More Topology Explaining

Follow-up to yesterday's post on the standard crappy method of explaining topology:

I was at a presentation about a year ago, where someone tried to explain basic topology concepts to non-mathematicians. Here's they went about it: "Consider a cube of cheese and a donut," they said. "They are different shapes. If you draw a small circle on the surface of the cube of cheese, it can be shrunk down to a point. If you draw a circle on the surface of the donut the right way, it cannot be shrunk down to a point. Strange but true."

I almost fell out of my chair when I heard that explanation.

There's a whole slew of things wrong with explanation: (1) Why a "cube" of cheese? That's only going to serve to confuse people into thinking that the geometric "cube" shape is somehow important to the description, when it's not. Again, the only important thing is that one has a hole and the other doesn't. Use some kind of curved shape to avoid tricking people into thinking that the square-ness has anything to do with what you're explaining. (2) Why "drawing a circle"? Yes, as mathematicians we know that's one way of visualizing the important Poincaré conjecture, but here we have to look at it from the perspective of the non-expert listener. Drawings of things don't shrink and expand, so that only promotes further confusion. Use something from daily life that naturally expands and contracts for your analogy. (3) How the heck would anyone accomplish "drawing on a cube of cheese" in the first place?

Here's how I would explain this.

"Consider an orange and a donut. In topology, the only important difference in their shapes is that one has a hole and the other doesn't. Here's how a mathematician would demonstrate that: With the orange, if you wrap a rubber band around it, you can always flick the rubber band aside so it falls off. With the donut, there's a way to connect a rubber band through the hole-in-the-middle part so there's no way to just flick it off. (You'd have to cut & glue the rubber band back together, but then it would be always hang onto the donut.) Doing this mathematically is one way to detect exactly which shapes have holes in them."