Klein bottle

A Klein bottle is a three-dimensional topological entity in which there can be no distinction between the "inside" and the "outside" of the object. The Klein bottle was first described in 1882 by Felix Klein. It is closely related to the Moebius Strip.

Picture a bottle with a hole in the bottom. Now extend the neck. Curve the neck back on itself, insert it through the side of the bottle (A true 4D Klein bottle would not require this step, but it's necessary to represent it in three dimensions), and connect it to the hole in the bottom.

Unlike a drinking glass, this object has no "rim" where the surface stops abruptly. Unlike a balloon, a fly can go from the outside to the inside without passing through the surface (so there isn't really an "outside" and "inside").

Why are these important? I'm not sure; will work on it or maybe someone will add to this.

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Links

Acme Klein Bottles Has actual Klein bottles! (Or at least 3D projections of them).

http://www.geom.umn.edu/~banchoff/Klein4D/Klein4D.html To view an animated 3D-picture of a Klein bottle

http://www.lipsons.pwp.blueyonder.co.uk/mathlego.htm has some Lego constructions of Moebius Strip and Klein_bottle structures.