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Fractals Information selected by Liliana Usvat A In mathematics, a fractal is a subset of a Euclidean space for which the fractal dimension strictly exceeds the topological dimension. Fractals appear the same at different levels, as illustrated in successive magnifications of the Mandelbrot set; because of this, fractals are encountered ubiquitously in nature. Many things in nature come in the shape of fractals. They come as fragmented geometrical shape split into parts, this parts further are, by each appearance, reduced in size and a copy of the whole. We can found those characteristics in snowflakes, crystals, rivers, and some plant structures. Knowing nature enables us to transform their properties and challenges into usefulness for human projects. One such approach to nature lies in the biomimicry. Fractals are extremely complex, sometimes infinitely complex - meaning you can zoom in and find the same shapes forever.Amazingly, fractals are extremely simple to make.A fractal is made by repeating a simple process again and again. Fractals are found all over nature, spanning a huge range of scales. We find the same patterns again and again, from the tiny branching of our blood vessels and neurons to the branching of trees, lightning bolts, and river networks. Regardless of scale, these patterns are all formed by repeating a simple branching process. A fractal is a picture that tells the story of the process that created it. Neurons from the human cortex. The branching of our brain cells creates the incredibly complex network that is responsible for all we perceive, imagine, remember. Scale = 100 microns = 10-4 m.
Purely geometric fractals can be made by repeating a simple process. The Sierpinski Triangle is made by repeatedly removing the middle triangle from the prior generation. The number of colored triangles increases by a factor of 3 each step, 1,3,9,27,81,243,729, etc.See the Fractivity on page 15 to learn to teach elementary school students how to draw and assemble Sierpinski Triangle The Koch Curve is made by repeatedly replacing each segment of a generator shape with a smaller copy of the generator. At each step, or iteration, the total length of the curve gets longer, eventually approaching infinity. Much like a coastline, the length of the curve increases the more closely you measure it.
HOW DOES THE MANDELBROT SET WORK We start by plugging a value for the variable ‘C’ into the simple equation below. Each complex number is actually a point in a 2-dimensional plane. The equation gives an answer, ‘Znew’ . We plug this back into the equation, as ‘Zold’ and calculate it again. We are interested in what happens for different starting values of ‘C’. Generally, when you square a number, it gets bigger, and then if you square the answer, it gets bigger still. Eventually, it goes to infinity. This is the fate of most starting values of ‘C’. However, some values of ‘C’ do not get bigger, but instead get smaller, or alternate between a set of fixed values. These are the points inside the Mandelbrot Set, which we color black. Outside the Set, all the values of ‘C’ cause the equation to go to infinity, and the colors are proportional to the speed at which they expand. The first image in the upper left comes from the same equation as the Mandelbrot Set, Z = Z2 + C. When we raise the ex-ponent to Z3 (i.e. Z*Z*Z), the Julia Set takes on a 3-fold symmetry, and so on. The degree of symmetry always corresponds to the degree of the exponent.
Water Fractals
Star Forts Fractals? Star forts, or as they are commonly called "Stellar" fortresses - one of the most disturbing minds of contemporaries of riddles - continues to raise more and more questions that have not been answered or not. And the most unpleasant thing is that, most likely, in our lifetime these answers will not be found. Now it is quite obvious that all the albums of the nineteenth century with drawings and plans, a description of the construction of these mysterious, not even structures, but rather formations, are gross forgeries. Man, most likely, has nothing to do with the creation of stellar forts. At least a person who is a representative of civilization, to which we relate ourselves, now living. And in favor of this argument there is a considerable amount of evidence. The first and most convincing is the number of such forts scattered across all continents. Even the erection of one of them requires a huge amount of resources and time. And if you summarize all the "stars", then they would require millions of man-hours of work to create them. The construction of such a scale simply did not make sense, and even information about it could not have been lost in history. But nothing has survived. Well ... Not that it is not preserved at all. Preserved. Only this was, rather, a reconstruction or an alteration.
Churches Fractal Patterns Sagrada Familia Barcelona. Do fractal pattern have an influence on human psyche?
other churches windows have fractal patterns
To understand this, we need to first briefly review the findings of Biophila and Attention Restoration Theory (A.R.T.)
The word Biophilia, used by psychologist Erich Fromm and popularized by E.O. Wilson, means “love of life or living systems”. Modern day psychologists have acquired empirical evidence in support of the Biophilia Hypothesis, which states that human beings have an innate biological predisposition to react positively to nature. In the words of Roger Ulrich of The University of Texas A&M, “There are well over 60 published scientific studies….which have consistently shown” that people will experience profound psychological and physiological stress relief by merely viewing natural scenery, and that “this effect is fairly generalized across different systems, from the brain, to the heart, to the sympathetic nervous system, to breathing, to stress hormones, and so on”. Attention Restoration Theory (A.R.T.) asserts that exposure to nature instantly and unconsciously produces significant improvements in cognitive functioning. Experiments conducted at the University of Michigan by psychologists Stephan and Rachel Kaplan and their colleagues are part of a rapidly growing body of evidence for A.R.T., showing a myriad of ways in which exposure to nature can contribute to brain health.
The geometry of nature – in fractals. Researchers at universities around the world have shown that looking at fractal shapes of mid-range complexity generates a sense of well-being and peace in the observer. Measurements of skin conductivity, as well as evaluations of EEG’s and fMRI’s, have led scientists to the conclusion that it is the fractal geometry which underlies the stress reduction and the many positive health benefits of viewing natural scenery. Links https://fractalfoundation.org/resources/what-are-fractals/ https://digitaldesignarch.wordpress.com/2018/11/11/natural-creativity/ https://fractalfoundation.org/fractivities/FractalPacks-EducatorsGuide.pdf |

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