The satisfying math of folding origami - Evan Zodl

Translator: Gavin Thomas Reviewer: Shelley Tsang 曾雯海

As the space telescope prepares to snap a photo,

當太空望遠鏡準備影相嗰陣，

the light of the nearby star blocks its view.

附近恆星嘅光芒遮咗佢嘅視線。

But the telescope has a trick up its sleeve:

但望遠鏡有個秘招：

a massive shield to block the glare.

用一個巨大嘅板嚟阻擋強光。

This starshade has a diameter of about 35 meters—

呢個遮星板直徑約為35米 ——

that folds down to just under 2.5 meters,

摺起嚟卻只有2.5米以下。

small enough to carry on the end of a rocket.

細到可以裝喺火箭嘅尾端。

Its compact design is based on an ancient art form.

佢緊湊嘅設計源自一種古老嘅藝術形式。

Origami, which literally translates to “folding paper,”

摺紙，字面意思係「摺叠紙張」，

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

呢項日本習俗，至少可追溯到17世紀。

In origami, the same simple concepts

摺紙藝術入面，同樣簡單嘅概念， 可以摺成任何物品，

yield everything from a paper crane with about 20 steps,

從大約20個步驟摺成嘅紙鶴。

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

到呢條需要超過1000個步驟嘅龍， 再到一個遮星板。

A single, traditionally square sheet of paper

只要一張紙，傳統上係方形紙，

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

僅僅透過摺叠，就可以變成幾乎任何形狀。

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

攤平嗰張紙，就會有個線條構成嘅圖案。

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

每條綫都代表一個凹陷嘅谷摺或凸起嘅山摺。

Origami artists arrange these folds to create crease patterns,

摺紙藝術家通過摺叠去形成摺痕圖案，

which serve as blueprints for their designs.

作為佢哋設計嘅藍圖。

Though most origami models are three dimensional,

雖然多數摺紙作品係立體嘅，

their crease patterns are usually designed to fold flat

佢哋嘅摺痕圖案通常被設計為平折，

without introducing any new creases or cutting the paper.

而唔會產生任何新摺痕或需要切割紙張。

The mathematical rules behind flat-foldable crease patterns

可以平折嘅摺痕圖案背後嘅數學規則

are much simpler than those behind 3D crease patterns—

比立體摺痕圖案嘅規則簡單好多——

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

先創建抽象嘅平面設計， 然後將佢塑造為立體形式會更容易。

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

任何可以平摺嘅摺痕圖案 都必須遵守四個規則：

First, the crease pattern must be two-colorable—

第一，摺痕圖案必須可用雙色上色 ——

meaning the areas between creases can be filled with two colors

即係摺痕之間嘅區域 只用兩種顏色就可以填滿，

so that areas of the same color never touch.

同色嘅區域因此永遠唔會接觸。

Add another crease here,

喺呢到添加一個摺痕，

and the crease pattern no longer displays two-colorability.

摺痕圖案就唔可以兩種顔色上色。

Second, the number of mountain and valley folds

第二，山摺同谷摺嘅數量

at any interior vertex must differ by exactly two—

喺每一個內部嘅頂點位都必須相差 2 ——

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

就例如呢到嘅三個谷摺同一個山摺咁。

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

我哋仔細睇下喺呢個頂點摺叠會發生啲咩：

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

如果喺呢個頂點加上一個山摺， 就會有三個谷摺兩個山摺。

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

如果加上一個谷摺，就會有四個谷摺同一個山摺。

Either way, the model doesn't fall flat.

無論用邊種方式，摺紙作品都摺唔平。

The third rule is that if we number all the angles

第三條規則係，

如果我哋將頂點内所有嘅角 都順時針或者逆時針編號，

at an interior vertex moving clockwise or counterclockwise,

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

偶數角加起，總數必須係180度，

as must the odd-numbered angles.

奇數角都必須一樣。

Looking closer at the folds, we can see why.

仔細觀察摺痕，我哋就睇得出點解。

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

如果喺呢個頂點增加一個摺痕並幫新嘅角編號。

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

偶數角同奇數角各自嘅角度總和 就唔再係180度，

and the model doesn’t fold flat.

作品就無法摺成平嘅。

Finally, a layer cannot penetrate a fold.

最後，一層紙唔可以穿過一個摺皺。

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

一個可摺疊嘅2D平面底座往往是一種抽象的表現形式

通常係最終嘅3D立體形狀嘅抽象代表。

of a final 3D shape.

Understanding the relationship between crease patterns, 2D bases,

摺紙藝術家理解摺痕圖案、平面底座， 同最終立體形狀嘅關係，

and the final 3D form allows origami artists

所以可以設計出不可思議嘅複雜形狀。

to design incredibly complex shapes.

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

以摺紙藝術家Robert J. Lang創作嘅摺痕圖案爲例。

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

呢個摺痕圖案可分成一隻生物嘅腿、尾巴同其他附肢。

tail, and other appendages.

When we fold the crease pattern into this flat base,

當我哋將摺痕圖案摺成呢個平面基礎，

each of these allocated areas becomes a separate flap.

每個劃分好嘅區域都會變成獨立嘅一層。

By narrowing, bending, and sculpting these flaps,

通過將佢縮窄、彎曲、變形，

the 2D base becomes a 3D scorpion.

平面基礎就變成左立體嘅蠍子。

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

咁如果我哋想喺同一張紙上，折出7朵咁嘅花呢？

If we can duplicate the flower’s crease pattern

如果我哋可以重複花嘅摺痕圖案，

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

並以滿足四條規律嘅方式將佢哋各自連接起嚟，

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

我哋就可以創建一個密鋪，即係一個重複嘅形狀圖案

that covers a plane without any gaps or overlaps.

無任何縫隙而且唔會重複地覆蓋一平面。

The ability to fold a large surface into a compact shape

將一個大嘅平面摺疊成緊湊形狀嘅能力，

has applications from the vastness of space

可以應用喺浩瀚無垠嘅宇宙，

to the microscopic world of our cells.

都可以用喺細胞嘅微觀世界。

Using principles of origami,

利用摺紙嘅原理，

medical engineers have re-imagined the traditional stent graft,

醫學工程師已經重新塑造傳統嘅脈覆膜支架，

a tube used to open and support damaged blood vessels.

係一條用於張開同支撐受損血管嘅管。

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

透過密鋪，剛性嘅管狀結構摺疊成緊湊嘅薄片

about half its expanded size.

體積大細約係張開時嘅一半左右。

Origami principles have been used in airbags, solar arrays,

摺紙原理已被用喺安全氣囊、太陽能板

self-folding robots, and even DNA nanostructures—

自我摺疊嘅機器人，甚至係DNA納米結構

who knows what possibilities will unfold next.

邊個知下個可能嘅發展係咩？