Review for Week of Feb 20

I was thinking about how the skew in our affine transformation makes it technically not self-similar. I refreshed the definition we got on 1/27 and realized that a critical component was that for every individually indexed subset of the larger set there exists a transformation that gives back the whole set. We spent a lot of time this week talking about overlap and proving that our skew does not have it. But what that means is that there are no points in the fractal set that map back to points on the boundaries of the fractal [eg (0,2) and (1,1)]. This implies that there is not a transformation that gives back the whole set, since there is no transformation that gives back (0,2). I looked over our original definition and decided that it helped to see the definition in the looser terms we got today – that it applies only to linear transformations that rotate, translate, and uniformly contract the input.

We started talking about balls a little bit, but we haven’t explored them enough to know what more we can know about dimensions.

Note: All Homework must be posted by next Friday for credit.