Simplicity demands underlying complexity


I’ll be writing about some of the things I saw in DC at some point soon. But I wanted to touch on one of the things that I did on the trip down and the trip back. Specifically, I was playing with an iPad app called Geo Designer. Basically, this is a little geometry program that allows one to build computer graphics like this one of typical ruler-and-compass constructions, such as these circles nested in an angle, tangent to the angle’s sides. Pretty cool.

But the thing that gets to me is the underlying complexity. I mean, sure. Go ahead. Draw an angle and then nest some circles inside it. That’s pretty cool. But to do it “right”, mathematically and geometrically, requires a whole lot of attention to detail, and precise geometry. And there’s no royal road to any of this, to borrow a phrasing from Euclid. There’s no way to fake this with a few rules of thumb: want a few circles nested along the bisector of an angle? Sure: learn some geometry.

Now here I am, twenty years after I loved basic geometry and wanted more, and then it devolved into algebra because, really, aren’t ruler and compass just so old-fashioned and out of touch? but it turns out that they’re not irrelevant. All this stuff is relevant to the world that kids live in today. That we all live in, today. Because simplicity is a byproduct of complexity, not the other way around. And yet we’ve done a great disservice to children by simplifying things for them, rather than complexifying them.

Maybe nobody uses these kinds of circles any more. It’s possible. But I know that my own intellectual powers were grown and challenged by the creation of this image, and I wonder how this ancient lesson can be applied again today.

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