In 1882, Felix Klein imagined sewing two MÃ¶bius 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…Related