Sphere Inside out Part - I

What people said

Strong rec Steven Lu · 2011-08 · Mathematics Stack Exchange · on “Other awesome topology related videos like this one?”

Turning a sphere inside-out: [this video] And part 2: [LINK] This is really, really cool. They describe things in really simple terms though (like referring to curvatures with smiles and frowns) and sometimes I wish I could connect it to the math that I know if relevant terms were brought up. The animations are simply amazing, and I can't believe this was made in 1994 ! I was wondering if anyone knows if these people made other videos like this using their techniques... So cool!

Strong rec Pietro Majer · 2011-09 · MathOverflow · on “An example of a beautiful proof that would be accessible at the high school level?”

This is maybe ambitious, for the details are obviously not completely accessible to the high school level; but the beauty of the ideas is, and this video is really a superb example of divulgation. Smale's theorem on the eversion of the 2-sphere and Thurston's construction [this video] .

Cites Randomblue · 2009-10 · MathOverflow · on “Best online mathematics videos?”

I know of two good mathematics videos available online, namely: Sphere inside out ( part I [this video] and part II [LINK] ) Moebius transformation revealed [LINK] Do you know of any other good math videos? Share.

Cites Alex R. · 2010-12 · MathOverflow · on “What would you want to see at the Museum of Mathematics?”

1) A high quality 3D movie (with glasses!) of sphere eversion, like this one [this video] 2) A Let's Make a Deal game show room, where people can play the game to death on a computer until they believe that they should switch doors. Offer candy prizes. 3) A scaled down Bridges of Koinsberg room, where you can try to walk across each bridge only once. 4) A large transparent (working!) replica of an Enigma machine. 5) A Velcro covered life-size Mobius strip which you can walk on with Velcro shoes (I hope you have good insurance)

Cites Alon Amit · 2011-01 · Mathematics Stack Exchange · on “what tools are good at making mathematic movies?”

There is no "perfect" software for making math visualization videos. The better movies of this kind were likely made using standard 3D animation software such as Cinema4D, Maya, 3d studio, XSI and such. You'll probably have to export data from a math application such as Mathematica into these suites and do the renderings in there. It's possible to render with just POV-Ray but working with a 3D suite can get you much better results. Some examples: Not Knot, part 1 [LINK] and part 2 [LINK] Moebius Transformations [LINK] Sphere Eversion Part 1 [this video] and Part 2 [LINK]

Cites Martin Worsek · 2011-07 · Mathematics Stack Exchange · on “Sphere eversion and Smale-Hirsch theorem”

For two manifolds $M^m$ and $N^n$ with $m<n$ the Smale-Hirsch theorem says that the differential map $d:\operatorname{Imm}(M,N)\to\operatorname{Mon}(TM,TN)$ is a weak homotopy equivalence, where $\operatorname{Imm}(M,N)$ is the space of immersion from $M$ to $N$ with the compact-open $C^{\infty}$ topology and $\operatorname{Mon}(TM,TN)$ is the space of vector bundle maps which are monomorphisms on their fibers from the tangent bundle $TM$ of $M$ to $TN$. How exactly does this now imply that there is a regular homotopy from $\operatorname{id}:S^2\to\mathbb{R}^3$ to $-\operatorname{id}:S^2\to\mathbb{R}^3$? The original paper by Smale mentions the use of the Stiefel manifold $V_{3,2}$ through $

Cites Patrick Da Silva · 2014-02 · Mathematics Stack Exchange · on “Immersions of the Klein Bottle”

If this is the case, you're essentially trying to reduce this to a two-dimensional problem. But the circle cannot flip itself inside out without "non-smooth points". A diffeomorphism of a smooth closed curve to another in the plane preserves its turning number. [this video]