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A beautiful picture of this varying Cantor set appears in [Ma2, p. 81]. A set in the unit square is drawn so that slicing it horizontally at height r gives a Cantor set of dimension r. This "curtain"-like set is redrawn and animated in Figure 7. Such figures even without the animation can be thought of as "movies" of Cantor sets: as r changes we can see just how C changes. In this paper we give such movies with dimension up to 2 (or even 3) using real animation: as dimension increases we will show curves of those dimensions that smoothly change, ending up as a space-filling curve with dimension 2 (or 3).
Figure 7. Mandelbrot's Cantor Curtains,
and 15 of 17.
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If we take the Cartesian product of a middle rth Cantor set with itself, we get a Cantor set like the ones drawn in Figure 5 in the previous section. This doubles the similarity dimension since D = -log(n)/log(s), where s is still (1-r)/2, but n has gone from 2 to 4=22.