# Sergei Yakovenko's blog: on Math and Teaching

## Sunday, March 23, 2008

### 2-Sphere eversion in 3D-space

Filed under: links — Sergei Yakovenko @ 12:23
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If a smooth curve  embedded in the plane $\mathbb R^2$  is deformed allowing self-intersections but remaining smooth, then there is a natural integral invariant, the rotation number, which prevents eversion of a circle (deformation of the oriented circle into another circle with an opposite orientation). For two-dimensional surfaces smoothly embedded in $\mathbb R^3$ a similar invariant of deformations exists, yet this invariant does not preclude eversion of the sphere inside out.

The possibility of such deformation was discovered bt S. Smale in 1958. Relatively recently W. Thurston invented a general algorithm of smoothening, which yields an explicit sphere eversion. All these spectacular things are discussed on the level accessible to high school students in the most fascinating animation (21 min.) discovered on the web by Dmitry Novikov (thanks!). A much shorter animation (mere 22 sec.) does not easily reveal the mistery, so the longer one is really worth its time!

## Saturday, November 10, 2007

### Visualization of holomoprhic maps

Filed under: links — Sergei Yakovenko @ 11:05
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Though it is difficult to visualize holomorphic maps and objects, some enthusiasts succeeded in this very impressively.

1. Conformal maps  (interactive Java applet) by Jonathan Foote.
2. Many more Java applets written by Terrence Tao, Fields prize winner, great mathematician and terrific math blogger.
3. Animated gifs of conformal maps by Douglas Arnold.

Take a tour: it is funny and very instructive.

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