Re: KLEIN BOTTLE

|> As far as I can tell, you actually *can* store stuff in this
|> 3-dimensional version, but not very easily: you need to pour it in the
|> bottom, then tip it round slowly so it goes throught the spout and ends
|> up in the `inside' of the bottom.

Zackly so.  In fact, I have done it with the glass Klein bottle my colleague
has in his office.  It's fun getting the last few bubbles out!  Mind you,
it's a bummer if you want to get the "inside" dry again in a hurry!

Of course the 3-dimensional glass KB is only a physical model, not the
real thing, which cannot be embedded in E^3, though it *can* be immersed,
(allows self-intersection).  The glass version isn't even that - there is
a slab of glass missing: the little circle where the small part of the tube
passes through the larger part.   But it'll do.


|> Why you would want to do that, I have absolutely no idea.

For the same basic reason you'd want to do ANY math:-  IT'S FUuuunnnn...


|> If you feel like making a Klein bottle, you can do it by taping two
|> Moebius strips to each other, all the way along both sides.

If you casn mange this with paper ones, you're a better man than I am,
Gunga Din!  You might succeed with some cloth ones, by sewing them together,
if you've carefully arranged for the edges to be equal lengths.   But a cloth
one won't show the essential features.

It's got to be glass!

My colleague even has a glass model of a Boy's surface!  (Immersion of the
projective plane into E^3 without singular points.)  Now that is *seriously*
bizarre!  It was also the hardest job the University glassblower ever had
to do, he said.  My colleague also has a home-made wire-netting version.

Neato.   But I'll leave someone else to try the ascii version of that!

But an ascii sphere-with-crosscap shouldn't be *too* difficult.  
(Same surface but differently immersed - it has two "pinch points").


Anyone want to give that a try?

 _____   ____________________________________________________________   _____
|X| . |      Bill Taylor.         W.T...@math.canterbury.ac.nz       |X| . |
|~ . .|  -----------------------------------------------------------   |~ . .|
|___._|               Entropy - where energy goes to die.              |___._|
         ===========================================================

                         /|
                        / |
                 ______|  |_____
                |      | |      |
                |    __| |___   |
                |   /  |/   /   |
                |__/       /____|
                  /_______/
                       |/

      This is the central core of the Boy's surface...  the easy part!
=============================================================================

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