Fractals – Mathematical Philosophy

The word ‘geometry’ conjures up circles, cubes, cylinders, and other regular or smooth objects. Familiar artefacts, such as buildings, furniture, or cars, make wide use of such shapes. However, many phenomena in nature and science are anything but regular or smooth. For example, a natural landscape may include bushes, trees, rugged mountains, and clouds, which are far too intricate to be represented by classical geometric shapes.

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Surprisingly, apparently complex and irregular objects can often be described in remarkably simple terms. Fractal geometry provides a framework in which a simple process, involving a basic operation repeated many times, can give rise to a highly irregular result. Fractal constructions can represent natural objects but also give rise to a vast array of other shapes, which may be of extraordinary complexity. The phrase ‘the beauty of fractals’ is often heard, a phrase that reflects the unending intricacy of fractal designs alongside the simplicity, which underlies their ever-repeating form.

Indeed, complex but attractive fractal pictures have become an art from in their own right, with exhibitions, competitions, and their use on designer clothing.

Since ancient times, mathematics and science have developed alongside each other, with mathematics used to describe and often explain, observed natural and physical phenomena. In many areas this marriage has been highly successful, indeed much of what we enjoy in modern life is a consequence of its success. For example, the mathematical methods and laws introduced by Isaac Newton underlie the operation of almost everything mechanical, from riding a bicycle to the orbit of the spacecraft.

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A picture on a computer screen can only be created to remove the round using a mouse because a great deal of geometrical calculation has gone into designing the software.

Nevertheless, there are many phenomena which, although governed by the basic laws of science, were historically regarded as too irregular or complex to be described or analyzed using traditional mathematics.

Classical geometry concentrated on smooth or regular object such as circles, ellipses, cubes, or cones. The calculus, introduced by Newton and Leibniz in the second half of the 17th century, was an ideal tool for analyzing smooth objects and rapidly became so central both in mathematics and science that any attempt to consider irregular objects was sidelined. Indeed, many natural phenomena were overlooked, perhaps deliberately, because their irregularity and complexity made them difficult to describe in the form that was mathematically manageable.

Reference:

Fractals, a very short introduction

Oxford University Press, Sep 26, 2013

all documents are copyrights © Elias El Soueidi

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