where everything has length,

一切事物都有長度、

width,

寬度、

and height.

和高度

But what if our world were two-dimensional?

但如果我們的世界是二維的會怎麼樣呢？

We would be squashed down

我們會被壓扁

to occupy a single plane of existence,

於一個存在的單一平面上

geometrically speaking, of course.

以幾何學來講，理所當然

And what would that world look and feel like?

那個世界看起來及感覺起來像怎麼樣呢？

This is the premise

這是一個假設

of Edwin Abbott's 1884 novella, Flatland.

由愛德溫‧艾勃特在1884年的 短篇小說《 平面國》中提出

Flatland is a fun, mathematical thought experiment

平面國是一個有趣的數學思維實驗

that follows the trials and tribulations of a square

敘述一個正方形遇到的種種考驗與磨練

exposed to the third dimension.

在歷經第三維度的時候

But what is a dimension, anyway?

但什麼是維度呢？

For our purposes, a dimension is a direction,

從我們的角度出發，一維是指一個方向

which we can picture as a line.

我們可以想成一條線

For our direction to be a dimension,

把我們的方向當作是一維

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

它必須與所有其他的維度都形成直角

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

所以，一維空間就是一條線

A two-dimensional space is defined

二維空間

by two perpendicular lines,

由兩條相互垂直的直線所定義

which describe a flat plane

它們建構了一個平面

like a piece of paper.

就像一張紙一樣

And a three-dimensional space

三維空間

adds a third perpendicular line,

增加第三條垂直線

which gives us height

它提供我們高度

and the world we're familiar with.

及那個我們熟悉的世界

So, what about four dimensions?

那四維呢？

And five?

五維？

And eleven?

甚至十一維？

Where do we put these new perpendicular lines?

我們要將這些新的垂直線放在哪呢？

This is where Flatland can help us.

這就是平面國可以幫助我們的地方

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

讓我們來看一下正方形主角的世界

Flatland is populated by geometric shapes,

平面國居住著各種幾何圖形

ranging from isosceles trianges

從等腰三角形、

to equilateral triangles

等邊三角形、

to squares,

正方形、

pentagons,

五角形、

hexagons,

六角形、

all the way up to circles.

一直到圓形

These shapes are all scurrying around a flat world,

這些圖形都在一個 平面的世界上到處跑來跑去

living their flat lives.

過著它們平面的生活

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

在它們臉的前方有一隻眼睛

and let's see what the world looks like

讓我們來看看從它們的角度上 這個世界看起來像甚麼樣

from their perspective.

實質上它們看到的是一維

What they see is essentially one dimension,

也就是一條線

a line.

但在艾勃特的平面國中

But in Abbott's Flatland,

越接近的物體看起來越明亮

closer objects are brighter,

這就是它們如何看到深度

and that's how they see depth.

所以三角形看起來與正方形不同、

So a triangle looks different from a square,

看起來與圓形不同

looks different a circle,

諸如此類

and so on.

它們的腦袋無法理解第三維度

Their brains cannot comprehend the third dimension.

事實上，它們極力否認第三維度的存在

In fact, they vehemently deny its existence

因為那根本完全不存在於它們的世界

because it's simply not part of their world

或經驗中

or experience.

但事實證明，它們所需要的

But all they need,

只是一點小小的刺激

as it turns out,

有一天，一個球體出現在平面國中

is a little boost.

拜訪我們的正方形英雄

One day a sphere shows up in Flatland

這是當球體經過平面國時看起來的樣子

to visit our square hero.

從正方形的角度來看

Here's what it looks like

這完全顛覆了它小小正方形的思想

when the sphere passes through Flatland

之後球體將正方形提升

from the square's perspective,

進入第三維

and this blows his little square mind.

也就是高度方向 一個平面國國民以前從未到過的地方

Then the sphere lifts the square

向正方形展示了它的世界

into the third dimension,

從這個高度，正方形可以看到所有事物

the height direction where no Flatlander has gone before

建築物的形狀、

and shows him his world.

所有隱藏在世界中珍貴的寶物、

From up here, the square can see everything:

甚至於它朋友的內部

the shapes of buildings,

這可能有點尷尬

all the precious gems hidden in the Earth,

不幸的正方形一接受第三維度後

and even the insides of his friends,

就乞求球體幫助它

which is probably pretty awkward.

探索第四或更高的維度

Once the hapless square

但球體感到非常生氣

comes to terms with the third dimension,

對於超過三維的看法

he begs his host to help him

並把正方形逐回平面國

visit the fourth and higher dimensions,

球體的憤怒是可以理解的

but the sphere bristles at the mere suggestion

第四維度很難

of dimensions higher than three

和我們在這世界的經歷達成一致

and exiles the square back to Flatland.

沒有被來訪的超立方體提升到第四維度

Now, the sphere's indignation is understandable.

我們無法體會

A fourth dimension is very difficult

但我們可以接近

to reconcile with our experience of the world.

你回溯到當球體

Short of being lifted into the fourth dimension

第一次到訪第二維時

by visiting hypercube,

它看起來像一連串的圓圈

we can't experience it,

當它碰觸到平面國時起始於一個點

but we can get close.

越變越大直到它穿越一半時

You'll recall that when the sphere

然後又萎縮變小

first visited the second dimension,

我們可以視此次拜訪

he looked like a series of circles

為三維物體的一連串橫截面

that started as a point

我們可以同樣對待

when he touched Flatland,

在第三維度的四維物體

grew bigger until he was halfway through,

我們說超球體

and then shrank smaller again.

是一個四維物體，等同於三維的球體

We can think of this visit

當這四維物體經過第三維度

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

它會看起來像這樣

Well, we can do the same thing

我們來看看另一個表現四維物體的方式

in the third dimension with a four-dimensional object.

我們有一個點，一個零維圖形

Let's say that a hypersphere

現在我們把它延伸一吋

is the 4D equivalent of a 3D sphere.

於是我們有了一個一維線段

When the 4D object passes through the third dimension,

把整個線段向外延伸一吋

it'll look something like this.

於是我們得到一個二維正方形

Let's look at one more way

把整個二維正方形向外延伸一吋

of representing a four-dimensional object.

於是我們得到一個三維立方體

Let's say we have a point,

你可以看見我們做了什麼

a zero-dimensional shape.

把整個立方體向外延伸一吋

Now we extend it out one inch

這一次與所有存在的三個維度相互垂直

and we have a one-dimensional line segment.

然後我們得到一個超立方體

Extend the whole line segment by an inch,

也叫四維超正方體

and we get a 2D square.

我們都知道

Take the whole square and extend it out one inch,

可能有四維生物存在於某個地方

and we get a 3D cube.

偶爾探頭到我們繁忙的三維世界

You can see where we're going with this.

看看有什麼大驚小怪的事情

Take the whole cube

事實上，可能有其他的四維世界

and extend it out one inch,

超越我們所能察覺的範圍

this time perpendicular to all three existing directions,

因為我們感知的特性 導致我們永遠看不到

and we get a 4D hypercube,

那不會顛覆你小小的腦袋嗎？

also called a tesseract.

For all we know,

there could be four-dimensional lifeforms

somewhere out there,

occasionally poking their heads

into our bustling 3D world

and wondering what all the fuss is about.

In fact, there could be whole

other four-dimensional worlds

beyond our detection,

hidden from us forever

by the nature of our perception.

Doesn't that blow your little spherical mind?