Make:

Very interesting. I'm curious to hear about a practical application of this theory. Anyone ?

Spoken like a true capitalist. Well, practical shmactical!

Of course you can turn a sphere inside out (eversioh). The interesting thing is that it can be done smoothly, without creating any singularity (creases, etc.).

So what does the math look like in doing something like this?

I had/have the same question as Shiva. Cool that it can be done, but why do it? Allowing self-intersection seems like it would limit practical application.

I watched this movie being created at the Geom Center when it was still at the U of Minnesota. ca. 1992 or so. My buddy Daeron Meyer was one of the techs who worked there for a while and helped with some of the rendering tasks.

It's a rather old math/topology problem. Some very smart people came up with a way to do it quite a while ago. Before fancy schmancy computers came along. :) You can search for "sphere eversion" and you'll see some of the drawings they came up with to explain how they did it. This one looks like about the simplest of any that I've looked at.

As to the question of practicality or why, I'm not sure either. I'm sure somebody much smarter than I am can see how this applies to other math problems or even real problems.

This may make me sound like a troll, but I too wonder about practical applications. Not that I don't find it interesting, but I'm having a hard time finding peace with myself knowing that what I'm doing isn't somehow helping the greater good. Aren't there hugely unjust wars going on? Isn't our pacification with intense mental challenges (aka games) like these somewhat responsible for their continuation? There's got to be a bigger picture I'm missing, but no one will explain and I'm left sitting here wondering what's wrong with myself that I just don't get it. Nevermind the rest of us who won't stop us...

This is cool, though. If I didn't worry about stuff like this and trying to pay bills I might be interested in exploring topology since I've seen this.

Why would you pick this of all things posted on Make to complain about furthering the greater good? Do you think that sticking LEDs in things will solve more conflicts than this?

At least math may have unforeseen uses down the line.

maybe that's why... I don't know. Maybe I figured people interested in a complicated topic like topology might have a more meaningful explanation than "gah shut up and live a little" like LED-sticker-inners might.

Either way, this has gone too far already. Apologies for ruining a potentially interesting thread with my own problems.

There're some good questions up there worth answering. I guess string theorists have learned a lot from topology... And DGM saw the video being produced!

Whoah, that was cool!

Is hte geometry Center still active, and do they produce videos like these anymore?

Very Interesting, even borders on friggin' NEAT!

This may make me sound like a troll, but I too wonder about practical applications. Not that I don't find it interesting, but I'm having a hard time finding peace with myself knowing that what I'm doing isn't somehow helping the greater good. Aren't there hugely unjust wars going on? Isn't our pacification with intense mental challenges (aka games) like these somewhat responsible for their continuation? There's got to be a bigger picture I'm missing, but no one will explain and I'm left sitting here wondering what's wrong with myself that I just don't get it. Nevermind the rest of us who won't stop us...
is.

This IS the greatest good of all! If there were no one theorizing about stuff like this, we wouldn't have all those cool nukes sitting in silos, subs, and on aircaft, ready to demonstrate to the world the superiority of Jesus over all other Gods.

TD

Off the top of my head, knowing how to manipulate surfaces without breaking smoothness at any point is useful for computer graphics, think about that the next time the Predator inverts an Alien's skull on some far-off planet ;-)

In the simple case of inverting a circle, the circle is manipulated in 2-space, thus when the "guy" tried to invert it by flipping it in 3-space is not allowed. However, the description of the material (being able to pass through each other) actually represents the 2-D circle placed in 3-space. The creases is as if the circle must be cut and reattached after it's flipped around. The inversion through flipping in 3-space actually represents the compounded math required to express the material as both ghostly and wave-y. If you allow a circle to exist in 3-space but not ghostly, you'd have to rotate the material even though you follow the wave method. Notice how the rotation of kinks are in pairs. This pairing represents the need to pair both a second of the curve to need both a wave-y aspect and the ghostly feature in order to successfully invert the curve.

I believe that the simple case represents the method in which a 2-dimension object can be manipulated in 3-space as a whole. And the sphere eversion represents the manipulation of a 3-dimensional object in 4-space. An application would be traversing through say parallel dimensions without creating creases in space (black holes, or pairs of black and white holes if the creasing does indeed happen in pairs).

Actually the sphere eversion occurs entirely within normal 3D Euclidean space; no fourth dimension is involved. Since it is an abstract surface, it can intersect itself. Again, the only tricky thing is that the transformation is done without ever creating any singularity (point at which curvature is infinite) in the surface. If you just try poking one side directly through the other, you soon see that you get a singular ring around the periphery; it's a non-trivial problem.

As to "applications", singularity-free transformations are quite important in many areas of mathematics, but if you have a concrete-bound mentality you won't appreciate the power of abstractions. We use conformal mapping a lot in solving certain kinds of electrical engineering problems, and it is a similar (but simpler) notion. History is replete with examples of abstract mathematical results eventually finding applications; the better we understand the nature and properties of patterns, the better we can identify opportunities to apply that knowledge. They ought to teach about this in schools, but I suspect they don't these days. The recent high-school math texts I've seen are so atrocious that nobody is likely to learn much from them, let alone be inspired by ideas.

Now how do you do it if the material cant pass though its self?

exactly. The material cannot pass through itself. That is why I introduce manipulation of a 2-Dimension object in a 2+1 Dimensional space. I believe that is how we can physically view eversions of curves.

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