There's a thing in mathematics called ``duality''. You can
probably get a feel for what it is by just the name. One of the
best examples of it in mathematics is that in Euclidean (normal,
run-of-the-mill, you-learned-it-in-high-school) geometry the
concept of *point* is dual to the concept of *line*.
Every theorem in geometry has a dual theorem which you can
obtain by swapping the notions *point* and *line*.

Two points determine a line. Two lines determine a point.

Another place duality crops up readily is in category theory where a theorem has its dual by simply reversing all of the arrows (inverting all of the morphisms).

Duality appears all over the place in mathematics. Another
ubiquitous concept in mathematics is *Symmetry*. A symmetry
of a system is a transformation which takes the system back to itself.
For example, rotate a square by ninety-degrees or flip it over one
of its diagonals. Or, rotate a circle by any amount.

It occurs to me that *Duality* is just *Symmetry*
in the semantic space/ conceptual space/ what-have-you. In the case
of the *point*/*line* duality of Euclidean geometry,
this is something like flipping the square over one of its diagonals.
Technically, it's more like the the symmetries of a line segment.
There's a complete change of 180-degrees, but everything still works
(including another complete change of 180-degrees).

This change of 180-degrees is a discrete symmetry. You either execute the whole thing or not. You can't stop part way. This is different from the continuous symmetry in rotating the circle. You can rotate it as far as you want to around its center.

The *Duality* that most people think of in mathematics
if very much like the symmetry of the line segment. You can
swap the whole thing end to end. But, you can't stop part way.

Additionally, it's not like the symmetry of the rectangle for which flipping end-to-end or top-to-bottom is different from rotating 180-degrees. There's only one-possible move in the usual cases of duality.

Now, I want to find examples of dualities for other symmetry groups. Is there some set of three concepts which can be interchanged to go hit all of the permutations? Is there some pair of concepts that can be continuously transformed from one to the other?

I'd imagine that space-time is a good place to start.