Webs, trees and branches - Complex Webs: Anticipating the Improbable - page 67

the geometric self-similarity is replaced by the repetition of structure across time rather
than space. Finally, there are fractal statistics in which the properties of the distribution
function are repeated across multiple scales of either space or time, or both. There are
many topics on each kind of fractal, so here we give only a brief overview of each type.
This should provide enough information to stimulate the imagination and describe one
or more of the phenomena where data have a statistical distribution of the hyperbolic
form. One of the things we learn is that Gaussian statistics have scaling behavior, per-
haps the simplest statistical scaling. But more significantly the hyperbolic statistics we
discussed in the first chapter have a richer scaling behavior. The hyperbolic statistics in
space indicate a kind of clustering associated with the fluctuations, so there are clusters
within clusters within clusters, and it is this clustering behavior that is associated with
the fractal property. Most of the complex phenomena of interest to us have this scaling,
which is fundamentally different from the scaling in normal statistics.
We previously introduced the notion of self-similarity and for geometric fractals self-
similarity is based on the idea of a reference structure repeating itself over many scales,
telescoping both up and down in size. It is interesting to use this scaling argument to
test our intuitive definition of dimension.
A one-dimensional unit line segment can be covered by N line segments of length r ,
the linear scale interval or ruler size. A unit square can be covered by N areas r 2 and
finally a unit cube can be covered (filled) by N cubes of volume r 3 . Note that in each
of these examples smaller objects of the same geometric shape as the larger object are
used to cover the larger object. This geometric equivalence is the basis of the notion of
self-similarity. In particular, the linear scale size of the covering objects is related to the
number of self-similar objects required to cover an object of dimension D by

N 1 / D r = 1 / . (2.24)

This relation can be inverted by taking the logarithm of both sides of the equation and
defining the dimension of the object by the equation

log N
log

=

) .

D (2.25)

( 1 / r

This expression is mathematically rigorous only in the limit of vanishingly small lin-
ear scale size r

0. This is where the discussion becomes very interesting. Although
( 2.25 ) defines the dimension of a self-similar object, there is nothing in the definition
that guarantees that D is an integer, such as 1, 2 or 3. In fact, in general D is not an
integer.
It is one thing to define a non-intuitive quantity such as a non-integer dimension.
It is quite another to construct an object having such a dimension. So let us examine
how to construct a geometric object that has a non-integer dimension. Consider the unit
line segment depicted in Figure 2.13 over which mass points are uniformly distributed.
Now we remove the middle third of the line segment and redistribute the mass of the
removed section along the remaining two segments so that the total mass of the result-
ing set remains constant. At the next stage, cut the middle third out of each of these
two line segments, and again redistribute the mass of the removed sections along the

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