Wednesday, March 20, 2019
Fractal Geometry :: essays papers
Fractal GeometryThe world of mathematics usu onlyy tends to be thought of as abstract.building complex and imaginary numbers, real numbers, logarithms, functions, sometangible and others imperceivable. But these abstract numbers, exclusivelysymbols that conjure an image, a quantity, in our mind, and complexequations, take on a new meaning with fractals - a concrete one.Fractals go from being truly simple equations on a piece of paper tocolorful, extraordinary images, and near of all, offer an explanation tothings. The importance of fractal geometry is that it provides ananswer, a comprehension, to nature, the world, and the universe.Fractals occur in swirls of scum on the surface of moving water, thejagged edges of mountains, ferns, tree trunks, and batchyons. They can beused to model the growth of cities, detail medical procedures and partsof the merciful body, create amazing computer graphics, and compressdigital images. Fractals are nigh us, and our existence, and they arepr esent in every mathematical law that governs the universe. Thus,fractal geometry can be applied to a diverse palette of subjects inlife, and wisdom - the physical, the abstract, and the natural.We were all astounded by the sudden revelation that the output of avery simple, two-line generating formula does not have to be a change andcold abstraction. When the output was what is now called a fractal,no one called it artificial... Fractals suddenly broadened the realmin which understanding can be based on a plain physical basis.(McGuire, Foreword by Benoit Mandelbrot)A fractal is a geometric shape that is complex and detailed at every take aim of magnification, as well as self-similar. Self-similarity issomething looking the same over all ranges of scale, meaning a smallportion of a fractal can be viewed as a microcosm of the larger fractal.One of the simplest examples of a fractal is the snowflake. It isconstructed by fetching an equilateral triangle, and after many iterationsof adding smaller triangles to increasingly smaller sizes, resulting ina snowflake pattern, sometimes called the von Koch snowflake. Thetheoretical result of multiple iterations is the groundwork of a finitearea with an infinite perimeter, meaning the dimension isincomprehensible. Fractals, sooner that word was coined, were simplyconsidered above mathematical understanding, until experiments were donein the 1970s by Benoit Mandelbrot, the father of fractal geometry.Mandelbrot developed a method that treated fractals as a part ofstandard Euclidean geometry, with the dimension of a fractal being anexponent. Fractals strike an infinity into a grain of sand. This infinity appearswhen one tries to round them.