Fractalic Awakening - A Seeker's Guide

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A fractal dimension is an index for characterizing fractal patterns or sets by quantifying their complexity as a ratio of the change in detail to the change in scale.

Fractal - Wikipedia

The self-similarity lies in the infinite scaling, and the detail in the defining elements of each set. Fractal dimensions are used to characterize a broad spectrum of objects ranging from the abstract [1] [3] to practical phenomena, including turbulence, [5] :97–104 river networks, :246–247 urban growth, [10] [11] human physiology, [12] [13] medicine, [9] and market trends. Ultimately, the term fractal dimension became the phrase with which Mandelbrot himself became most comfortable with respect to encapsulating the meaning of the word fractal, a term he created.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.

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illustrates, traditional notions of geometry dictate that shapes scale predictably according to intuitive and familiar ideas about the space they are contained within, such that, for instance, measuring a line using first one measuring stick then another 1/3 its size, will give for the second stick a total length 3 times as many sticks long as with the first. Instead, a fractal dimension measures complexity, a concept related to certain key features of fractals: self-similarity and detail or irregularity. The concept of fractality is applied increasingly in the field of surface science, providing a bridge between surface characteristics and functional properties. Overall, fractals show several types and degrees of self-similarity and detail that may not be easily visualized. The concept of fractal dimension described in this article is a basic view of a complicated construct.As the length of the measuring stick is scaled smaller and smaller, the total length of the coastline measured increases (See Coastline paradox). Numerous surface descriptors are used to interpret the structure of nominally flat surfaces, which often exhibit self-affine features across multiple length-scales.