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The difference for fractals is that the pattern reproduced must be detailed. Self-similarity itself is not necessarily counter-intuitive e. If this is done on fractals, however, no new detail appears nothing changes and the same pattern repeats over and over, or for some fractals, nearly the same pattern reappears over and over. The feature of "self-similarity", for instance, is easily understood by analogy to zooming in with a lens or other device that zooms in on digital images to uncover finer, previously invisible, new structure. The mathematical concept is difficult to define formally, even for mathematicians, but key features can be understood with little mathematical background. The word "fractal" often has different connotations for the lay public as opposed to mathematicians, where the public are more likely to be familiar with fractal art than the mathematical concept. The consensus is that theoretical fractals are infinitely self-similar, iteratedand detailed mathematical constructs having fractal dimensions, of which many examples have been formulated and studied in great depth. Mandelbrot himself summarized it as "beautiful, damn hard, increasingly useful. There is some disagreement among mathematicians about how the concept of a fractal should be formally defined. Starting in the 17th century with notions of recursionfractals have moved through increasingly rigorous mathematical treatment of the concept to the study of continuous but not differentiable functions in the 19th century by the seminal work of Bernard BolzanoBernhard Riemannand Karl Weierstrass and on to the coining of the word fractal in the 20th century with a subsequent burgeoning of interest in fractals and computer-based modelling in the 20th century. Analytically, fractals are usually nowhere differentiable. However, if a fractal's one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily an integer. Likewise, if the radius of a sphere is doubled, its volume scales by eight, which is two the ratio of the new to the old radius to the power of three the dimension that the sphere resides in.
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Doubling the edge lengths of a polygon multiplies its area by four, which is two the ratio of the new to the old side length raised to the power of two the dimension of the space the polygon resides in. One way that fractals are different from finite geometric figures is the way in which they scale. Mandelbulb3D/Reference/Formulas/Formula types Fractals exhibit similar patterns at increasingly small scales called self similarityalso known as expanding symmetry or unfolding symmetry if this replication is exactly the same at every scale, as in the Menger sponge it is called affine self-similar.įractal geometry lies within the mathematical branch of measure theory. In mathematicsa fractal is a self-similar subset of Euclidean space whose fractal dimension strictly exceeds its topological dimension.įractals appear the same at different levels, as illustrated in successive magnifications of the Mandelbrot set because of this, fractals are encountered ubiquitously in nature.