如何把球面内外翻转：斯梅尔悖论

hbghlyj

A peek inside the mind of a mathematician

The video is about Thurston’s technique for everting a sphere. But the real gems are the little visual techniques (complete with sound effects), so important for this kind of mathematics. Thurston’s video not only makes them accessible, but provides a rare peek into the mind of one of the greatest mathematicians of our time.

zh.wikipedia.org/wiki/斯梅爾悖論
差拓扑结构中，球面外翻（Sphere eversion）是指在三维空间中，將球面從內向外翻。值得注意的是，我們有辦法在不割開、撕裂或製造摺痕的前提下，連續且光滑地將球面由內向外翻（有可能產生自交）。 這對非数学家甚至是瞭解定期同伦的人來說都十分意外，并可以被视为一种真詭論：乍看下是假，實際上為真。

更準確地说，令
\[f\colon S^2\to \mathbb R^3\]
為标准嵌入，則有一个定期同伦的浸入
\[f_t\colon S^2\to \mathbb  R^3\]
使得 $f_0=f$ 且 $f_1=-f$。

無摺痕球面外翻的存在性證明是由史蒂芬·斯梅爾於1957年率先完成。雖然已經有一些電腦動畫幫助人們想像，但很難提供這種翻轉的動畫片。第一个展示性的例子經過數位数学家的努力才完成，包括弗拉基米爾·阿諾爾德和盲人數學家伯纳德·莫兰。另一方面，证明这样的「翻轉」存在容易多了，这就是斯梅尔證明的事。

hbghlyj

Sphere Eversion By: Ricky Reusser

A brief history of sphere eversions

Visualizing a sphere eversion

A History of Sphere Eversions

Sphere Eversion - an Original Visualization

Sphere Eversion, a topology puzzle explained with pictures, including a bibliography of popular science books

About the Outside In Sphere Eversion

Sphere eversion movie – Illustrating Mathematics

Outside In: A brief history of sphere eversions

Try this software from the Geometry Center, which provides an even more hands on experience of turning a sphere inside out.

hbghlyj

surfaces.gotfork.net/
The first parameterization of Boy's Surface was found by Bernard Morin in 1978 and was used as a half-way model for sphere eversion.