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 C_{k} to get C_{k+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 C_{k} 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/log_{2}((1-r)/2).
This can be seen using the IFS given by f_{0}(x)=sx,
f_{1}(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=2^{2}.
Communications in Visual Mathematics, vol 1, no 1, August 1998.
Copyright © 1998,
The Mathematical Association of America.
All rights reserved.
Created: 18 Aug 1998 ---
Last modified: Sep 30, 2003 6:28:12 PM
Comments to: CVM@maa.org