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“sin²(?) is odious to me.” — Carl Friedrich Gauss

There's lots of interesting behavior to be found when iterating a function.  One thing I will show is that by putting a metric on the set of pictures in a plane, we can watch a randomized point reliably trace out a fractal.  Another has to do with cycles of functions on the real numbers, i.e. points x where fⁿ(x)=x.  It turns out that there's a not terribly complicated way to specify exactly when a continuous function with an n-cycle must also have an m-cycle, and I guarantee it won't be anything you're expecting (unless you've seen it before).  And if you'd like, when the talk is done, I can iterate it and give it again!  And again!  And...