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A Klein Bottle is a 4-dimensional shape that has neither an inside nor outside.
Like a Möbius strip, if you trace a path along its surface, you will travel on both sides of the surface before returning to the starting point.
In the mathematical field of topology, it is a 3D immersion of a closed, one-sided, non-orientable, boundary-free manifold.
A true Klein Bottle does not intersect itself, so it can only fully exist in 4 dimensions. But in the same way a 2D shadow is cast by a 3D object, this can be considered the 3D “shadow” of a 4D Klein Bottle—the closest we can get in our universe.
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Adam Savage Explains Möbius Strips and Klein Bottles!
The following square is a fundamental polygon of the Klein bottle. The idea is to 'glue' together the corresponding red and blue edges with the arrows matching, as in the diagrams below. Note that this is an "abstract" gluing in the sense that trying to realize this in three dimensions results in a self-intersecting Klein bottle.
To construct the Klein bottle, glue the red arrows of the square together (left and right sides), resulting in a cylinder. To glue the ends of the cylinder together so that the arrows on the circles match, one would pass one end through the side of the cylinder. This creates a curve of self-intersection; this is thus an immersion of the Klein bottle in the three-dimensional space.