A friend of mine just showed me this video online of an eversion of the sphere; that is, a regular homotopy from the sphere sitting inside 3 space to itself which reverses orientation.
As it is on google video, its not clear whether it is actually a public domain video or just standard internet crime, so I apolgize if I am linking to an ill gotten work (especially given that at least one credited contributor to the video has been known to frequent this site).
Tags: math.AT
August 30, 2007 at 1:23 am |
Outside In was an old video produced by the Geometry Center at the University of Minnesota. The Geometry Center is no more (its old website is hosted at UIUC), but I don’t know what that means for its intellectual property, or if they ever released it to the public domain.
I remember going to the Center long, long ago and playing around with some of the clay models they’d made of some of the intermediate stages of the eversion. I’d read through Smale’s original eversion in one of my father’s old Scientific American issues, but this one was a lot easier to follow. Still, the Smale version does have a very nice structure to it.
August 30, 2007 at 1:37 am |
[...] At The Everything Seminar, Greg Muller points out that Outside In is available on Google Video. This was a little video [...]
August 30, 2007 at 11:45 am |
I have somehow never seen this video before — it clears up a lot of the confusion I had about Thurston’s corrugations.
There is also a very cool collection of “optimal” sphere eversions that Rob Kusner developed at UMass. These are eversions of the sphere which distort the sphere a minimal amount.
These optimal eversions are based around a theorem of Bryant’s that the only embedded spheres which minimize the elastic energy (aka Willmore energy)
are the round ones. In that last equation, 
is the mean curvature: the average of the directional curvatures of your surface. Kusner improved this idea enough to get a gradient flow from the sphere to a halfway inverted surface, then back down to the inside-out sphere. You can see the resulting minimal sphere eversion in the video Optiverse, and read more about Kusner’s process of developing this eversion from his homepage.
The Willmore energy is pretty cool. Despite being a function of Euclidean invariants, it turns out to be invariant under Mobius transformations as well. This lets you pull off some neat tricks when you are doing differential geometry on elastic surfaces.
There is a nice website on minimax and other sphere eversions over on John M. Sullivan’s homepage as well.
August 30, 2007 at 4:25 pm |
Thanks for the link..
I have no idea how this licensing stuff goes – the Geometry
Center is no more but the video was sold by A.K. Peters
publishing, at
http://www.akpeters.com/product.asp?ProdCode=0466
so I assume they still have the rights. It sure was a lot of fun
to work on in any case (I spent all my summers as an undergrad
at the Geometry Center, at least initially to work on Outside In.)
The Geometry Center itself was a really wonderful place,
and provided a great introduction to math to a lot
of people who participated in their REUs and other activities (like me).
I was very sad to see it go.
October 1, 2007 at 5:43 am |
[...] Sphere Eversion Video [...]
October 20, 2007 at 8:46 pm |
[...] Credit where credit is due: The Everything Seminar. [...]
May 24, 2009 at 7:41 pm |
..] Более подробно можно ознакомиться по этой ссылке ..]
здесь видел ет gamebulletin.ru