I have seen renderings of such a thing (Klein Bottle). Can these things exist?

Waht is the role of a 4th dimension in a Klein bottle?

A globe is a three-dimensional object with one surface. So is a Möbius Strip.

A Klein bottle is not three-dimensional. The handle goes through the side to connect with the top without making any hole, something that is possible only with four spatial dimensions.

Let us note that the picture in Exapno Mapcase’s link is not a picture (in two dimensions) of a four-dimensional Klein bottle. It’s a picture (in two dimensions) of a four-dimensional Klein bottle that has already been collapsed down into three dimensions. By collapsing it down into three dimensions, it was necessary to have the “handle” portion go through the side of the bottle. In an actual four-dimensional Klein bottle, the handle would not go through the side. It would reach out into the fourth dimension and thus enter the interior of the bottle without going through the side.

In the usual definition for these purposes, a globe does have two surfaces, the interior and the exterior of the globe. In a four-dimensional Klein bottle, there is only one surface. As the handle passes into four dimensions and then connects with the top of the bottle, the exterior and interior surfaces join.

I’m guessing the OP is after a non-orientable surface, in which, roughly, its ‘upper’ and ‘under’ sides are the same – i.e. it has only one side; in which case the Möbius strip is an example (it’s, however, 2D, as is the Klein bottle – the difference being that the Klein bottle can’t be embedded in 3D space (and also that the Klein bottle has no boundary, i.e. it’s a ‘closed surface’; if that’s necessary to fulfil the OP’s criteria, I don’t think it’s possible)).

Perhaps I should have said that a globe does have two sides, rather than two surfaces, and thus the Klein bottle has one side.

If the question is “does there exist a closed, non-orientable surface that can be embedded in three-dimensional space (i.e., represented in three dimensions without intersecting itself)?”, then the answer is no. All non-orientable surfaces are connected sums of some number of projective planes, the projective plane can’t be embedded in three dimensions, and (hmm, here’s a loophole) I’m pretty sure that any surface that is the connected sum of unembeddable surfaces cannot itself be embedded in three dimensions. This last bit seems reasonable to me, though I don’t know for certain that it’s correct — anybody know for certain?

Depends on what you mean by “three-dimensional”: Do you mean the object itself has a three-dimensional surface, or that it is embedded in the third dimension, i.e., bends (or is curved) in it:

The Möbius strip has one two-dimensional surface which is bent in the third dimension. The Klein bottle is exactly analogous to that, one dimension upscale, by having a two-dimensional surface that is bent in the fourth dimension.

A simple loop drawn with a pen on a piece of paper would be the analogue to the Möbius strip one dimension downward, incidentally: A one-dimensional surface bent in the second dimension.

Another analogy is to a set of linked rings. In two-dimensions - a drawing on a piece of paper, say - you can’t intersect two rings without one of them cutting through the other in two places. But in three dimensions, the rings can link with no cuts or interruptions by passing though one another. No two-dimensional being could understand how that is possible, even though the math would show it. The Klein bottle intersects itself without a cut or interruption in a similar manner, but we in the third dimension can’t see how this is possible.

Does this hold if the three-dimensional space itself is not infinite unbounded Euclidean? In a 3D space analogous to a torus, I’d think you’d be able to embed a Klein bottle without it self-intersecting.

OK, that’s probably not what the OP had in mind…

I can’t think of any way to do this, if the embedding space is itself orientable. But you might be right.

Yep, I mean a Klein bottle can be embedded in a 2 dimensional manifold (e.g. itself).

I think the idea is that it can’t be embedded in a 3 dimensional Euclidean space. Any Euclidean space is non-compact.

Brainwave today during class. You can embed the Klein bottle in **RP**[sup]3[/sup] pretty easily. Just think of it as a ball with antipodes on the boundary identified and toss a cylinder in there and stretch it; cylinder goes out and out until it hits the boundary and comes back around the other side, and then you can glue together the two edges when they meet up. But the cylinder was turned inside out when you stretched it through the boundary of the ball, so you make a Klein bottle when you glue together the two ends.

I agree with **Chronos** that the embedding space probably has to be orientable, but I don’t have the machinery to guess why. I’ll ask my roommate later today.