Fractalic Awakening - A Seeker's Guide

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Every smaller piece is composed of an infinite number of scaled segments that look exactly like the first iteration. As is the case with dimensions determined for lines, squares, and cubes, fractal dimensions are general descriptors that do not uniquely define patterns.

Fractalic Awakening - New metaphysics, fractal, fractals

ldots } , a non-integer value that suggests the fractal has a dimension not equal to the space it resides in. 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. The earliest roots of what Mandelbrot synthesized as the fractal dimension have been traced clearly back to writings about nondifferentiable, infinitely self-similar functions, which are important in the mathematical definition of fractals, around the time that calculus was discovered in the mid-1600s. The two fractal curves described above show a type of self-similarity that is exact with a repeating unit of detail that is readily visualized.

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. In that paper, Mandelbrot cited previous work by Lewis Fry Richardson describing the counter-intuitive notion that a coastline's measured length changes with the length of the measuring stick used ( see Fig. Correlation dimension: D is based on M {\displaystyle M} as the number of points used to generate a representation of a fractal and g ε, the number of pairs of points closer than ε to each other. Fractal complexity may not always be resolvable into easily grasped units of detail and scale without complex analytic methods but it is still quantifiable through fractal dimensions.

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The examples discussed here were chosen for clarity, and the scaling unit and ratios were known ahead of time. Whereas the animation only shows a few iterations, the theoretical curve is scaled in this way infinitely. No small piece of it is line-like, but rather it is composed of an infinite number of segments joined at different angles. If the theoretical fractal dimension of a set exceeds its topological dimension, the set is considered to have fractal geometry.These include, as examples, strange attractors for which the detail has been described as in essence, smooth portions piling up, [17] :49 the Julia set, which can be seen to be complex swirls upon swirls, and heart rates, which are patterns of rough spikes repeated and scaled in time. The concept of fractality is applied increasingly in the field of surface science, providing a bridge between surface characteristics and functional properties. In terms of that notion, the fractal dimension of a coastline quantifies how the number of scaled measuring sticks required to measure the coastline changes with the scale applied to the stick. These works were accompanied by perhaps the most pivotal point in the development of the concept of a fractal dimension through the work of Hausdorff in the early 1900s who defined a "fractional" dimension that has come to be named after him and is frequently invoked in defining modern fractals. 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.

Fractalic Awakening - New metaphysics, fractals

However, the theoretical patterns that the images represent have no discrete pixel-like pieces, but rather are composed of an infinite number of infinitely scaled segments and do indeed have the claimed fractal dimensions. The first four iterations of the Koch snowflake, which has a Hausdorff dimension of approximately 1. To elaborate, a fractal line measured at first to be one length, when remeasured using a new stick scaled by 1/3 of the old may be 4 times as many scaled sticks long rather than the expected 3 ( see Fig.For sets describing ordinary geometric shapes, the theoretical fractal dimension equals the set's familiar Euclidean or topological dimension. It is found however that many physical surface phenomena cannot readily be interpreted with reference to such descriptors, thus fractal dimension is increasingly applied to establish correlations between surface structure in terms of scaling behavior and performance. There are several formal mathematical definitions of fractal dimension that build on this basic concept of change in detail with change in scale: see the section Examples.