"Chaos does provide a framework or a mindsetor point of view, but it is not as directly explanatory as germ theory or plate tectonics. Chaos is a behavior - a phenomenon - not a causal mechanism. [...] The situation with fractals is similar. The study of fractals draws one’s eye toward patterns and structures that repeat across different length or time scales. There is also a set of analytical tools - mainly calculating various fractal dimensions - that can be used to quantify structural properties of fractals. Fractal dimensions and related quantities have become standard tools used across the sciences. As with chaos, there is not a fractal theory. However, the study of fractals has helped to explain why certain types of shapes and patterns occur so frequently." (David P Feldman,"Chaos and Fractals: An Elementary Introduction", 2012)
"Fractals are different from chaos. Fractals are self-similar geometric objects, while chaos is a type of deterministic yet unpredictable dynamical behavior. Nevertheless, the two ideas or areas of study have several interesting and important links. Fractal objects at first blush seem intricate and complex. However, they are often the product of very simple dynamical systems. So the two areas of study - chaos and fractals - are naturally paired, even though they are distinct concepts." (David P Feldman,"Chaos and Fractals: An Elementary Introduction", 2012)
"Fractals' simultaneous chaos and order, self-similarity, fractal dimension and tendency to scalability distinguish them from any other mathematically drawable figures previously conceived." (Mehrdad Garousi, "The Postmodern Beauty of Fractals", Leonardo Vol. 45 (1), 2012)
"One of the most important artistic properties of fractals is the randomness governing the process of making them. Each fractal is essentially generated by a basic formula and one or more gradients that identify the colors of the fractal. Sometimes, however, fractals are generated by tens of different formulas and gradients." (Mehrdad Garousi, "The Postmodern Beauty of Fractals", Leonardo Vol. 45 (1), 2012)
"The concept of infinity embedded in fractals' identity provides an infinity of possibilities to explore in a single image. The repetition of a formula is the key to becoming more familiar with it. When trying a completely new formula, all fractal artists are engaged in the same activity - a random playing around." (Mehrdad Garousi, "The Postmodern Beauty of Fractals", Leonardo Vol. 45 (1), 2012)
[fractal:] "A fragmented geometric shape that can be split up into secondary pieces, each of which is approximately a smaller replica of the whole, the phenomenon commonly known as self similarity." (Khondekar et al, "Soft Computing Based Statistical Time Series Analysis, Characterization of Chaos Theory, and Theory of Fractals", 2013)
"Only at the edge of chaos can complex systems flourish. This threshold line, that edge between anarchy and frozen rigidity, is not a like a fence line, it is a fractal line; it possesses nonlinearity." (Stephen H Buhner, "Plant Intelligence and the Imaginal Realm: Beyond the Doors of Perception into the Dreaming of Earth", 2014)
"Geometric pattern repeated at progressively smaller scales, where each iteration is about a reproduction of the image to produce completely irregular shapes and surfaces that can not be represented by classical geometry. Fractals are generally self-similar (each section looks at all) and are not subordinated to a specific scale. They are used especially in the digital modeling of irregular patterns and structures in nature." (Mauro Chiarella, "Folds and Refolds: Space Generation, Shapes, and Complex Components", 2016)
"A quantum of action in a field creates the dimensions it inhabits. Fields contain waves of potential." (Rick Delmonico, "Fractals all the way down", 2018)
"At every level of the fractal hierarchy, variation enters the system in different ways, so there is no scale invariance. All of the information is stored in the relationships of and in each level in the computational geometry." (Rick Delmonico, "Fractals all the way down", 2018)
"How can unity and infinity share the same space? There is only one way, as a fractal." (Rick Delmonico, "Fractals all the way down", 2018)
"In a fractal nested hierarchy there are an infinite number of boundaries and boundary conditions, this is important because this is where emergence comes from and this is why there are no perfect symmetries."
"The computational geometry of space/time is fractal. The entire system is interwoven as a fractal and as each series of iterations occur, there is a dithering of sorts because the difference in possibilities can be matastable in many locations in the computation. Each layer of the system computes in a frequency that is probably following the phi ratio. Measurement is always described as ratio and scale may be, in a sense, an illusion. Light appears to be scale invariant and the cutoff frequency may be another illusion. If this is true, everything is fields and particles are just a smaller version of a field." (Rick Delmonico, "Fractals all the way down", 2018)
"The fractal nesting of the relationship between space and time at different scales causes phase transitions as the influence of one force changes with respect to another. From this we get the astonishing variety of behaviors in the material world." (Rick Delmonico, "Fractals all the way down", 2018)
"Waves travel according to their dimensionality. The idea is to find agreement in two unrelated fields that point to the same thing, a duality. Waves can't be quantized, interactions among the associated dimensions can. Fractal geometry and levels of description. It is the interaction of forces across scale, with ratio being the only thing that is discrete." (Rick Delmonico, "Fractals all the way down", 2018)
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