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.

## Dual.

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.