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If you were on a ship at sea, and were a topologist of sufficient strength to be able to conjure up a Klein bottle on the spot and then toss it into the sea, what would be the water-line of the floating Klein bottle, after all disturbing water waves had died out?

  • If it is made of lead, then it is the empty set. – Servaes Nov 04 '18 at 16:58
  • But interesting question. What are your thoughts on the problem? What progress have you made and where did you get stuck? – Servaes Nov 04 '18 at 17:00
  • Wouldn't that depend on the material? How would the answer differ from any other shape that doesn't keep water out? Why would a klein bottle float any differently than a sieve. – fleablood Nov 04 '18 at 17:07
  • @flea, there is an error in most glass models of the Klein bottle that one sees. It is immersed; most sculptors leave out the disk where the shape passes through itself. As an actual physical model, there is an interior filled with, say, air, where that disk is sort of a bottletop. – Will Jagy Nov 04 '18 at 17:15
  • @WillJagy If I understand your point (which I might not) those are not true Klein Bottles (which can't be made) and if we floated a false klein bottle there would be an air filled interior. In a true Klein bottle there'd be no such thing and the water would complete fill in (if the material is heavy enough). – fleablood Nov 04 '18 at 17:38
  • Would whoever posted that cute poem and then deleted it please re-post it? It gives an upper bound for the solution, since the three sailors would be weighing down the bottle, but could see the resulting water-line. This upper bound was not given explicitly, but might be able to be captured using the Implicit Function Theorem, and so I was going to up-vote the answer (but not accept it) for materially advancing the theory of Klein bottle immersions, when I found that it was deleted:( –  Nov 04 '18 at 17:58
  • @Servaes: The amount of thought I have given to this, beyond just asking the question, would fit into the navel of a flea, with room left over for a caraway seed. –  Nov 04 '18 at 18:02
  • @fleaflood Will Jagy's comment is relevant; since the actual Klein bottle may not be embedded in $\Bbb R^3$ it becomes nonsense to ask for where it would ultimately sit I'm the water. One can only talk about the image of the immersion. –  Nov 04 '18 at 20:08

1 Answers1

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$$\begin{array}{l}\text{Three jolly sailors from Blaydon-on-Tyne}\cr \text{They went to sea in a bottle by Klein.}\cr \text{Since the sea was entirely inside the hull}\cr \text{The scenery seen was exceedingly dull.}\end{array}$$

http://www.sf-encyclopedia.com/entry/winsor_frederick

https://www.amazon.com/Space-Childs-Mother-Goose/dp/1930900465

Will Jagy
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