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which stays finite as e approaches 0 is where p = d, the Koch curve is a fractal of
dimension d. See [Mand83] for an interesting connection between the Koch curve and
the coastline of Britain (they both have the same dimension). The Koch curve is only
one example of a space that can be defined by recursive substitution. See Glassner's
overview ([Glas92]) of constructions of this type and how they can create interesting
objects that can be viewed at different scales.
Looking at the concept of dimension from a slightly different point of view, one
notices that, in real life, dimension is relative to a particular context. Mandelbrot
refers to this everyday notion of dimension as the
effective dimension
. For example,
consider a 10-centimeter-wide ball of 1-millimeter thread. The dimension associated
to it would depend on the distance of the viewer to the object:
Distance to object
Effective dimension
•
0 (one sees a point)
10 cm
3 (one sees a solid ball)
10 mm
1 (one sees the threads)
0.1 mm
3 (the threads look like columns)
0.01 mm
1 (one sees the fibers in the threads)
Finally, what do fractals have to do with dynamical systems and chaos? In a sense,
not much. The Mandelbrot set above is a fractal, however, and so the connection
between the two is that the sets important to the study of a dynamical system, in par-
ticular chaotic ones, often are fractals.
22.4
Iterated Function Systems
This section gives a very brief overview of iterated function systems. A good reference
for this topic is [Barn88] and our discussion here basically touches on some high-
lights from that topic. We start with two examples.
Consider transformations w
1
, w
2
,..., w
k
of the plane of the form
()
=+,
w
pp p
A
i
i
i
where the A
i
are 2 ¥ 2 matrices and the
p
i
are fixed points. We are interested in what
happens to points as we repeatedly apply the transformations w
i
to them. For example,
consider the transformations w
1
, w
2
, and w
3
, where
05 0
005
.
===
Ê
Ë
ˆ
¯
AA A
(22.3)
1
2
3
.
and
=
()
=
(
)
=
(
)
p
11
,,
p
150
,
,
and
p
50 50
,
.
(22.4)
1
2
3
Consider Algorithm 22.4.1. What the algorithm does is start out with a white rectan-
gle [1,50] ¥ [1,50] that has a black boundary and then successively applies the trans-
