Graphics Reference
In-Depth Information
See [HurW48] for how this rather complicated definition can be used to prove
some standard facts about the dimension function. In particular,
R
n
has dimension n
in this new sense and a subset of
R
n
is n-dimensional if and only if it contains an open
subset of
R
n
. Boundaries of nice n-dimensional sets like manifolds are (n - 1)-
dimensional. Subsets of dimension less than n - 1 cannot disconnect
R
n
.
Another approach to the concept of dimension is via measure theory. This fact
should not be surprising when one considers how different words such as “length”
and “area” are used to measure the “size” of one- and two-dimensional sets, respec-
tively. Note also how length is obtained by approximations using finite segments. The
sum of the individual segment lengths (raised to the power 1) is the approximating
value. In the two-dimensional case, one could use a union of squares (or rectangles)
to approximate the region, so that the area would be approximated by a sum of terms
each of which is a length squared. A two-dimensional region could be thought of as
an infinite union of segments, but defining its area by adding up the lengths of these
segments (raised to the power 1) would give the value infinity. One could say that
a two-dimensional region has infinite “length” and zero “volume.” Only areas, the
measure of rectangles, lead to a nontrivial finite result. In general, the “correct”
measure is the power n that one needs so that adding up the individual measures
leads to a non-trivial result. In 3-space, we need cubes, but covering an area with
cubes would take an arbitrarily small volume.
In order to motivate the next definition of dimension, we change our viewpoint
slightly and also try to make things more precise. It takes 1/e intervals of length e to
cover the interval [0,1]. It takes 1/e
2
squares of width e to cover [0,1] ¥ [0,1]. More
generally, one would expect that the number, n(e), of d-dimensional cubes of width e
needed to cover a nonempty compact d-dimensional set
X
in
R
n
should be defined by
a formula of the form
d
1
Ê
Ë
ˆ
¯
()
=
nc
e
,
e
where c is a constant that depends on the set. Solving for d gives
()
-
()
ln
n
e
ln
c
d
=
.
ln
1
e
We are interested in what happens when e gets small. In that case the term
ln
ln
c
1
()
goes to zero. These intuitive comments lead to the following.
Definition.
Let e>0 and let n(
X
,e) be the smallest number of closed balls of radius
e whose union covers
X
. Define the
fractal dimension
of
X
, dim
F
X
, by
(
)
lim
ln
n
X
,
e
dim
X
=
,
F
ln
1
e
e
0
