Fractal Geometry :: essays papers

Fractal GeometryThe world of mathematics ordinarily tends to be thought of as abstract.Complex and imaginary numbers, real numbers, logarithms, functions, approximatelytangible and others imperceivable. But these abstract numbers, simplysymbols that conjure an image, a quantity, in our mind, and interlockingequations, take on a new meaning with fractals - a concrete one.Fractals go from being very simple equations on a piece of paper tocolorful, extraordinary images, and most of all, offer an explanation tothings. The grandeur 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 woful water, thejagged edges of mountains, ferns, tree trunks, and canyons. They can beused to model the exploitation of cities, detail medical procedures and partsof the human body, create amazing information processing system graphics, and compressdigital images. Fractals are about us, and our exis tence, and they arepresent in every mathematical law that governs the universe. Thus,fractal geometry can be applied to a diverse palette of subjects inlife, and science - the material, 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 dry andcold abstraction. When the output was what is this instant called a fractal,no one called it artificial... Fractals suddenly broadened the realmin which fellow feeling 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 everylevel of magnification, as well as self-similar. Self-similarity issomething looking the same over all ranges of scale, meaning a pocketableportion of a fractal can be viewed as a microcosm of the bigger fractal.One of the simplest examples of a fractal is the snowflake. It isconstructed by taking an equilateral triangle, and after some(prenominal) 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 creation of a finite compass with an infinite perimeter, meaning the dimension isincomprehensible. Fractals, before that word was coined, were simplyconsidered higher up 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 euclidian geometry, with the dimension of a fractal being anexponent. Fractals pack an infinity into a grain of sand. This infinity appearswhen one tries to measure them.