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138 surfing through hyperspace
Figure 5.10
Bennett.
Glass Klein bottle designed and manufactured by glassblower Alan
hand, a Klein bottle is a one-sided surface without edges. Unlike an ordinary bottle,
the "neck" is bent around, passing through the bottle's surface and joining the
main bottle from the inside. One way to build a physical model of a Klein bottle
in our 3-D universe is to have it meet itself in a small, circular curve.
Imagine your frustration (or perhaps delight) if you tried to paint just the
outside of a Klein bottle. You start on the bulbous "outside" and work your way
down the slim neck. The real 4-D object does not self-intersect, allowing you to
continue to follow the neck that is now "inside" the bottle. As the neck opens
up to rejoin the bulbous surface, you find you are now painting inside the bulb.
If an asymmetric Flatlander lived in a Klein bottle's surface, he could make a
trip around his universe and return in a form reversed from his surroundings.
Note that all one-sided surfaces are nonorientable. Figure 5.10 is a glass Klein
bottle created by glassblower Alan Bennett (see note 2 for more information).
Figure 5.11 is a more intricate Klein-bottle-like object.