Exploring other dimensions - Alex Rosenthal and George Zaidan

探索其它維度 - Alex Rosenthal 和 George Zaidan

譯者: Bernice Huang 審譯者: Qiwen Lu

We live in a three-dimensional world

我們住在一個三維世界

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

在歷經第三維度的時候

exposed to the third dimension.

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,

pentagons,

正方形、

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.

What they see is essentially one dimension,

實質上它們看到的是一維

a line.

也就是一條線

But in Abbott's Flatland,

但在艾勃特的平面國中

closer objects are brighter,

越接近的物體看起來越明亮

and that's how they see depth.

這就是它們如何看到深度

So a triangle looks different from a square,

所以三角形看起來與正方形不同、

looks different a circle,

看起來與圓形不同

and so on.

諸如此類

Their brains cannot comprehend the third dimension.

它們的腦袋無法理解第三維度

In fact, they vehemently deny its existence

事實上，它們極力否認第三維度的存在

because it's simply not part of their world

因為那根本完全不存在於它們的世界

or experience.

或經驗中

But all they need,

但事實證明，它們所需要的

as it turns out,

is a little boost.

只是一點小小的刺激

One day a sphere shows up in Flatland

有一天，一個球體出現在平面國中

to visit our square hero.

拜訪我們的正方形英雄

Here's what it looks like

這是當球體經過平面國時看起來的樣子

when the sphere passes through Flatland

from the square's perspective,

從正方形的角度來看

and this blows his little square mind.

這完全顛覆了它小小正方形的思想

Then the sphere lifts the square

之後球體將正方形提升

into the third dimension,

進入第三維

the height direction where no Flatlander has gone before

也就是高度方向 一個平面國國民以前從未到過的地方

and shows him his world.

向正方形展示了它的世界

From up here, the square can see everything:

從這個高度，正方形可以看到所有事物

the shapes of buildings,

建築物的形狀、

all the precious gems hidden in the Earth,

所有隱藏在世界中珍貴的寶物、

and even the insides of his friends,

甚至於它朋友的內部

which is probably pretty awkward.

這可能有點尷尬

Once the hapless square

不幸的正方形一接受第三維度後

comes to terms with the third dimension,

he begs his host to help him

就乞求球體幫助它

visit the fourth and higher dimensions,

探索第四或更高的維度

but the sphere bristles at the mere suggestion

但球體感到非常生氣

of dimensions higher than three

對於超過三維的看法

and exiles the square back to Flatland.

並把正方形逐回平面國

Now, the sphere's indignation is understandable.

球體的憤怒是可以理解的

A fourth dimension is very difficult

第四維度很難

to reconcile with our experience of the world.

和我們在這世界的經歷達成一致

Short of being lifted into the fourth dimension

沒有被來訪的超立方體提升到第四維度

by visiting hypercube,

we can't experience it,

我們無法體會

but we can get close.

但我們可以接近

You'll recall that when the sphere

你回溯到當球體

first visited the second dimension,

第一次到訪第二維時

he looked like a series of circles

它看起來像一連串的圓圈

that started as a point

當它碰觸到平面國時起始於一個點

when he touched Flatland,

grew bigger until he was halfway through,

越變越大直到它穿越一半時

and then shrank smaller again.

然後又萎縮變小

We can think of this visit

我們可以視此次拜訪

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

為三維物體的一連串橫截面

Well, we can do the same thing

我們可以同樣對待

in the third dimension with a four-dimensional object.

在第三維度的四維物體

Let's say that a hypersphere

我們說超球體

is the 4D equivalent of a 3D sphere.

是一個四維物體，等同於三維的球體

When the 4D object passes through the third dimension,

當這四維物體經過第三維度

it'll look something like this.

它會看起來像這樣

Let's look at one more way

我們來看看另一個表現四維物體的方式

of representing a four-dimensional object.

Let's say we have a point,

我們有一個點，一個零維圖形

a zero-dimensional shape.

Now we extend it out one inch

現在我們把它延伸一吋

and we have a one-dimensional line segment.

於是我們有了一個一維線段

Extend the whole line segment by an inch,

把整個線段向外延伸一吋

and we get a 2D square.

於是我們得到一個二維正方形

Take the whole square and extend it out one inch,

把整個二維正方形向外延伸一吋

and we get a 3D cube.

於是我們得到一個三維立方體

You can see where we're going with this.

你可以看見我們做了什麼

Take the whole cube

把整個立方體向外延伸一吋

and extend it out one inch,

this time perpendicular to all three existing directions,

這一次與所有存在的三個維度相互垂直

and we get a 4D hypercube,

然後我們得到一個超立方體

also called a tesseract.

也叫四維超正方體

For all we know,

there could be four-dimensional lifeforms

我們都知道

可能有四維生物存在於某個地方

somewhere out there,

occasionally poking their heads

偶爾探頭到我們繁忙的三維世界

into our bustling 3D world

and wondering what all the fuss is about.

看看有什麼大驚小怪的事情

In fact, there could be whole

事實上，可能有其他的四維世界

other four-dimensional worlds

beyond our detection,

超越我們所能察覺的範圍

hidden from us forever

因為我們感知的特性 導致我們永遠看不到

by the nature of our perception.

Doesn't that blow your little spherical mind?

那不會顛覆你小小的腦袋嗎？