Klein Bottle or Klein Jar? Either way, it's my grandson in Connecticut's 8th grade project to build one for me!

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.

In this sense, a Klein Bottle is a 2-dimensional manifold which can only exist in 4-dimensions!

But let's not mention that to my grandson in Connecticut. This should keep him off video games and mindless TV until he graduates from high school.

Hint to grandson: The trick will be in getting the nose of the bottle into the inside of the bottle from the side without poking a hole in the side of the bottle.

But I'm still left with the question: When did they change its name from "Klein Jar" to "Klein Bottle"? Look in any projective geometry book published before, say, 1960 and you will see the above "bottle" referred to as a "Klein Jar." Next thing you know, they will be talking about landing a human on a comet. Crazy world, this is!!