Fractal, in mathematics, a geometric shape that is complex and detailed in structure at any level of magnification.

Often fractals are self-similar—that is, they have the property that each small portion of the fractal can be viewed as a reduced-scale replica of the whole.

One example of a fractal is the "snowflake" curve constructed by taking an equilateral triangle and repeatedly erecting smaller equilateral triangles on the middle third of the progressively smaller sides.

Theoretically, the result would be a figure of finite area but with a perimeter of infinite length, consisting of an infinite number of vertices.

 In mathematical terms, such a curve cannot be differentiated (see Calculus).

 Many such self-repeating figures can be constructed, and since they first appeared in the 19th century they have been considered as merely bizarre.

A turning point in the study of fractals came with the discovery of fractal geometry by the Polish-born French mathematician Benoit B. Mandelbrot  in the 1970s.

Mandelbrot adopted a much more abstract definition of dimension than that used in Euclidean geometry, stating that the dimension of a fractal must be used as an exponent when measuring its size.

The result is that a fractal cannot be treated as existing strictly in one, two, or any other whole-number dimensions.

Instead, it must be handled mathematically as though it has some fractional dimension. The "snowflake" curve of fractals has a dimension of 1.2618.

Fractal geometry is not simply an abstract development. A coastline, if measured down to its least irregularity, would tend toward infinite length just as does the "snowflake" curve.

Mandelbrot has suggested that mountains, clouds, aggregates, galaxy clusters, and other natural phenomena are similarly fractal in nature, and fractal geometry's application in the sciences has become a rapidly expanding field.

Benoit B. Mandelbrot founded a new branch of mathematics, fractal geometry.

In conventional geometry, an object’s dimension is expressed in whole numbers; a line, for example, is one-dimensional and plane has two dimensions.

In fractal geometry, objects may have "fractional" dimensions.

For example, a fractal image may have a border that is infinitely detailed, and thus, a dimension between one and two. ^[1]

 

 In addition, the beauty of fractals has made them a key element in computer graphics.

Fractals have also been used to compress still and video images on computers.

In 1987, English-born mathematician Dr. Michael F. Barnsley discovered the Fractal Transform^TM which automatically detects fractal codes in real-world images (digitized photographs).

The discovery spawned fractal image compression, used in a variety of multimedia and other image-based computer applications.

 

Contributed By:
Benoit B. Mandelbrot
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