sgr ms thesis - University of Maine

therefore it is possible to relate a breccia to its mechanism of formation by
analyzing these self-similar characteristics.
5.2. Fractal Theory
When an object shows self-similar properties, it is described as fractal.
Fractal theory sprouted from the desire to quantitatively describe geometries
observed in nature. Unlike Euclidean geometry, fractals refer to complex shapes
defined by a fractional, or fractal, dimension (D) (Mandelbrot, 1967, 1983; Urtson,
2005). Founded by Benoit Mandelbrot in 1967, fractal theory has since been
applied to many scientific problems. The self-similarity of fractals implies that
patterns tend to repeat themselves at all scales, and for a true fractal, the
number of scales of natural patterns is infinite. For the initial purposes of this
thesis, it is best to consider the fractal dimension in terms of a repeated pattern
of size distribution. Consider the repeated pattern in Figure 5.2 (Sammis et al.,
1987). Sections of a cube are repeatedly split into smaller and smaller
components, producing a distribution of various sizes. This pattern is quantified
by
(5.2)
Assuming that the pattern of size distribution produced by breaking the cube is
fractal, the fractal dimension is a function of the number of cubes with side
length . For the broken cube, the pattern of size distribution is fractal, and =
2.58. This value is unique to this size distribution and any change from this
pattern would produce a different
. Interest lies in the self-similar characteristics
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