I♡PM ▶ Dragon Boat 2021

Would you admire making model dragon boats or rice dumplings out of a piece of paper?

If yes, read on. If no, enjoy a movie【M1】, or this theme song from the movie.

Last time we looked at the math of numbers. This time we shall see the math of shapes. The simplest shapes are: triangles, squares, circles, etc. which you all know. That's geometry, a branch of math that can be traced back to antiquity.

In Chinese, there is an ancient saying, "Without a right-angle ruler and a compass, one cannot draw squares or circles." (「不以規矩，不能成方圓。」) Western geometry follows the Greek tradition, using a compass (圓規) and a straightedge (直尺). These are standard tools for the study of geometry.

In my schooldays, I always have ruler and compass in my pencil box. Now my pencil box is nowhere to be found. Can we play with geometry without these tools? It turns out that there is a lot to learn about geometry by another ancient skill: paper folding. The art of paper folding is called origami (折り紙、摺紙).

利用紙張，摺出模型龍舟或粽子，你會欣賞嗎？

若然，請繼續閱讀。若否，請欣賞電影【M1】，或欣賞電影中這首主題曲。

上次講數字的數學，今次談形狀的數學。眾所周知，最簡單的形狀是：三角形、正方形、圓形等。這是幾何，是數學的分支，起源可以追溯到遠古年代。

中國古語有云：「不以規矩，不能成方圓。」，其中「規」指圓規，「矩」指直角曲尺。西方的幾何，遵循希臘傳統，使用圓規和直尺。這些都是研究幾何的標準工具。

小時候上課，總帶著筆盒，內有間尺和圓規。時至今日，筆盒已不知所踪。沒有這些幾何工具，能夠玩一玩幾何嗎？原來，還有另一種歷史悠久的技巧：摺紙，可用作幾何學習。摺紙的藝術，日文稱為：折り紙 (origami)。

Origami Fun 摺紙樂趣

Hong Kong stamps in the series 'Paper Folding Fun', released on May 22, 2008. Click to enlarge (click again to close)
2008年5月22日發行的香港郵票「摺紙樂」系列。 點擊放大(再擊關閉)
In 2006 the Hong Kong Post organised the “Children Stamps –- Paper Folding Fun Design Competition”. The winning entries were featured in a set of stamps in 2008【A1】.

It is not hard to fold a paper boat. With some ingenuity one can fold a dragon boat【A2】. On the other hand, you can try folding 3D geometric shapes, like square-based pyramids, which can be colored to be nice models of the rice dumpling of the Dragon Boat Festival【A3】.

I would like to tell you:

Welcome to the world of origami! Let us start with playing some basic geometry by paper folding.
2006年，香港郵政舉辦「兒童郵票：摺紙樂設計比賽」。獲獎作品多姿多彩，用作發行一套2008年兒童郵票【A1】。

摺一艘紙船很容易。憑著一些創意，可以摺成一艘龍舟【A2】。另外，可嘗試摺出立體幾何形狀，例如方底金字塔，其後着色，成為端午節粽子的漂亮模型【A3】。

我想告訴你：

歡迎來到摺紙世界！讓我們透過摺紙，耍玩一些基本幾何。現在開始！

Angle Bisection 二等分角

mode bisect origin 50 240 width 180 angle 62 stage 1 mode bisect origin 50 240 width 180 angle 62 stage 2 mode bisect origin 50 240 width 180 angle 62 stage 3
Folding a sheet of paper to bisect an angle. 摺紙完成二等分角。
bisect 75
Take a piece of paper, square or rectangular. Fold a side from a corner and unfold. The crease will mark an angle (shown above, left diagram). To bisect this angle, simply fold the angle base to the angle arm. Unfold to reveal the crease, which is the angle bisector (shown above, right diagram).

If you are using a modern browser【*】, click the button "Animate" to watch an animation of angle bisection by paper folding.

Now take another piece of paper and either drawing a triangle, or make one by folding up some sides. For each angle of the triangle, draw its angle bisector, or get one by simple folding as described. You will find that,

取一張紙，正方形或長方形均可。從一邊隨一角斜摺然後展開，摺痕將標記一個角度（上左圖）。要把該角二等分，只要將角基線摺疊到角臂線便可。展開即顯示摺痕，是該角的二等分線（上右圖）。

若正在使用新一代瀏覽器【*】，請單擊右圖「Animate」按鈕，觀看二等分角的摺紙動畫。

現在，再拿一張紙，畫一個三角形，或者將某些側邊摺起來做成一個三角形。對於三角形的每個角度，繪出其角度二等分線，或者通過上述簡單摺紙得到每個角的二等分線。你會發現：

bisect A 250 90 B 66 186 C 263 234
Drag vertices of the triangle by mouse or finger. 用滑鼠或手指，拖動三角形的角點。    
The point is called the in-center of the triangle, and the circle is the inscribed circle. Can you see: The answers to both questions are intimately related【B1】.

Again, if you are using a modern browser【*】, you can play with the triangle on the right: using mouse or finger, drag the vertices of the triangle to check that the 3 angle bisectors always meet at the in-center. Click "Circle" button to show the inscribed circle. To restore to the original triangle, click "Reset" button.

該點稱為三角形的內接心，圓形稱為內接圓。你能否明白： 兩個問題的答案，互相關係密切【B1】。

同樣，如果使用新一代瀏覽器【*】，則可以把玩右圖的三角形：使用滑鼠或手指，拖動三角形的角點，檢查三個角的二等分線，始終相交在內接心。單擊「Circle」按鈕，以顯示內接圓。要回復到原來的三角形，請單擊「Reset」按鈕。

Angle Trisection 三等分角

mode trisect origin 50 240 width 180 angle 72 stage 1 mode trisect origin 50 240 width 180 angle 72 stage 2 mode trisect origin 50 240 width 180 angle 72 stage 3
mode trisect origin 50 240 width 180 angle 72 stage 4 mode trisect origin 50 240 width 180 angle 72 stage 5 mode trisect origin 50 240 width 180 angle 72 stage 6
Folding paper to trisect an angle (adapted from a method by the Japanese folder and mathematician Hisashi Abe). 摺紙完成三等分角（根據日本摺紙家和數學家：阿部恒的方法改編而成）。
trisect 70
Take another sheet of paper, square or rectangular, and make an angle by folding a side around a corner (shown above, first diagram). It is possible to divide such an angle into three equal parts (shown above, last diagram):
  1. Fold the top down freely to make a horizontal yellow crease above the middle.
  2. Fold the bottom up to meet the yellow crease, making a halfway green crease.
  3. The yellow crease ends at E on the left, and the corner below E is B. (second diagram)
  4. Make an inclined fold such that E lands on the angle arm, and B lands on the green crease.
  5. Unfold to reveal the inclined crease, which cuts the green crease at V, and the base at T. (third diagram)
  6. Fold the line BT to BV, the crease is one trisector of the original angle. (fourth diagram)
  7. Fold the line BV to the angle arm, the crease is another trisector of the original angle. (fifth diagram)
  8. Unfold completely to reveal the two trisectors of the original angle. (sixth diagram)

In a modern browser, click "Animate" above to watch this method in action. Can you figure out why this folding sequence gives the angle trisectors? You'll need to know a bit of elementary geometry【C1】.

換另一張紙，正方形或長方形的紙。然後將一邊從一角折摺，以形成一個角度（上第一圖）。可以把該角三等分（上最後圖），步驟如下：
  1. 將頂邊隨意往下摺，使中位上方出現水平的黃色摺痕。
  2. 摺起底邊，以重疊黃色摺痕，形成半中腰的綠色摺痕。
  3. 黃色摺痕的最左端是E點，而E的垂直底角是B點。（圖二）
  4. 進行傾斜的摺疊，使E點落在角臂上，而B點則落在綠色摺痕上。
  5. 展開顯示傾斜摺痕，該摺痕與綠色摺痕相交於V點，與底邊相交於T點。（圖三）
  6. 將BT線摺疊至BV線，摺痕為原本角的一條三等分線。（圖四）
  7. 將BV線摺疊到角臂上，摺痕是原本角的另一條三等分線。（圖五）
  8. 完全展開，顯示原本角的兩條三等分線。（圖六）

在新一代瀏覽器，單擊上方的「Animate」按鈕，觀看此摺法的實際過程。能否弄清楚，為什麼這摺法次序，會把角度三等分？你需要具備一些基本幾何知識【C1】。

morley A 165 69 B 19 234 C 265 270
Drag vertices of the triangle by mouse or finger. 用滑鼠或手指，拖動三角形的角點。    
A triangle has 3 angles. If the trisectors from each angle is drawn, the neighbouring trisector pairs meet at 3 points. These 3 points form the vertices of an equilateral triangle, no matter what is the shape of the original triangle. For a modern browser, you can verify this by dragging the vertices of the triangle on the right. Click "Circle" button to show the inscribed circle, which encloses the equilateral triangle centrally. To restore to the original triangle, click "Reset" button.

Isn't this wonderful? Such a beauty is hidden in geometry!

Indeed, this marvelous result is buried for almost 2,000 years. Following the Greek tradition, geometric constructions allow only a compass and an unmarked ruler. Euclid, in his great work Elements, could easily bisect an angle, but could not trisect an angle. His book was the bible of geometry, hence no one look into angle trisectors. Even Archimedes, who succeeded to trisect any angle by a compass and a ruler with marks【C2】, did not consider anything interesting about angle trisectors.

Frank Morley was a Cambridge mathematician migrated to Pennsylvania, USA. He found this wonderful equilateral triangle formed by the angle trisectors around 1899. He was not an expert in geometry, and he thought that this must be a well-known geometric result. Therefore, he only talked to his friends about this result, hoping that one of them would tell him the name of the theorem. After a while, he became convinced that this is a new result, and published his discovery. This special equilateral triangle is now called Morley's triangle.

There are various proof for this Morley's triangle theorem. Perhaps the most elegant is one given by John Conway in 2006【C3】.

三角形具有三個角。如果從每個角度繪製三等分線，則相鄰的每對三等分線，將相交於三個點。無論原本三角形的形狀是如何，這三個相交點均會形成一個等邊三角形。在新一代瀏覽器，你可以通過拖動右圖的三角形的角點，來驗證這一結果。單擊「Circle」按鈕，顯示內接圓，該圓也將把等邊三角形圍在中央。要回復到原來的三角形，請單擊「Reset」按鈕。

如此優美的結果，隱藏在幾何中，這不是很奇妙嗎？

的確，這項出色的結果，埋藏了近2000年。由於謹守希臘數學的傳統，幾何作圖只允許使用圓規和未標記的直尺。Euclid（歐幾里得）在他的偉大作品《幾何原本》中，很容易把一角作二等分，但未能將一角作三等分。他的著作，被視為幾何學的聖經，因此沒有人研究角的三分線。即使是 Archimedes（阿基米德），他利用圓規和尺上標記成功將一角作三等分【C2】，但也不認為角的三等分線具任何有趣之處。

Frank Morley（弗蘭克·莫利）是劍橋數學家，移居美國賓夕法尼亞州。他在1899年左右，發現由三角形三等分線組成的奇妙等邊三角形。但他不是幾何專家，認為這必定是著名的幾何結果。因此，他只向朋友談論提及此事，希望其中一位會告訴他定理的名稱。一段時間後，他確信這是嶄新結果，並發表自己的發現。這個特殊的等邊三角形，現稱：莫利三角形（Morley's triangle）。

莫利三角形定理，有各式各樣的證明。其中，以 John Conway 在2006年發表的證明最精采【C3】。

Origami Math 摺紙數學

A beautiful geometric design with origami. Click to enlarge (click again to close)
一個美麗的幾何設計與摺紙。 點擊放大(再擊關閉)
If the story that no one suspected any theorems about angle trisectors for 2000 years is not strange enough, the story of the intricate connection between paper folding with math is even more incredible.

People skilled in paper folding are called origamists, they don't know about math. Mathematician can fold papers from instructions, but they don't see the math. They are two separate groups of people, for a very long time.

Mathematicians, especially those skilled in geometry, tried hard to trisect the angle using the classical tools: a compass and an unmarked ruler. They failed, and eventually they suspected that this is an impossible task. But they cannot tell why this is impossible.

This impossibility was finally proved by Pierre Wantzel in 1837, related to the fact that the Greek instruments cannot be used to solve cubic equations in general.

Felix Klein was a well-known German mathematician with interest in many topics. In one of his popular math lectures, he mentioned a book Geometric Exercises in Paper Folding【D1】, published in 1893. The author was Tandalam Sundara Rao, an Indian inspired by the use of origami in kindergarten. Using elementary techniques, he showed how to fold regular polygons. Even circles and ellipses can formed by folding the tangents to these curves. Basically he demonstrated paper folding can achieve classical geometric constructions -- those involve solving quadratic equations.

However, origamists knew long ago that, besides the basic folds, one can make a fold so that two points will meet two lines (see, for example, the third figure in angle trisection). In 1936, Margherita Piazzolla Beloc, an Italian female mathematician, showed that this fold can be used to solve cubic equations.

This prompted origamists to look for exact angle trisection by paper folding. The method used in our trisection animation is adapted from Hisashi Abe, a Japanese origamist and mathematician. He devised his method in 1970s, and reported around 1980【D2】.

Nowadays, there are books, lectures, journals and conferences on the math behind origami. Moreover, origami becomes modular: each piece of paper folds into a unit, to be part of a whole complex structrue. Such modular origami advances the art and science of paper folding.

Tom Hull is a math professor. Through his research, he uncovered the neglected math of paper folding. He wrote books and arranged courses on the underlying math of origami. His work in modular origami displays the union of math and art【D3】.

也許十分奇怪：2000多年來，為何沒有人想過，關於角三等分線的任何定理？更不可思議的，是摺紙與數學之間的千絲萬縷。

摺紙大師當然摺紙技術純熟，但他們對數學一無所知。數學家能夠跟隨步驟進行摺紙，但他們看不出數學。長久以來，這兩個群組互不問津。

數學家，尤其是幾何大師，努力嘗試使用經典工具（直尺和圓規）把角度三等分。他們屢試不爽，最終意識到這任務是不可能完成。但無法理解不可能的原因。

這個三等分角不可能性，最終在1837年由 Pierre Wantzel 給予證明，原因與以下事實有關：希臘傳統的幾何工具，不能用於求解三次方程。

Felix Klein 是一位著名德國數學家，對許多課題都感興趣。他在一次頗受歡迎的數學講座中，提及一本書《摺紙中的幾何練習》【D1】。此書1893年出版。作者是一名印度人 Tandalam Sundara Rao。他從幼兒園中摺紙學習得到啟發，使用基本摺紙技術，展示如何摺出正規多邊形。通過將切線摺痕形成曲線，甚至摺出圓形和橢圓形。基本上，他證明了摺紙可以實現經典的幾何操作 -- 涉及求解二次方程。

然而，摺紙大師早已知道，除基本的摺折外，還有這樣的摺法：使兩個點疊貼到兩條線（例如，參見三等分角中的圖三）。1936年，意大利女數學家 Margherita Piazzolla Beloch 證明了，這一摺法可用於求解三次方程。

這促使摺紙大師們，尋找把角度準確三等分的摺法。上述三等分角動畫中使用的方法，是改編自日本摺紙大師和數學家：阿部恒（Hisashi Abe）的創作。他在1970年代設計這摺法，並在1980年前後發表報告【D2】。

如今，關於摺紙蘊藏的數學，有許多書籍、講座、期刊和會議探討。此外，摺紙變得模塊化：每張紙摺成一個單元，組成一整個複雜的結構。這種模塊化摺紙（modular origami），推進了摺紙的藝術與科學。

數學教授 Tom Hull 通過研究，發掘被忽略的摺紙數學。他撰寫書本，說明摺紙的基礎數學，並設計課程授學。他的模塊化摺紙作品，展示數學與藝術的結合【D3】。

Origami Art 摺紙藝術

A colorful flock of origami cranes. A pair of foxes by wet folding, by Vietnamese origami artist Hoang Tien Quyet. Click to enlarge (click again to close)
一群五顏六色的摺紙鶴。 濕折完成的一對狐狸，越南摺紙藝術家阮泓強作品。 點擊放大(再擊關閉)
Origami is paper folding with an artistic touch, just like painting is drawing with artistic style. Most likely originated in China, then spread around the world, paper folding gradually became an art form in Japan. Regarded as one of national quintessence (國粹) , the oldest existing book on origami was published in 1798 in Japan.

Origami is a minimalist art form, using the least material to create the most complicated forms. It is also a metamorphic art, transforming a 3D shape from a single piece of paper, no cutting, no glue. It is amazing to see the transformation, from a plain sheet to fairly complex creatures, especially those with intricate details.

摺紙，是具有藝術美感的紙張摺折，正如繪畫，是具有藝術風格的紙上作圖。摺紙極有可能源自中國，然後傳遍世界各地。在日本，摺紙漸漸發展成一種藝術形式，奉為國粹。現存最古老的摺紙書籍，於1798年在日本出版。

摺紙藝術，是一種極簡主義的表現：使用最少的材料，創造最複雜的形式。摺紙也是一門變形藝術：從平面張紙變成立體形狀，沒有切割，沒有膠粘。看著一張普通的紙，演變成相當複雜的生物，尤其是那些精緻細節，真是太神奇了！

Traditional origami has straight creases, giving delicate and exquisite cranes, butterflies, beetles, etc. Modern origami takes a break from tradition. This owes much to Akira Yoshizawa（吉澤章）【E1】, who pioneered the technique of wet-folding, turning origami into sculptures with smooth lines. In 1954 he founded the International Origami Centre in Tokyo, and publicized origami art in worldwide exhibtions.

The work of Akira Yoshizawa inspired the next generation of origami artists, including:

Each artist has a distinctive style.

In the words of Akira Yoshizawa, the acknowledged grandmaster of modern creative origami:

You can fold a simple quadrilateral paper into any shape as you want.
I wished to fold the laws of nature, the dignity of life,
and the expression of affection into my work ...
Folding life is difficult, because life is a shape or an appearance caught in a moment,
and we need to feel the whole of natural life to fold one moment.
傳統摺紙藝術，摺痕筆直，摺成精緻細膩的仙鶴、蝴蝶、甲蟲等。現代摺紙藝術，突破傳統，很大程度上要歸功於吉澤章（Akira Yoshizawa）【E1】。他開創了濕摺技術，將摺紙改變成線條流暢的雕塑。1954年，他在東京創立了國際摺紙中心，並在世界各地的展覽中，宣傳摺紙藝術。

吉澤章的作品，啟發了下一代的摺紙藝術家，包括：

他們風格獨特，各有千秋。

吉澤章是公認的現代創意摺紙之父，他曾說：

你可以把一張簡單的四邊形紙，摺成任何你想要的形狀。
我希望在作品中，摺出自然的法則，生命的尊嚴，
以及自己投入感情的表達 ...
摺出生氣很難，因為生氣是一瞬間捕捉到的形態。
我們是需要感受整個自然界的生氣，來摺得片刻。
Many Japanese museums have origami exhibitions. Their webpages have a wonderful display of attractive origami forms by various artists, see【E5】and【E6】.

How to fold a paper dragon? There are two forms of dragon: western and oriental. Both can be achieved quite life-like. See【E7】and【E8】for instructions.

On a more practical side, you can fold a mobile stand for your smartphone【E9】.

許多日本博物館都有摺紙展覽。他們精美的網頁，展示了多種藝術家的摺紙作品，各有特式，請參見【E5】和【E6】。

如何摺成紙龍？有兩種：西方龍和東方龍，均可以摺得栩栩如生。有關步驟，請參見【E7】和【E8】。

更實用的是，可以摺出移動支架，放上你的智能手機【E9】。

Origami Science 摺紙科學

The Miura fold（ミウラ折）, proposed by the Japanese astrophysicist Koryo Miura in 1985. Comparison of the mirror size of Hubble Telescope and James Webb Telescope. Click to enlarge (click again to close)
三浦折（ミウラ折），1985年由日本天體物理學家三浦公亮提出。 Hubble（哈勃）望遠鏡和 James Webb（詹姆斯·韋伯）望遠鏡的鏡面尺寸比較。 點擊放大(再擊關閉)
Before the days of smartphones, to travel abroad you need a map. How to fit a large map into your pocket? There are various techniques, applying the principles of origami【F1】. In particular the Miura fold (shown on right), named after the Japanese astrophysicist Koryo Miura, can pull to expand, and push to contract, suitable for map folding【F2】. Miura invented his fold for the deployment of solar panels in a Japanese satellite.

The principles of origami are based on origami math. Whenever there is a need to pack something in a small space, this art of paper folding can be appled. These are some examples of everyday use of origami:

Robert Lang, a retired NASA engineer, was a physicist but now an origami artist. He explains what modern origami looks like【F6】. He was responsible for the design the gigantic mirrors for the successor to the Hubble Space Telescope.

The Hubble Space telescope, which was launched in 1990, after many spectacular pictures of the universe, is now eagerly waiting for its replacement. The power of a telescope is measured by the size of its mirror, thus the new one, called John Webb Space Telescope (JWST) will have a mirror 6 times larger. The problem is: how to pack such a gigantic mirror into the rocket capsule to be launched into space?

The solution is to design a flexible mirror: fold it up first, then unfold it in space. This requires some design using principles of origami【F7】.

This origami space telescope is scheduled to be launched in October 2021【F8】.

Update: The James Webb Space Telescope was finally launched on Christmas Day 2021. Over a year, it has been sending in many stunning images of the early universe【F9】.

在未有智能手機的年代，出外旅行要靠地圖。如何將整幅地圖放入口袋內？方法有很多，都是應用摺紙原理【F1】。特別是三浦折（右圖示），以日本天體物理學家三浦公亮命名，可以拉動展開、推動收縮，適合地圖折疊【F2】。三浦公亮發明他的三浦折，用作部署日本衛星的太陽能電池板。

摺紙原理，具有數學基礎。凡是需要在狹小空間裝嵌東西，這門藝術就派上用場。以下是摺紙在日常應用的一些示例：

美國宇航局退休工程師 Robert Lang，之前是一位物理學家，現在是摺紙藝術家。他解釋了現代摺紙的模樣【F6】，也負責設計巨型反射鏡，取代哈勃太空望遠鏡。

哈勃太空望遠鏡於1990年發射升空，此前它拍攝了許多壯觀的宇宙照片，現正急切等待替換。望遠鏡的功能，取決于鏡面的大小。因此，韋伯太空望遠鏡（JWST = James Webb Space Telescope）的新鏡面將會大6倍。問題是：如此巨大的鏡面，如何裝進火箭艙並發射到太空？

解決方案，是設計一個靈活的鏡面：先將其摺疊起來，然後在太空展開。這就需要應用摺紙的原理，進行一系列設計【F7】。

這款摺疊式太空望遠鏡 JWST，計劃將於2021年10月發射升空【F8】。

更新：JWST 太空望遠鏡終於在2021年聖誕節發射升空。一年多以來，它發送了多張早期宇宙令人驚嘆的圖像【F9】。

Origami Movie 摺紙電影

Kubo and The Two Strings (2016), directed by Travis Knight and produced by Laika. Click to enlarge (click again to close)
《酷寶：魔弦傳說，2016年》，由 Travis Knight 執導，Laika 動畫室製作。 點擊放大(再擊關閉)
Origami can produce very life-like models of paper creatures, a static model. If you shoot the static model frame by frame, each frame changing the model by a little bit, you can make the static model come to life. This is called stop-motion animation.

Of course, this is tedious. It seems much easier to get real actors and move real objects. But stop-motion animation can create fantastic worlds. Imagine a scene with a paper boat hanging a paper sail, sailing on paper lake --- this is much more interesting that the real world!

This stop-motion movie by Laika, Kubo and The Two Strings（酷寶：魔弦傳說） 2016, has a story based on a legend enacted by origami.

Origami is about creation under restriction. This is the same for stop-motion movies. Try this: how to make waves without wind? Of course, the water is fake and the waves just look real. The challenge is to have the effect as realistic as possible.

It is easy to film a doll with strands of hair blowing in the wind. It is hard to film a doll with strands of hair fixed by wax and moved by hand frame by frame to appear to be blowing in the wind.

If you have watched this movie, watch videos about how stop-motion makes this movie. Or watch these informative videos on origami【G1】【G2】【G3】.

摺紙製作的人物模型，栩栩如生，乃靜態模型。如果逐格逐格拍攝，而每一格稍作改變，可使靜態模型動若脫兔，稱為定格動畫。

這是相當費時的。聘用真正演員、移動實物，似乎容易得多。但是定格動畫，可以創造夢幻般的世界。想像一下場景：一艘紙船懸掛紙帆，在紙湖上航行，這比現實世界有趣得多！

《酷寶：魔弦傳說》是2016年由 Laika 製作出品的定格動畫電影，故事引伸自一則傳奇，由摺紙演譯。

摺紙，是在受限制下的創作。定格動畫，也是一樣。試想想：如何做到無風起浪？當然，浪是虛的，水是假的。要使效果盡可能呈現逼真，甚具挑戰性。

這很容易：在風中拍攝洋娃娃的頭髮。但這很難：用蠟固定髮絲，一格一格地稍移髮絲拍攝；結果看起來，就像隨風飄揚的頭髮。

如看過這齣電影，請觀看此動畫定格製作的視頻。或可看其他有關摺紙藝術的視頻【G1】【G2】【G3】，增廣見聞。

Epilog 後記

As discussed, origami and stop-motion animation are creations under restriction. So is geometry.

Practical geometry is the math of measurements, but Greek geometry is the math of ideas. Ideas are ideal: points without size, lines without width, planes without thickness. They are restricted to be simple, yet the whole geometry is erected from the properties of points, lines and planes.

When shooting a stop-motion animation, 'rain', wind' or 'water' are stopped to take shots. The result: virtual legends are brought to life, with special effects matching those of a live movie.

Very often, beauty blooms when something is created out of restricted conditions.

The world's turning and we're learning.
I♡PM (13 June, 2021)

如前所述，摺紙和定格動畫，都是受限制的創作。幾何也是如此。

實用的幾何，是測量的數學。但希臘的幾何，是理念的數學。理念有理想：點沒有大小，線沒有寬度，平面沒有厚度。它們僅限於簡單，但整個幾何學都是由點、線和平面的特性構成。

拍攝定格動畫時，要暫停「雨」、「風」或「水」來逐格拍。結果是：虛擬的傳奇世界變得逼真，直迫現場拍攝電影的特技。

很多時候，條件雖受限制，仍可有所發揮，綻放出美麗。

世界正在轉，我們在研鑽。
I♡PM (2021年6月13日)

Notes and Links 說明及連結