“Non-orientable surface” is the thing. The surface of that thing is not orientable. That means you can’t draw some normal vector to define the surface. When you traverse through the surface with a normal vector you can end up in the same point with that vector pointing the opposite direction in such a surface. That gives two directions for the same point or in other words you can’t define the surface like that with a normal vector. Its the surface and not the shape itself that is non-orientable

If you follow the surface with your finger, starting from the outside, you can end up on the inside, without traversing the surface. There is only one surface, no concept of outside or inside, contrary to a good old cube for instance.

Similar to a Moebius strip but with a higher dimension. A MS is easier to understand since you can easily make one to try to run your finger on the surface and end up anywhere.

But infinite recursion hurts my tummy

But I’m allergic to almonds and gluten intolerant!

Also obligatory link to the Numberphile video about Cliff’s Klein bottles and his crawl-space robot warehouse. https://www.youtube.com/watch?v=-k3mVnRlQLU

Also obligatory link to his really fascinating and engaging book about being one of the first people to catch a hacker: https://en.wikipedia.org/wiki/The_Cuckoo's_Egg_(book)

Yeah but it was written on a Möbius strip and we’re still not sure where the beginning and where the end was.

ETA: the paper https://people.eecs.berkeley.edu/~sequin/PAPERS/2013_JMA_Klein-bottles.pdf