Graphics Reference
In-Depth Information
The IFS is said to be
overlapping
if it is neither just touching nor totally disconnected.
22.4.7 Theorem.
Let ((
R
n
,d),W) be an IFS where the elements of W are similarity
transformations. Let W = {w
1
,w
2
,...,w
k
}, let c
i
be the contraction factor of w
i
, and let
D, D Œ [0,•) , be the unique solution to the equation
k
Â
D
c
i
=
1.
i
=
1
Let
A
be the attractor of the IFS and let dim
F
A
and dim
H
A
be its fractal and Haus-
dorff-Besicovitch dimension.
(1) If the IFS is totally disconnected or just-touching, then D £ n and D = dim
F
A
= dim
H
A
.
(2) If the IFS is overlapping, then dim
F
A
£ D .
Proof.
See [Barn88].
Barnsley explains how one can use Theorem 22.4.7 to estimate the fractal dimen-
sion of an attractor of an IFS. The integer D in the theorem actually has a name.
A set
X
Õ
R
n
is said to be
invariant
for a set {w
1
,w
2
,...,w
k
} of contrac-
Definition.
tions of
R
n
if
k
U
()
X
=
w
i
X
.
i
=
1
If the contractions are similarity transformations of
R
n
and if for some integer s,
the Hausdorff s-dimensional measure m
s
(
X
) > 0 but
(
()
()
)
=
m
w
XX
,
w
0
,
for
i
π
j
,
s
i
then the set
X
is said to be
self-similar
and the
similarity dimension
of
X
, dim
S
X
, is
the integer D defined by the equation
k
D
Â
c
i
=
1,
(22.5)
i
=
1
where c
i
is the contraction factor of w
i
.
For a more thorough discussion of self-similarity and a proof of the fact that the
similarity dimension is well-defined see [Falc85]. Intuitively, a self-similar set is a set
that is built up out of parts that are similar to the entire set. For example, a cube is
self-similar because it can be divided into four smaller cubes, each of which can be
divides into four smaller cubes, and so on. The triadic Koch curve is also a self-similar
set. The next corollary, which states that the fractal, Hausdorff-Besicovitch, and sim-
