In the market for a Klein Bottle or just a wool hat shaped like one? Check out the selection at Acme Klein Bottle.

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.

My favorite is their $18,000 meter-tall Giant Klein Bottle… see the fascinating build photos on Acme’s site…