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The Klein bottle was first described in 1882 by the German mathematician Felix Klein.
Mathematically, the Klein bottle is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined.
Felix Klein
Informally, it is a one-sided surface which, if travelled upon, could be followed back to the point of origin while flipping the traveller upside down.
It's a one-sided closed surface which cannot be embedded in three-dimensional Euclidean space, but it may be immersed as a cylinder looped back through itself to join with its other end from the "inside". It may be embedded in Euclidean space of dimensions 4 and higher.
The above text is from Wikipedia; similarly I can't better the following excellent videos, from Hannover Uni and the folk at www.klein-bottle-film.com, in conveying the essentials of The Klein Bottle. First though, here's an illustration of an unusual hex klein bottle that is 3D printable
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A STEAMPUNK NOVEL, FULL OF
ANARCHIC EXPERIMENTAL SCIENCE
"Hodges emitted a scream the like of which
I hadn't heard since his scrotum was burned off
Unrelated to this post, below is an example of
eclectic science esoterica
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