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The Klein Bottle is a mathematical joke: a surface with only one side.  It's just about to enclose a space like a regular bottle, then it curves back with a twist so the inner side of the surface merges with the outer, and suddenly it doesn't enclose any volume at all.  The inside and outside spaces become one, like the sides of a Möbius strip.

This neat trick of space can't quite exist in three dimensions, since the surface has to cross through itself, but we can still make models to see how it works.  This one is 3D printed in stainless steel with some bronze for color.  It's durable, washable, and ergonomic, sized to fit the hand sweetly. 

The perfect finishing touch for any math fan's kitchen, and you don't have to study geometry to enjoy it. 
Yes, it really opens bottles!
 

Klein Bottle Opener  3" tall - $72

Add a message or monogram.  Up to 20 characters (assuming they're not all M and W), production time is 4 weeks, let me know your message in the message box at checkout. Klein Bottle Inscription - $25 If you'd like a quantity with the same message, please write, there may be price breaks.

Want one that isn't a bottle opener? Right here, in steel or a variety of fine plastics.

Perhaps of historical interest, the first version of this page.  I might have opened a bottle too many before I wrote it.