Information Technology Reference
In-Depth Information
Figure 6.22.
Here is depicted the procedure of generating a Koch line segment resulting from expanding the
middle third of each line segment in going from iteration
n
to iteration
n
+
1. In the limit
n
→∞
the resulting line segment is one that is infinitely crinkled and has a fractal dimension
D
=
/
ln 4
ln 3.
4
3
L
L
(
n
)
=
(
n
−
1
).
(6.122)
Inserting (
6.121
) into both sides of (
6.122
) yields the relation
4
3
3
(
n
−
1
)(
D
−
1
)
,
3
n
(
D
−
1
)
=
(6.123)
which simplifies to the value for the fractal dimension
2
ln 2
D
=
ln 3
≈
1
.
26
;
(6.124)
this is twice the dimension of the fractal dust found in an earlier chapter. The origi-
nal line segment has been crinkled to such a degree that the distance along the curve
between the end points becomes infinite. This divergence in length occurs because
1
−
D
<
0, so that in the limit
n
→∞
the length of the line segment given by (
6.121
)
becomes infinite, that is, lim
A more extensive discussion of the
consequences of this scaling is given by West and Deering [
37
].
Unfortunately, the web in Figure
6.21
has little clustering [
11
]. The authors of [
11
]
propose an alternative deterministic web yielding
0
L
(η)
→∞
.
η
→
1 and large clustering. The web
is realized according to the prescription illustrated in Figure
6.23
. In this web the root
(a university president) interacts with two vice presidents, one for science and one for
the humanities. The two vice presidents, in turn, work with the help of two subordinate
α
≈
