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Figure 2.13.
A Cantor set constructed from infinitely many trisecting operations of the unit interval is depicted
for the first six generations. The fractal dimension of the resulting “fractal dust” is 0.6309.
remaining line segments so that none is lost. In this example the parameter
a
is the
ratio of the total mass to the mass of each segment after one trisecting operation. Since
each segment receives half the mass of its parent
a
2. The second parameter
b
is the
ratio of the length of the original line to the length of each remaining segment. Since
this is a trisection
b
=
3. The parameter
a
gives us an idea of how quickly the mass is
being concentrated and the parameter
b
gives us a comparable idea of how quickly the
available space is being thinned out.
Using the definition (
2.25
) we see that, for the trisection Cantor set at each stage
n
,
the number of line segments increases by two so that
N
=
2
n
; also the scaling ratio
decreases by a factor of 1/3 at each stage so that the ruler size becomes a function of
n
,
r
=
3
n
. Consequently, the fractal dimension is given by
=
1
/
2
n
log
[
]
log 2
log 3
=
D
=
]
=
0
.
6309
,
(2.26)
3
n
log
[
a
n
independently of the branching or generation number
n
. In general
N
=
and
b
n
, so the fractal dimension
D
of the resulting Cantor set is the ratio of the
logarithms,
r
=
1
/
log
a
log
b
D
=
(2.27)
in terms of scaling parameters. Note that this is the same ratio as that previously
encountered in our discussion of scaling.
As West and Goldberger [
93
] point out, there are several ways in which one can intui-
tively make sense of such a fractional dimension. Note first of all that in this example
D
is greater than zero but less than one, the dimension of a line in Euclidean geome-
try. This makes sense when one thinks of the Cantor set as a physical structure with
mass: it is something less than a continuous line, yet more than a vanishing set of
points. Just how much less and how much more is given by the ratio of the two param-
eters. If
a
were equal to
b
, the structure would not change no matter how much the
original line were magnified; the mass would clump together as quickly as the length
scaled down and a one-dimensional line would appear on every scale. However, if
a
were greater than
b
a branching or flowering object would appear, namely an object
that develops finer and finer structures under magnification such as the fractal trees
