geometry so intuitive that we are taught in school, based on lines, points and surfaces is really a great effort idealized abstraction because these elements do not exist in the everyday world. A real line or surface irregularities are filled overlook to abstract its essence and translate it into simpler concepts as straight and flat.
With fractals, in a way, get rid of this abstraction, and we come a little closer to the real thing. Benoit Mandelbrot uses the simple example of a real object, such as the coastal countries, to approximate fractals. Are broken lines still look like when we change of scale. Precisely these two properties are what define a fractal discontinuity (break, fracture, hence the name) and self-similarity with the change of scale. We measure the degree of fracture and irregularity with a single number called fractal dimension. Reviewing intuitively the concept of dimension, note that a point has no extent (size zero) to a line we measure in meters or linear feet, which means assigning a dimension (a single measure: length), to a surface must be measured in meters or square centimeters (size two: length and width) and a volume as measured in meters or cubic centimeters (three dimensions: length by width by height). A fractal generally have a dimension (fractal dimension) which is between zero and one, between one and two or between two and three.
Suppose the simplest case, a line represented by a thread fractal crumpled, and imagine that it has fractal dimension 1.25. If another thread has fractal dimension 1.35, the simple comparison of their fractal dimensions implies that the second thread is more wrinkled than the first, presents more irregularities. The whole part of the fractal dimension (in this case 1) is telling us that the object with which we deal is a line, we measured the fractional part of their degree of irregularity.
The fractal dimension also gives the ability for the purpose of occupying the space . The fractal dimension 1.35 thread is able to fill the plane dimension better than 1.25. In fact, if we wrinkles will increase the fractal dimension close to 2 when we managed to fill, almost entirely to surface thread. A classic fractal this type is called the Peano curve.
Fractals are basically simple objects, are easily generated by computer . By programming very few orders, and from a minimum number of data, create real wealth and wonders of extraordinary complexity. The Mandelbrot fractal is an example. As we try to expand, using information technology, any of its parts we find a new landscape similar to the original but with new and surprising details. We can go on as we want and allow us to power our computer, we continue to show a new fantasy world that is never repeated, with each new expansion. A world emerged from almost nothing, a simple expression that binds and new data feeds. Curiously, the expression is so simple: posterior value = (old) 2 + constant (with a restrictive condition).
The observation of these fractals created by computer, always reminds us of some natural object near unknown but possibly because the economy of means to achieve complexity is a very typical feature of Nature. It is the strategy adopted to achieve better distribution of blood vessels throughout the body, the optimum arrangement of the branches of trees or in the folds of the brain to get the largest area in the smallest space. Real
wonders of fractal art.
(*) In my work with Book of Notes, the monthly column cienciasyletras.
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