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# The Klein Bottle in Four-Space

The Klein bottle is a non-orientable surface obtained by identifying
the ends of a cylinder with a twist. This representation is
constructed from two pieces, one a tube around a figure eight curve
and the other a surface of revolution of a piece of that curve. There
is a curve of self-intersection in three-dimensional space, but the
intersection points have different fourth coordinate so the surface is
actually embedded, without self-intersection, in four-space.
To view the surface rotating in three-space, select the picture.

To see how rotation in four-space can produce local
self-intersections, select the second picture.

The Klein bottle can be thought of as two Möbius bands
identified along their boundaries. To see the Klein bottle separate
into these pieces and recombine, select the third picture.

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