And now to start my second week of this summer mathematics A to Z challenge. This time I’ve got another word that just appears all over the mathematics world.
The word “dual” turns up in a lot of fields. The details of what the dual is depend on which field of mathematics we’re talking about. But the general idea is the same. Start with some mathematical construct. The dual is some new mathematical thing, which is based on the thing you started with.
For example, for the box (or die) you create the dual this way. At the center of each of the flat surfaces (the faces, in the lingo) put a dot. That’s a corner (a vertex) of a new shape. You should have six of them when you’re done. Now imagine drawing in new edges between the corners. The rule is that you put an edge in from one corner to another only if the surfaces those corners come from were adjacent. And on your new shape you put in a surface, a face, between the new edges if the old edges shared a corner. If you’ve done this right, you should get out of it an eight-sided shape, with triangular surfaces, and six corners. It’s known as an octahedron, although you might know it better as an eight-sided die.
And now a neat thing. You can take the dual again, building another mathematical construct based on this thing you created. It’ll be whatever thing you started with.
Take the eight-sided, six-cornered figure and follow the same old rules about making a dual. You’ll get back a box, just like you started with. Well, it might be smaller, but it’ll be the same shape. Just sit closer to it and you can’t tell it from the original.
Now you probably wonder, why do this? What is the point?
Here the example of boxes betrayed me. The dual of a solid shape like that is a solid shape and while some of these new shapes are interesting, there doesn’t seem to be much you can do with the dual that you can’t do with the original shape. But in general, the dual to a mathematical object will be a different kind of mathematical object. And there’s the point.
It can happen that something we want to do is awkward or clumsy or tedious if we try doing it to one kind of mathematical object, but is easy on some other kind of object. If the kinds of objects are duals, then, joy! Instead of the original problem that’s hard, we look at the dual, where the problem is easy. Then once we’ve found the solution, using the dual, we take the dual of that and we have the answer.
This sort of shifting between kinds of mathematical objects, looking for the ones that make a problem easy, is another of the stock tricks of the mathematician.
Meanwhile, drawing solid shapes and working out their duals is a fun pastime. If you need inspiration, try working from a set of tabletop gaming dice, or the Platonic solids. At least one of these shapes is its own dual.