ChaoticMarket 45geometry was developed, however, these integral dimensions were all
wehadtogoon.
Fractal geometry was developed initially by the mathematician
and scientist Benoit Mandelbrot, who won the 1993 Wolf Prize in
physics.^9 He observed that natural objects are not as simple as the
descriptions offered by Euclidean geometry: “Clouds are not spheres
[and] mountains are not cones.” For example, how would we classify
a piece of paper crumpled up an infinite number of times in terms
of Euclidean geometry?
It is not three-dimensional because it is not a pure solid form (it
has creases and crevices). (In mathematical terms, it is not com-
pletely differentiable across its entire surface.) It is also not two-
dimensional because it has depth. In fact, its dimension is between
two and three. That property makes the crumpled paper a fractal:
Its dimension is a fraction (two point something).
With respect to time-series data, dimensionality depends on
whether the system from which the data are taken is random or
nonrandom. If a system is random, time-series data taken from it
will reflect that randomness and have as large a dimension as can
possibly be. In the case of data being presented on a sheet of paper,
the highest possible dimension is two (the dimension of the paper
itself). In any case, the data will fill a plane.
If a system is nonrandom, time series of data taken from it will
reflect that nonrandomness and show a fractal dimension: The data
will not fill the plane but will clump together. That clumping to-
gether reflects the correlations influencing the data (i.e., causing it
to be nonrandom).
These properties distinguishing random from nonrandom time
series may be conceptualized in a different way. For example, our
conception of a crumpled piece of paper as a three-dimensional ob-
ject can be regarded as embedding a fractal in a dimension greater
than itself. That greater dimension is called the embedding dimen-
sion.
Fractals retain their fractal dimension when placed in an em-
bedding dimension; random distributions do not. Thus, unlike non-
random distributions, random distributions fill their space the way
gas fills a volume: The gas spreads out because there is nothing to
bind the molecules together. This is, of course, the defining char-
acteristic of Brownian motion as discussed earlier.
Peters calculated the fractal dimension of the S&P 500 as 2.33
and did the same for other global stoc kmar kets, all of which also