The Klein Bottle. I remember first learning about it, and it was, well, hard to visualize. (“Self intersecting? The outside and inside all the same? Hanging out in four-dimensional space? Ummm, okay.”) Then I discovered ACME Klein bottles, and they made my life happy because I could sort of understand what a Klein bottle was, and because their web page is fun to read. Plus, they sell Klein Steins:
(Godzilla finished his beverage before I got a chance to photograph it.)
One feature of a Klein bottle is that it can be made from two Möbius strips. Another thing that was hard for me to picture (though again Clifford Stoll at ACME comes to the rescue), but I just ran across this video of a Klein bottle that had been made by zipping together two Möbius strips, and I thought it was so neat that I wanted to share it. I might just have to figure out how to work the sewing machine just so I can make one.
(And really, I’d just intended this post to end here, but then I found the following clip and couldn’t resist adding it!)
ACME Klein Bottle photo by Lethe, published under GNU Free Documentation License.