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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 4I don’t tell them the problems are impossible, of course, as that would ruin the fun.  Usually it engages them, confuses them, and makes them suspicious of me.  That’s a win-win-win in my book.

So recently, 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 tube where one end meets the other to seal the object up, but somehow gets turned inside out in the process.  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 doesn’t really intersect itself.

A nice example of impossibly creative student work!