The Mudcat Café TM
Thread #59418 Message #1524449
Posted By: GUEST,Bee-dubya-ell
20-Jul-05 - 01:45 PM
Thread Name: BS: The Mother of all BS threads
Subject: RE: BS: The Mother of all BS threads
Au contraire, mon Rapaire! The Acme Klein bottle website to which you yourself provided a link says:
A Klein Bottle is locally 2-dimensional ... every small patch follows the laws of 2-dimensional Euclidean geometry. In this sense, a Klein Bottle is a 2-dimensional manifold, and its inside is the same as its outside. But although it's a 2-D manifold, it can only exist in 4-dimensions!
Alas, our universe has only 3 spatial dimensions, so even Acme's dedicated engineers can't make a true Klein Bottle.
A photograph of a stapler is a 2-dimensional immersion of a 3-dimensional stapler. In the same way, our glass Klein Bottles are 3-D immersions of the 4-D Klein Bottle. Our Klein Bottle is a 3-dimensional photograph of a "true" Klein Bottle.
A Klein Bottle cannot be embedded in 3 dimensions, but you can immerse it in 3-D. (An immersion may have self-intersections; Embeddings have no self-intersections. Neither an embedding nor an immersion has folds or cusps.)
We represent a Klein Bottle in glass by stretching the neck of a bottle through its side and joining its end to a hole in the base. Except at the side-connection (the nexus), this properly shows the shape of a 4-D Klein Bottle. And except at the nexus, any small patch follows the laws of 2-dimensional Euclidean geometry.