We'll consider finite collections
of similarities of Rm (usually with m=2)
that are contractions - for each fk there is a shrinking
factor sk between 0 and 1 such that for
all x and y we have
In fact, all our functions will have the same contraction,
This is an example of an iterated function system (IFS) and must
have a unique compact attractor: a set A with the property
that it equals the union of all the fk(A). (For
details, see [Ba].)
In addition, our IFS's will always satisfy the open set condition
(also called Moran's open set condition: see [Sag, p. 158], [Ed, p. 160], [Hu, p. 735], or [Ba, p. 129]); there will
be an open set U such that:
(The open set condition prevents "overlap" among the functions of the
In this situation one defines the similarity dimension of the
attractor A of our system of n functions with contraction
constant s to be the unique number D such that
n s D=1
So D = -log(n)/log(s).
To see that this definition of dimension makes sense, note these examples:
- For k different from j, fk(U) and
fj(U) are disjoint, and
- For all k, fk(U) is a subset of U.
This is perhaps the easiest definition of dimension.
Its drawback is that it only applies to sets defined as attractors in this
However, for such sets, it gives the same value as the very general
concept of Hausdorff dimension ([Hu]).
- In R1, A=[0,1],
f1(x)=1-x/2, U=(0,1), s=1/2,
so D=1 and a segment has similarity dimension 1.
- In R2, A=[0,1]×[0,1], and using
complex variable notation, f0(z)=z/2,
so D=2 and a square region has similarity dimension 2.
- In R3, we cut a cube (A) into 8
congruent cubelets with half the edge length, so n=8, s=1/2,
so D=3 and a cube has similarity dimension 3.
In all our examples, U can be taken to be the interior of the
convex hull of A.
For example, for the snowflake curve, U will be the inside of a
triangle, while for the Hilbert curve U will be the inside of a
The Cantor set will play a role in many of our
The standard Cantor set, C, can be constructed by starting with the
unit interval, C0=[0,1], deleting the open middle
third to get C1 the union of [0,1/3] and
[2/3,1], deleting the open middle thirds of these two intervals to
get C2, etc.
Then C is the intersection of all these Ck.
All homeomorphs of C are called Cantor sets and are characterized
as nonempty metric, compact, totally disconnected, perfect (no isolated
points) sets ([HY,
p. 99]). In particular, as we will use here, instead of deleting
thirds we can delete any part between 0 and 1 and still get
a Cantor set. And the Cartesian product of two of these Cantor sets will
be a Cantor set.
Figure 5 in
Section 5 shows such Cantor sets.
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: 18 Aug 1998 23:59:59
Comments to: CVM@maa.org