I sometimes get science educator envy. I watch them convey ideas with hatching chickadees, snapping plants, fiery explosions, counter-intuitive gyroscopic dances, color-changing liquids, erupting volcanoes, whirling water spouts, sizzling arcs, glistening gems, and gravity-defying stunts like this recent demonstration of quantum levitation that’s making the rounds:

How cool is that?

Yes, math educators do have their attention-getting demos like pretty soap films, topological magic tricks, perfect shuffles, hypnotic juggling patterns, tilings and tessellations, etc. But so much mathematics lives as a pure abstraction. The Klein bottle is a good example. Sure, you can make very suggestive models of the Klein bottle, like this nifty Acme Klein Bottle, but the model is an imperfect representation of the actual concept (because the model self-intersects whereas the true Klein bottle does not).

Instead of handing over magnets and Buckeye Balls, mathematicians must rely on language (and fascinating gestures that would impress even the Italians) to hand over their objects. No wonder mathematical language has developed into such a precise and rich one, full of adjectives and nouns that exist no place else, like and .

And no wonder mathematicians are often seen sitting completely still, head cocked to the side, and staring intently, but at nothing in particular. What on earth are they staring at? Well, in fact, nothing…on earth.