FRACTAL GEOMETRY
The formal mathematical definition of fractal geometry is by Benoit Mandelbrot. It says that a fractal is a set for which the Haudorff Besicovitch dimension strictly exceeds the topological dimension .However, this is a very abstract definition. Generally, we can define a fractal as a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced -size copy of the whole. Fractals are generally self -similar and independent of scale.
Fractal geometry is recent synthesis of old mathematical constructs. It is a misunderstood idea that is quickly becoming buried under grandiose terminology that serves no purpose. Its essence is induction using simple geometric constructs to transform initiating objects. The fractal objects that we create with this process often resemble natural phenomenon.
The strength of Mandelbrot ‘s
findings was his research into the findings of the earlier mathematicians and development of a practical application of their theory. Mandelbrot coined the term fractal to describe a class of functions first discovered by Cantor, Koch and Piano. He showed these functions yield valuable insight into the creation of models for natural objects such as coast lines and mountains. Mandelbrot popularized the notion of fractal geometry for these types of objects. Although he did not invent the ideas he presents,Mandelbrot must considered important because of his synthesis of the theory at a time when science was reaching out for new more accurate models to describe it's process.
The following are some properties of fractals :A fractal is a geometric figure or natural object that combines the following characteristics
a) It's parts have the same form or structure as the whole, except that they are at a different scale and may be slightly deformed
b) It's form is extremely irregular or fragmented and remains so, what ever the scale of examination
c) It contains "distinct elements "whose scales are very varied and cover a large range
d) Formation by itration
Fractal geometry is based on the idea of self similar forms. To be self similar, a shape must be able to be divided into parts that are smaller copies which are more or less similar to the whole. Because of the smaller similar divisions of fractals.,they appear similar at all magnification. However while fractals are self similar, not all self similar forms are fractals. Fractals often have a finite boundary that determines the area that it can take up, but the perimeter of the fractal continuously grows and is finite.
The formal mathematical definition of fractal geometry is by Benoit Mandelbrot. It says that a fractal is a set for which the Haudorff Besicovitch dimension strictly exceeds the topological dimension .However, this is a very abstract definition. Generally, we can define a fractal as a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced -size copy of the whole. Fractals are generally self -similar and independent of scale.
Fractal geometry is recent synthesis of old mathematical constructs. It is a misunderstood idea that is quickly becoming buried under grandiose terminology that serves no purpose. Its essence is induction using simple geometric constructs to transform initiating objects. The fractal objects that we create with this process often resemble natural phenomenon.
The strength of Mandelbrot ‘s
findings was his research into the findings of the earlier mathematicians and development of a practical application of their theory. Mandelbrot coined the term fractal to describe a class of functions first discovered by Cantor, Koch and Piano. He showed these functions yield valuable insight into the creation of models for natural objects such as coast lines and mountains. Mandelbrot popularized the notion of fractal geometry for these types of objects. Although he did not invent the ideas he presents,Mandelbrot must considered important because of his synthesis of the theory at a time when science was reaching out for new more accurate models to describe it's process.
The following are some properties of fractals :A fractal is a geometric figure or natural object that combines the following characteristics
a) It's parts have the same form or structure as the whole, except that they are at a different scale and may be slightly deformed
b) It's form is extremely irregular or fragmented and remains so, what ever the scale of examination
c) It contains "distinct elements "whose scales are very varied and cover a large range
d) Formation by itration
Fractal geometry is based on the idea of self similar forms. To be self similar, a shape must be able to be divided into parts that are smaller copies which are more or less similar to the whole. Because of the smaller similar divisions of fractals.,they appear similar at all magnification. However while fractals are self similar, not all self similar forms are fractals. Fractals often have a finite boundary that determines the area that it can take up, but the perimeter of the fractal continuously grows and is finite.
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