## Duality, fundamental and profound, but here’s a starter for you.

This week in the mathematics A-To-Z I’ve been writing I mentioned duals. I asserted there were duals all over mathematics, but gave only one example, that of turning solid shapes into other solid shapes. It happens that two weeks ago HowardAt58’s “Saving School Math” blog ran a post, with pictures, about creating duals to lines, and to points on the plane. If you’re careful to set out the rules you start with, you can match any straight line to a point in the plane. And you can match any point in the plane to a straight line. And so … well, read, and see some ways to look at lines and points and other shapes which may be mind-expanding.

Duality, how things are connected in unexpected ways. The simplest case is that of the five regular Platonic solids, the tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron. They all look rather different, BUT…..

take any one of them and find the mid point of each of the faces, join these points up, and you get one of the five regular Platonic solids. Do it to this new one and you get back to the original one. Calling the operation “Doit” we get

tetrahedron –Doit–> tetrahedron –Doit–> tetrahedron
cube –Doit–> octahedron –Doit–> cube
dodecahedron –Doit–> icosahedron –Doit–> dodecahedron

The sizes may change, but we are only interested in the shapes.

This is called a Duality relationship, in which the tetrahedron is the dual of itself, the cube and octahedron are duals of each other, and the dodecahedron and icosahedron are also duals of each other.

Now we will look at lines…

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## A Summer 2015 Mathematics A To Z: dual

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.