Fractalic Awakening - A Seeker's Guide

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Fractal dimensions were first applied as an index characterizing complicated geometric forms for which the details seemed more important than the gross picture. [16] For sets describing ordinary geometric shapes, the theoretical fractal dimension equals the set's familiar Euclidean or topological dimension. Thus, it is 0 for sets describing points (0-dimensional sets); 1 for sets describing lines (1-dimensional sets having length only); 2 for sets describing surfaces (2-dimensional sets having length and width); and 3 for sets describing volumes (3-dimensional sets having length, width, and height). But this changes for fractal sets. If the theoretical fractal dimension of a set exceeds its topological dimension, the set is considered to have fractal geometry. [17] That is, for a fractal described by N = 4 {\displaystyle N=4} when ε = 1 3 {\displaystyle \varepsilon ={\tfrac {1}{3}}} , such as the Koch snowflake, D = 1.26185 … {\displaystyle D=1.26185\ldots } , a non-integer value that suggests the fractal has a dimension not equal to the space it resides in. [3]

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D 2 = lim M → ∞ lim ε → 0 log ⁡ ( g ε / M 2 ) log ⁡ ε {\displaystyle D_{2}=\lim _{M\to \infty }\lim _{\varepsilon \to 0}{\frac {\log(g_{\varepsilon }/MUnlike topological dimensions, the fractal index can take non- integer values, [18] indicating that a set fills its space qualitatively and quantitatively differently from how an ordinary geometrical set does. [1] [2] [3] For instance, a curve with a fractal dimension very near to 1, say 1.10, behaves quite like an ordinary line, but a curve with fractal dimension 1.9 winds convolutedly through space very nearly like a surface. Similarly, a surface with fractal dimension of 2.1 fills space very much like an ordinary surface, but one with a fractal dimension of 2.9 folds and flows to fill space rather nearly like a volume. [17] :48 [notes 1] This general relationship can be seen in the two images of fractal curves in Fig.2 and Fig. 3 – the 32-segment contour in Fig. 2, convoluted and space filling, has a fractal dimension of 1.67, compared to the perceptibly less complex Koch curve in Fig. 3, which has a fractal dimension of approximately 1.2619.

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Figure 6. Two L-systems branching fractals that are made by producing 4 new parts for every 1/3 scaling so have the same theoretical D {\displaystyle D} as the Koch curve and for which the empirical box counting D {\displaystyle D} has been demonstrated with 2% accuracy. [8]

In mathematics, a fractal dimension is a term invoked in the science of geometry to provide a rational statistical index of complexity detail in a pattern. A fractal pattern changes with the scale at which it is measured. D 0 = lim ε → 0 log ⁡ N ( ε ) log ⁡ 1 ε . {\displaystyle D_{0}=\lim _{\varepsilon \to 0}{\frac {\log N(\varepsilon )}{\log {\frac {1}{\varepsilon }}}}.}