Math for Little People

I'm 22. There's a part of me that's asking: "Why haven't you proved any great theorems yet?" But I can't beat myself up too much over this--with Republicans in the White House, and with so much HBO magic happening lately, the odds are stacked against me.

But there's a different kind of math anxiety creeping up: one of these days, I might have kids. Those kids are going to be curious about the world around them, and they're going to look to me for answers--at least when their mother isn't around.

For the most part, I'll be happy to fend them off with a mix of idle speculation and propaganda. But... what if they ask why honeycombs have six sides?

What if they want to know why scooting towards the middle of the see-saw makes you go up? What if they ask why the rubber chain-links on the swing don't ever come apart, if they're not 'connected'? Could I in good conscience give sloppy answers to questions I know involve beautiful math?

I know the Feds wouldn't come and take them away; I know the kids won't be too disappointed; I know their appetite for rigorous proof will come slowly, if at all. But could I live with myself? No, I've got to get a better grasp of the 'basic facts' of life in our spatial world before I help bring someone new along, if only for my own sake.

But what are these? It's up for grabs, but here's the category I've had in mind mostly: qualitative features of structures, arrangements, and movement, observable by 'medium-sized' agents like children, and generated by/consistent with a naive 'block-world' understanding of matter and physical law (not referring to the AI environment of that name, but in a similar spirit). This is the conceptual space in which I've lived most of my life (and thought about discrete math and computer science), and if it's good enough for me it's good enough for my kids. Molecules and lightning bolts I won't sweat so much, but Block World I want to get right.

So then... how to pick out, think about, and explain its most important facts?

Some of them are, I think, topological, as I've alluded with the swing example. But kids won't swallow homotopy theory any more than they'll eat lima beans, even if I can still remember it by then. Anyways, there's a potential save here: Block World is basically discrete, so the space of 'topological' deformations of objects is much smaller and more well-behaved (I've already posted on discretization in topology).

A good place to start would be the Jordan Curve Theorem, famed for its difficulty in spite of its surface obviousness: what could be clearer than the fact that a non-self-intersecting loop in the plane divides the plane into two components, 'inside' and 'outside'?

As one discrete formulation, consider the loop as a path on the grid, where each step in any direction is of even length. Then it appears that the inside is path-connected, also by grid paths (without the even-length restriction), and separated from the outside. For example:



I think this problem is excellent fodder for thought, and I would encourage anyone who's been scared away by the Theorem in the past to work on it in this friendlier setting.

No hints, though--would your kids respect you?

(Clarification for concerned parties: there are no children on the horizon; I do believe in responsible parenting; and I don't get HBO, although I do swear by Curb Your Enthusiasm.)