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:

Not bad at all, for someone who is dimensionally challenged.  Here’s a nice representation for comparison, although it’s still a cheat:  the Klein bottle really doesn’t intersect itself.

A nice example of student work!