Klein Bottles

The Swedish paper Ny Teknik, published by the Engineer's Organization, recommended these Klein Bottles as christmas gifts for those who "already had everything" and an interest in mathematics too. I didn't notice it in time for christmas, maybe next year?



Here is a short background from the vendor's site:

In 1882, Felix Klein imagined sewing two Möbius Loops together to create a single sided bottle with no boundary. Its inside is its outside. It contains itself.

Take a rectangle and join one pair of opposite sides -- you'll now have a cylinder. Now join the other pair of sides with a half-twist. That last step isn't possible in our universe, sad to say. A true Klein Bottle requires 4-dimensions because the surface has to pass through itself without a hole.

It's closed and non-orientable, so a symbol on its surface can be slid around on it and reappear backwards at the same place.You can't do this trick on a sphere, doughnut, or pet ferret -- they're orientable.