three existing directions,

这次垂直于所有的三个已存在的方向

and we get a 4D hypercube,

我们就有了一个四维的超立方体

also called a tesseract.

又被称作正八胞体

For all we know,

根据我们现在所知道的

there could be four-dimensional lifeforms

很有可能有一种四维的生命体

somewhere out there,

在宇宙中某个遥远的地方

occasionally poking their heads

偶尔把它们的头探入

我们繁忙的三维世界

into our bustling 3D world

and wondering what all the fuss is about.

奇怪我们都在瞎忙什么

In fact, there could be whole

其实，可能有许多其它的

other four-dimensional worlds

四维世界

beyond our detection,

无法被我们发现

hidden from us forever

因为我们感知的能力

by the nature of our perception.

永远隐藏在我们找不到的地方

Doesn't that blow your little spherical mind?

这有没有彻底颠覆你的三维观念？