The satisfying math of folding origami - Evan Zodl

415,675 views ・ 2021-02-11

TED-Ed


Please double-click on the English subtitles below to play the video.

Translator: Gavin Thomas Reviewer: Shelley Tsang 曾雯海
00:07
As the space telescope prepares to snap a photo,
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當太空望遠鏡準備影相嗰陣,
00:11
the light of the nearby star blocks its view.
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附近恆星嘅光芒遮咗佢嘅視線。
00:14
But the telescope has a trick up its sleeve:
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但望遠鏡有個秘招:
00:17
a massive shield to block the glare.
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用一個巨大嘅板嚟阻擋強光。
00:21
This starshade has a diameter of about 35 meters—
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呢個遮星板直徑約為35米 ——
00:25
that folds down to just under 2.5 meters,
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摺起嚟卻只有2.5米以下。
00:29
small enough to carry on the end of a rocket.
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細到可以裝喺火箭嘅尾端。
00:32
Its compact design is based on an ancient art form.
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佢緊湊嘅設計源自一種古老嘅藝術形式。
00:37
Origami, which literally translates to “folding paper,”
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摺紙,字面意思係「摺叠紙張」,
00:42
is a Japanese practice dating back to at least the 17th century.
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呢項日本習俗,至少可追溯到17世紀。
00:47
In origami, the same simple concepts
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摺紙藝術入面,同樣簡單嘅概念, 可以摺成任何物品,
00:49
yield everything from a paper crane with about 20 steps,
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從大約20個步驟摺成嘅紙鶴。
00:53
to this dragon with over 1,000 steps, to a starshade.
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到呢條需要超過1000個步驟嘅龍, 再到一個遮星板。
00:58
A single, traditionally square sheet of paper
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只要一張紙,傳統上係方形紙,
01:01
can be transformed into almost any shape, purely by folding.
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僅僅透過摺叠,就可以變成幾乎任何形狀。
01:05
Unfold that sheet, and there’s a pattern of lines,
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攤平嗰張紙,就會有個線條構成嘅圖案。
01:09
each of which represents a concave valley fold or a convex mountain fold.
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每條綫都代表一個凹陷嘅谷摺或凸起嘅山摺。
01:15
Origami artists arrange these folds to create crease patterns,
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摺紙藝術家通過摺叠去形成摺痕圖案,
01:20
which serve as blueprints for their designs.
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作為佢哋設計嘅藍圖。
01:23
Though most origami models are three dimensional,
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雖然多數摺紙作品係立體嘅,
01:26
their crease patterns are usually designed to fold flat
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佢哋嘅摺痕圖案通常被設計為平折,
01:29
without introducing any new creases or cutting the paper.
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而唔會產生任何新摺痕或需要切割紙張。
01:33
The mathematical rules behind flat-foldable crease patterns
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可以平折嘅摺痕圖案背後嘅數學規則
01:37
are much simpler than those behind 3D crease patterns—
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比立體摺痕圖案嘅規則簡單好多——
01:42
it’s easier to create an abstract 2D design and then shape it into a 3D form.
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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。
01:48
There are four rules that any flat-foldable crease pattern must obey.
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任何可以平摺嘅摺痕圖案 都必須遵守四個規則:
01:54
First, the crease pattern must be two-colorable—
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第一,摺痕圖案必須可用雙色上色 ——
01:57
meaning the areas between creases can be filled with two colors
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即係摺痕之間嘅區域 只用兩種顏色就可以填滿,
02:01
so that areas of the same color never touch.
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同色嘅區域因此永遠唔會接觸。
02:05
Add another crease here,
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喺呢到添加一個摺痕,
02:07
and the crease pattern no longer displays two-colorability.
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摺痕圖案就唔可以兩種顔色上色。
02:11
Second, the number of mountain and valley folds
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第二,山摺同谷摺嘅數量
02:14
at any interior vertex must differ by exactly two—
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喺每一個內部嘅頂點位都必須相差 2 ——
02:19
like the three valley folds and one mountain fold that meet here.
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就例如呢到嘅三個谷摺同一個山摺咁。
02:24
Here’s a closer look at what happens when we make the folds at this vertex.
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我哋仔細睇下喺呢個頂點摺叠會發生啲咩:
02:29
If we add a mountain fold at this vertex, there are three valleys and two mountains.
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如果喺呢個頂點加上一個山摺, 就會有三個谷摺兩個山摺。
02:34
If it’s a valley, there are four valleys and one mountain.
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如果加上一個谷摺,就會有四個谷摺同一個山摺。
02:39
Either way, the model doesn't fall flat.
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無論用邊種方式,摺紙作品都摺唔平。
02:42
The third rule is that if we number all the angles
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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,
02:46
at an interior vertex moving clockwise or counterclockwise,
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02:49
the even-numbered angles must add up to 180 degrees,
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偶數角加起,總數必須係180度,
02:55
as must the odd-numbered angles.
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奇數角都必須一樣。
02:58
Looking closer at the folds, we can see why.
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仔細觀察摺痕,我哋就睇得出點解。
03:02
If we add a crease and number the new angles at this vertex,
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如果喺呢個頂點增加一個摺痕並幫新嘅角編號。
03:06
the even and odd angles no longer add up to 180 degrees,
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偶數角同奇數角各自嘅角度總和 就唔再係180度,
03:11
and the model doesn’t fold flat.
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作品就無法摺成平嘅。
03:14
Finally, a layer cannot penetrate a fold.
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最後,一層紙唔可以穿過一個摺皺。
03:18
A 2D, flat-foldable base is often an abstract representation
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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。
03:23
of a final 3D shape.
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03:25
Understanding the relationship between crease patterns, 2D bases,
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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,
03:30
and the final 3D form allows origami artists
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所以可以設計出不可思議嘅複雜形狀。
03:34
to design incredibly complex shapes.
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03:37
Take this crease pattern by origami artist Robert J. Lang.
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以摺紙藝術家Robert J. Lang創作嘅摺痕圖案爲例。
03:41
The crease pattern allocates areas for a creature's legs,
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呢個摺痕圖案可分成一隻生物嘅腿、尾巴同其他附肢。
03:45
tail, and other appendages.
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03:47
When we fold the crease pattern into this flat base,
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當我哋將摺痕圖案摺成呢個平面基礎,
03:50
each of these allocated areas becomes a separate flap.
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每個劃分好嘅區域都會變成獨立嘅一層。
03:55
By narrowing, bending, and sculpting these flaps,
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通過將佢縮窄、彎曲、變形,
03:58
the 2D base becomes a 3D scorpion.
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平面基礎就變成左立體嘅蠍子。
04:02
Now, what if we wanted to fold 7 of these flowers from the same sheet of paper?
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咁如果我哋想喺同一張紙上,折出7朵咁嘅花呢?
04:08
If we can duplicate the flower’s crease pattern
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如果我哋可以重複花嘅摺痕圖案,
04:10
and connect each of them in such a way that all four laws are satisfied,
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並以滿足四條規律嘅方式將佢哋各自連接起嚟,
04:15
we can create a tessellation, or a repeating pattern of shapes
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我哋就可以創建一個密鋪,即係一個重複嘅形狀圖案
04:19
that covers a plane without any gaps or overlaps.
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無任何縫隙而且唔會重複地覆蓋一平面。
04:23
The ability to fold a large surface into a compact shape
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將一個大嘅平面摺疊成緊湊形狀嘅能力,
04:27
has applications from the vastness of space
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可以應用喺浩瀚無垠嘅宇宙,
04:30
to the microscopic world of our cells.
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都可以用喺細胞嘅微觀世界。
04:33
Using principles of origami,
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利用摺紙嘅原理,
04:35
medical engineers have re-imagined the traditional stent graft,
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醫學工程師已經重新塑造傳統嘅脈覆膜支架,
04:40
a tube used to open and support damaged blood vessels.
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係一條用於張開同支撐受損血管嘅管。
04:43
Through tessellation, the rigid tubular structure folds into a compact sheet
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透過密鋪,剛性嘅管狀結構摺疊成緊湊嘅薄片
04:48
about half its expanded size.
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體積大細約係張開時嘅一半左右。
04:51
Origami principles have been used in airbags, solar arrays,
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摺紙原理已被用喺安全氣囊、太陽能板
04:56
self-folding robots, and even DNA nanostructures—
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自我摺疊嘅機器人,甚至係DNA納米結構
05:01
who knows what possibilities will unfold next.
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邊個知下個可能嘅發展係咩?
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