Math and Art: An Impossible Construction
A favorite pastime of mine is offering impossible problems to students as extra credit, like asking them to find the smallest perfect square that has a remainder of 3 when divided by 4. I don’t tell them the problems are impossible, of course, as that would ruin the fun. Usually it engages and confuses them, and it makes them suspicious of me. That’s a win-win-win in my book.
So while discussing some three-dimensional geometry, I offered extra credit to anyone who could build a model of a Klein bottle. The Klein bottle is a hard-to-imagine surface that has neither an inside nor an outside; it’s like a bag that is sealed up, but somehow the bag is inverted in on itself. If you are familiar with the Mobius strip, the Klein bottle is basically a Mobius strip, one dimension up.
One reason that the Klein bottle is hard to visualize is that it can’t exist in three dimensions. It needs a fourth dimension in order to twist around on itself, kind of like the way the Mobius strip (which itself is two-dimensional) needs that third dimension to twist through before you tape it back together. So, I was pretty impressed with the student who made this:
A nice example of student work!