The satisfying math of folding origami - Evan Zodl

譯者: 至磊Zi Le 黃Ng 審譯者: Helen Chang

As the space telescope prepares to snap a photo,

太空望遠鏡準備拍照時，

the light of the nearby star blocks its view.

鄰近的星光遮蔽了它的視線。

But the telescope has a trick up its sleeve:

但望遠鏡有個祕密招數：

a massive shield to block the glare.

用巨大的板子擋住亮光。

This starshade has a diameter of about 35 meters—

遮星板直徑長達約 35 公尺，

that folds down to just under 2.5 meters,

摺起來卻只有 2.5 公尺以下，

small enough to carry on the end of a rocket.

小得可以裝在火箭的尾端。

Its compact design is based on an ancient art form.

它緊湊密實的設計源自 一種古老的藝術形式。

Origami, which literally translates to “folding paper,”

Origami 翻譯成「摺紙」，

is a Japanese practice dating back to at least the 17th century.

這項日本風俗至少 可追溯到十七世紀。

In origami, the same simple concepts

摺紙藝術中，同樣幾條簡單的原則

yield everything from a paper crane with about 20 steps,

可以從摺大約二十個步驟的紙鶴，

to this dragon with over 1,000 steps, to a starshade.

到需要超過一千個步驟的龍，

再到遮星板。

A single, traditionally square sheet of paper

只要一張紙，傳統上是正方形，

can be transformed into almost any shape, purely by folding.

幾乎可以轉化為任何形態， 而且只需要用摺的。

Unfold that sheet, and there’s a pattern of lines,

攤平那張紙，就有線構成的圖樣，

each of which represents a concave valley fold or a convex mountain fold.

每條都代表凹陷的谷摺 或凸起的山摺。

Origami artists arrange these folds to create crease patterns,

摺紙藝術家安排摺法， 製作摺痕的圖樣，

which serve as blueprints for their designs.

當成他們的設計藍圖。

Though most origami models are three dimensional,

儘管多數摺紙作品是立體的，

their crease patterns are usually designed to fold flat

它們的摺痕圖樣通常會 設計成可以摺成平的，

without introducing any new creases or cutting the paper.

而且不需要新的摺痕或剪裁紙張。

The mathematical rules behind flat-foldable crease patterns

可以摺平的摺痕圖樣背後的數學規則

are much simpler than those behind 3D crease patterns—

比立體摺痕圖樣的規則簡單多了，

it’s easier to create an abstract 2D design and then shape it into a 3D form.

做出抽象的平面設計， 再塑造為立體形式比較簡單。

There are four rules that any flat-foldable crease pattern must obey.

可以摺平的摺痕圖樣 必須遵循四條規則。

First, the crease pattern must be two-colorable—

第一，摺痕圖樣必須 可用兩種顏色上色，

meaning the areas between creases can be filled with two colors

意思是摺痕之間的區域 只用兩種顏色就能填滿，

so that areas of the same color never touch.

同色的區域永遠不會接在一起。

Add another crease here,

若在此多加一條摺痕，

and the crease pattern no longer displays two-colorability.

摺痕圖樣就不能用兩種顏色上色了。

Second, the number of mountain and valley folds

第二，山摺和谷摺的數量

at any interior vertex must differ by exactly two—

在每一個內部的頂點 都必須剛好差 2 ，

like the three valley folds and one mountain fold that meet here.

例如這裡是三個谷摺和一個山摺。

Here’s a closer look at what happens when we make the folds at this vertex.

我們把這個頂點的摺法看仔細一點。

If we add a mountain fold at this vertex, there are three valleys and two mountains.

如果在這個頂點多加一個山摺， 就會是三個谷摺和兩個山摺。

If it’s a valley, there are four valleys and one mountain.

如果多加一個谷摺， 就是四個谷摺和一個山摺。

Either way, the model doesn't fall flat.

無論何者，都無法摺成平的。

The third rule is that if we number all the angles

第三條規則是

如果我們將某個內部頂點周圍的角 順時針或逆時針編號，

at an interior vertex moving clockwise or counterclockwise,

the even-numbered angles must add up to 180 degrees,

偶數角加總必須是 180 度，

as must the odd-numbered angles.

奇數角也一樣。

Looking closer at the folds, we can see why.

把摺法看仔細一點就知道為什麼了。

If we add a crease and number the new angles at this vertex,

如果我們多加一條摺痕， 重新幫這個頂點的角編號，

the even and odd angles no longer add up to 180 degrees,

偶數角和奇數角各自的 總和就不會是 180 度，

and the model doesn’t fold flat.

作品就無法摺成平的。

Finally, a layer cannot penetrate a fold.

最後，不能有某一層 貫穿整個摺紙作品。

A 2D, flat-foldable base is often an abstract representation

一張平面且可以摺平的基礎

常是最終的立體外形的抽象代表。

of a final 3D shape.

Understanding the relationship between crease patterns, 2D bases,

摺紙藝術家理解平面摺痕圖樣 和最終立體外形的關聯，

and the final 3D form allows origami artists

所以可以設計出 不可思議的複雜形體。

to design incredibly complex shapes.

Take this crease pattern by origami artist Robert J. Lang.

以摺紙藝術家羅伯特‧ J ‧朗 創作的摺痕圖樣為例。

The crease pattern allocates areas for a creature's legs,

這張摺痕圖樣可分成 一隻生物的腿、尾、其他附肢。

tail, and other appendages.

When we fold the crease pattern into this flat base,

當我們將摺痕圖樣摺成平面基礎，

each of these allocated areas becomes a separate flap.

每個劃分好的區塊 都會變成獨立的一層。

By narrowing, bending, and sculpting these flaps,

將其縮窄、彎曲、變形，

the 2D base becomes a 3D scorpion.

平面基礎就成了立體的蠍子。

Now, what if we wanted to fold 7 of these flowers from the same sheet of paper?

那如果我們想用同一張紙 摺七朵這種花呢？

If we can duplicate the flower’s crease pattern

如果我們能重複這朵花的摺痕圖樣，

and connect each of them in such a way that all four laws are satisfied,

並以滿足四條規則的方式連接，

we can create a tessellation, or a repeating pattern of shapes

就可以製造出密鋪， 也就是重複的形狀圖樣

that covers a plane without any gaps or overlaps.

沒有縫隙且不會重疊地覆蓋一平面。

The ability to fold a large surface into a compact shape

將巨大平面摺成緊緻形體的能力

has applications from the vastness of space

可以用在寬廣的宇宙，

to the microscopic world of our cells.

也能用在細胞的微觀世界。

Using principles of origami,

利用摺紙的原則，

medical engineers have re-imagined the traditional stent graft,

醫學工程師已經重新塑造 傳統的主動脈覆膜支架，

a tube used to open and support damaged blood vessels.

這是一種可以張開 並支撐損毀血管的管子。

Through tessellation, the rigid tubular structure folds into a compact sheet

透過密鋪，固定的管狀結構 可以摺成緊密的薄片，

about half its expanded size.

大小只有張開時的一半左右。

Origami principles have been used in airbags, solar arrays,

摺紙的原則已經用在 安全氣囊、太陽能板、

self-folding robots, and even DNA nanostructures—

自我摺疊的機器人， 甚至是 DNA 的奈米結構，

who knows what possibilities will unfold next.

誰知道下一個可能的發展是什麼呢？