What would walking the Klein Bottle look like to a 3D being?

[AtlasSniperma]

Hey folks,
I'm not certain if this is the right board to put this in(I couldn't figure where else itd go).

I've been thinking about the Klein bottle and had a curious thought. Since the 'bottle' is a 4d object with only one face, it can be navigated in its entirity via two dimensions. You can walk on the surface and visit every point(Yes I know this is generally understood), but what would that look like?

The way I see it is that it's basically a torus in 4d, so on the inside it's constantly the same, so at some points walking in a direction would create absolutely no change in perspective leading to the assumption that no movement has been made. While movement in the parallel direction would shift the entire perceived shape of the 'bottle', without changing the property that walking in the first direction causes no change.
If I'm explaining this well; does it make sense and do you agree with it?
If I'm not explaining this well; How would you imagine it would look to a 3d being, navigating a 4d Klein bottle by moving in 2 dimensions?

 

[andrewkirk]

As a 3D being you could not see all of the Klein bottle at once. At any time t, if you are at location x in the 4D space at that time, you could only see the parts of the Klein bottle that were part of a 3D hypersurface of the 4D space, which we would call S(t,x) because it may depend on both t and x.

Various different such functions S could be invented. The only obvious constraint is that S(t,x) must contain the spacetime location (in the 5D spacetime manifold) with coordinates (t,x), since you need to be in the same spacetime location as yourself.

You would see boundaries where the Klein bottle seems to just end. These would be the points where the Klein bottle wanders out of the hypersurface S(t,x). These boundaries would change as you walked around the Klein Bottle, in a way that might not be altogether dissimilar to how the horizon changes as you travel the Earth (that analogy is only very loose though, so don't try to do too much with it).

 

[Svein]

https://en.wikipedia.org/wiki/Klein_bottle

 

[sophiecentaur]

Just a thought, but has anyone ever actually constructed a Klein bottle, large enough to walk around inside/outside?? It would involve walkways and ladders, perhaps.
There are many examples of large scale optical illusions in Science parks, so could there be one somewhere?

Sign on the wall with one arrow, marked "Way out and way in".

 

[sophiecentaur]

I found this Klein Bottle House but I wonder how easy it would be to appreciate where you are in it and 'see it as a whole. I was thinking of something more curved, like the small ones you see. Top marks to the builder for trying, though.

 

[Vanadium 50]

I found this Klein Bottle House

The owner tried to paint "just the outisde". He's still painting.

 

[Svein]

Not exactly a Klein Bottle, but still interesting: Check out Theodore Sturgeons story: http://livskunskap.dyndns.org/veidos/english/literary/What dead men tell.pdf.

 

[AtlasSniperma]

Just a thought, but has anyone ever actually constructed a Klein bottle, large enough to walk around inside/outside?? It would involve walkways and ladders, perhaps.

This is actually similar to the reason I ask. I want to make a program that allows you to walk around a Klein bottle and want to understand how it would behave before I do it. By andrewkirk's rationale though it would just be like walking in any infinite plane, which makes me concerned that it isn't possible.

 

[andrewkirk]

Walking around a Klein bottle is very different between a 2D creature that lives in the surface and a 3D creature that walks on the surface. My comments above are for the latter. It is much simpler for a 2D creature. You can model it as a square where going out one edge makes you reappear from the opposite edge. The arcade game of Asteroids uses this approach with a torus mapping. A Klein Bottle mapping is the same except the direction of the top side is switched, so that exiting at the bottom one cm from the bottom-left corner causes a reappearance from the top one cm from the top-right corner.