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The topology of turning a Sphere inside out without puncturing it

sine

04:03 / 21.07.04

an open call for anyone who knows or can reference the math

Jack Vincennes

11:56 / 21.07.04

Is this anything like what you're looking for? No mention of extra dimensions, but it's about turning spheres inside out.

DecayingInsect

14:00 / 22.07.04

pix here

Note that the intermediate stages are only required to be smooth immersions of the 2-sphere in 3-space --self-intersections are allowed

Does this makes it less counter-intuitive?

sine

22:41 / 24.07.04

Yeah, that looks like it must be it; thanks, both of you.

Not sure where I got the higher dimensions idea from...

Perfect Tommy

07:46 / 27.07.04

I'm talking out my ass here, but I think this might be somewhere in the neighborhood of Poincare's Conjecture, which turned out to be much easier to prove in 4+ dimensions than 3. (Like, it may have just been proven for 3, but the jury's still out.) Maybe you were thinking of that at the time. I think there was also something interesting proven about 20+ dimensional sphere packing kind of recently.

Atyeo

09:06 / 27.07.04

I think you are right there Tommy.

There was a feature in a recent New Scientist about the possible solving of the 3-sphere.

According to Poincare, the easiest way to describe the topolgy of the universe is that we are all living on a four dimensional sphere.

I've never been able to fully understand that one. Anyone?

DecayingInsect

11:56 / 10.08.04

Linux/IRIX users can obtain a sphere eversion animation program here

to get it to compile under linux I needed vroot.h

Good luck: it's very pretty if you can get it to work!

sine

19:34 / 10.08.04

Beautiful! Thanks, Decaying Insect.

Lionheart

14:26 / 16.09.04

Maybe I'm understanding it wrong but doesn't the video cheat by showing one side of the sphere literally going through another? Thus you can invert anything if you can make one physical layer go through another physical layer.

DecayingInsect

15:01 / 16.09.04

As you say anthing could be turned inside-out in this way.

I think that the point that is supposed to be surprising is that there are no kinks or tears at any intermediate stage of the eversion.

If the sphere was not allowed to self-intersect there is no way it could be everted: the interior of the sphere would be well-defined at each stage and would have to remain bounded as the sphere deformed.

Lionheart

14:07 / 18.09.04

DecayingInsect: You're right. It wouldn't be possible in 3 spatial dimensions. But it would be possible with 4 spatial dimensions. The thing is that this thread states that it is possible to turn a sphere inside out in only 3 spatial dimensions. Which, from watching the video, I can see isn't true cause they damn cheat by having layers going through other layers.

DecayingInsect

16:59 / 21.09.04

I agree that it's a bit misleading to say that the sphere is being 'turned inside out' when it is allowed to pass through itself, sort of like a soap bubble, but even with this 'cheating' you can't just do this any old way because you have to keep the surface smooth at each step.

For example if you were to reach through the sphere, grab the south pole and pull it through the north pole you would end up with a kink at some point, which is not allowed.

Lionheart

16:04 / 23.09.04

Technically, if you could put it through a 4th spatial dimensions (little side note: you can apply the pythagorean theorem to any amount of spatial dimensions. 2 dimensional theorem is: a^2 + b^2 = c^2. 3 dimensions: a^2 + b^2 + c^2 = d^2. 4 spatial dimensions: a^2 + b^2 + c^2 + d^2 = e^2) you could invert it without changing its topography. How? Basically in 3 spatial dimensions you can flip objhects by physically turning them around. In the 4th spatial dimension turning 3d objects inside out will be like flipping them.

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