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Klein bottle mystery

In the book "Basic Topology" by M.A.Armstrong I found an explanation about how to construct a Klein bottle. I had to reread it 5 times and I was still not quite convinced. I am retelling it here.

Begin with a sphere, remove two discs from it, and add a Möbius strip in their places. 
A Möbius strip has after all a single circle as boundary, and all that we are asking 
is that the points of its boundary circle be identified with those of the boundary 
circle of the hole in the sphere. One must imagine this identification taking place 
in some space where there is plenty of room (euclidean four-dimensional space will do). 
This cannot be realized in three dimensions without having each Möbius strip 
intersect itself. The resulting closed surface is called the Klein bottle. 

I was scratching my head around how this procedure actually produces a Klein bottle until I found this question in MathSE.

Klein-bottle-as-two-Möbius-strips

This picture there in one of the answers is really really nice, it really shows what happens if we cut a Klein bottle in half - we really get two Möbius strips as a result. The cut is done by a plane "parallel to the handle" which cuts the bottle in two symmetric parts.





So... it's really for a reason they say "a picture is worth a thousand words".





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