Imagine a long bottle--or cylinder--with an opening at both the top and bottom. Curve the bottom opening back on itself and insert it through the side of the bottle, and extend it up inside the bottle until it connects with the mouth. The result will be a fair approximation of a Klein bottle, an object with no distinction between the "inside" and "outside" surfaces. Unlike a beer bottle, a Klein bottle has no "rim" where the surface stops abruptly; a fly can go from the outside to the inside without passing through the surface.
First described in the late 19th century by the German mathematician Felix Klein, the Klein bottle is closely related to the Möbius strip. In fact, a Klein bottle can be created--mathematically--by attaching two Möbius strips along their "boundary" circles.
In reality, a requirement of a true Klein bottle is that it must not intersect itself at any point, and this is impossible to achieve within the limitations of ordinary three-dimensional space. However, a Klein bottle can be realized in 4-dimensional space by "lifting" the part of the bottle that is about to intersect itself into the fourth dimensional axis. How this can be illustrated visually, I'm not sure; it's hard enough to create a 2D representation of the 3D approximation above.
Copyright © 2018 Walter B. Myers. All rights reserved.