VSFCF - CVM 1.1

# Connection to the Cantor Set

In a sense we simplify now by reducing dimension by one. Instead of looking at subsets of the two-dimensional plane we'll look at subsets of the one-dimensional line. In Section 3 we constructed a standard Cantor set by removing the middle third of the intervals in Ck to get Ck+1. If instead we remove the middle rth of each subinterval (0<r<1), we will still get a Cantor set. And because the measure of Ck will then be (1-r)k, the measure of C will be zero (taking the limit as k goes to infinity) for any such r. However, C has similarity dimension -1/log2((1-r)/2). This can be seen using the IFS given by f0(x)=sx, f1(x)=s(x-1)+1, for s=(1-r)/2 where the open set condition is satisfied by U=(0,1).

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, frames 9 and 15 of 17. Also available is an animation (98 kb) and a speed controlled version. Use the browser back button to return. Source code is available.

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.

Communications in Visual Mathematics, vol 1, no 1, August 1998.