Up: Thomas Banchoff's Project List

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.

Up: Thomas Banchoff's Project List