Page 170 - Carbonate Sedimentology and Sequence Stratigraphy

APPENDIX A

Fractals

The term “fractal” was coined by Mandelbrot (1967) for
objects or sets of fractional dimension. For want of a gener-
ally accepted definition of fractals, I will characterize them
by listing essential properties and examining some classic
examples. We will then proceed to methods for determining
the fractal dimension and the fractal character of natural
phenomena. For more information, the reader is referd to
Falconer (1990) for mathematical background, Schroeder
(1991) for a discipline-crossing overview of fractals, and to
Turcotte (1997) and Hergarten (2002) for overviews directed
to earth scientists.

What is a fractal?
A fractal is an object or set of objects that
➤ is self-similar, i.e. looks the same at all scales,
➤ has a fine structure,
➤ has a fractal dimension larger than its topological di-
mension.

The topological dimension is, for most practical purposes,
equal to the Euclidean dimension. Several different formu-
las are used to calculate the fractal dimension; all of them
quantify the fine structure of the fractal, for instance by mea-
suring the degree to which a wiggly curve fills an area, or a
series of aligned dots starts to form a one-dimensional line.
For natural scientists, it is important to recognize the dif-
ference between abstract fractals and natural fractals. Fig.
A.1 shows an example of an abstract fractal conceived in Fig. A.1.— The Koch curve – a geometric fractal. It is gener-
1904 by the Swedish mathematician H. von Koch. It is cre- ated by erecting an equilateral triangle over the middle third of a
ated by erecting an equilateral triangle over the middle third line segment and repeating the operation at smaller and smaller
of a straight line segment and repeating this operation with segments to infinity. After Falconer (1990), modified.
smaller and smaller line segments to infinity. The result is a
wiggly curve that looks the same at all scales. This means
that any part of the curve properly enlarged looks like the
whole. The curve is self-similar because of its fine structure
and has a fractal dimension between 1 and 2, between the
Euclidean dimension of a line and that of an area. We may
say that the curve is so wiggly that it starts to fill part of an
area. The topological dimension of the curve is one. Topol-
ogy is also known as “rubber-sheet geometry”. If the Koch
curve in Fig. A.1 were made of a rubber string then it would
be possible to change it by continuous transformation with-
Fig. A.2.— The Cantor bar (or Cantor set) – a mathematical frac-
out any cutting or pasting into a straight line or any other
tal well known in stratigraphy. The initiating step is a line segment.
smooth curve. In the generating step, the middle third of the line is removed and
Fig. A.2 shows another mathematical fractal – the Cantor this operation is repeated to infinity. After Schroeder (1991), modi-
set, also called Cantor bar, conceived by the German mathe- fied.

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