CHAPTER 5
FRACTAL ANALYSIS
One of the most difficult tasks facing morphologists is that of quantifying and
measuring size and shape. Traditionally, parameters such as length and volume were
used to try to quantitatively describe and compare morphological characteristics. In
Euclidean geometry linear measures are considered one dimension, smooth surfaces are
two dimensions, and volumes and weights are three dimensions. Objects that occur in
nature, however, seldom have edges that are straight or surfaces that are smooth (Long
1985). Some objects in nature possess certain qualities that can be described by a non-
Euclidean fractional dimension, which lies between the values of one and two
(Mandelbrot 1977). These objects are known as fractals. Fractals are geometric objects
that are self-similar in nature. Self-similarity means that the fractal object is composed of
smaller units that possess the same shape as the whole object. Fractals have complex
edges or surfaces that increase linearly as the resolution of the units used to measure them
increase (Hartwig 1991). Fractal analysis is a technique used to interpret the geometric
complexities of fractals.
Several researchers believe some cranial sutures are fractal objects (Long 1985,
Hartwig 1991, Long and Long 1992, Gibert and Palmqvist 1995, Montiero and Lessa
2000, Yu et al. 2003). Long (1985) explored the idea of whether or not complex sutures
exhibit fractal properties such as self-similarity and a dimension between one and two.
To address this question, Long (1985) examined the sutures on the shells of extinct
ammonites and cranial sutures of white-tailed deer. The sutures in both of these