[pbmserv] concerning klein bottles

[Bill Taylor <W.Taylor@math.canterbury.ac.nz>]

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->OK, is anyone here NOT a mathematician?

HAHAH!  Choice.  For us mathies, you might like to ponder this surface - 
(it may be what the original paper was about?) - which could be
called nested Klein bottles.

Take a (slightly thick-surfaced) Klein bottle, and slice it into
two locally parallel sheets, all the way round its surface.

If you did this to a sphere, (or any orientable surface) you'd just
finish up with two nested spheres (or whatever).  But if you do it
to a Klein bottle, you get what looks like two nested surfaces,
but is in fact just a *single* surface.  Kool! 
 
Exercise for the reader: what IS this surface?

BTW if one does this to a Boyes' surface, (smooth projective plane,
or Mobius strip with the single edge contracted down to a point
and smudged over), one gets....

....a Klein bottle!

-------------------------

A mathematician named Klein
Thought the Mobius strip was divine.
        So he said: "If you glue
        Up the edges of two,
Then you'll get a weird bottle like mine!"

----- kiwibill