Fractals

The word “fractal” derives from the Latin word “frangere,” which means “to break.” The field marks a whole new geometrical modeling paradigm that can remarkably capture the complexity of shapes and textures found in nature, such as clouds, forests, trees, flowers, galaxies, leaves, feathers, rocks, mountains, coastlines, and even blood vessels. The study of fractals involves the notions of self-similarity, repetitive iteration, and fractional dimension. In particular, fractal geometry generalizes ordinary notions of length, scale, and dimension in interesting and subtle ways. For example, the 1-D length of a coastline depends on how finely it is measured, a 2-D spiral can be of either infinite or finite length yet occupy a finite area, and a 3-D space can be densely filled without using an ordinary solid.

Fractals require the generalization of the concept of dimension (allowing it to be noninteger) that in turn is intimately tied to the notion of scale. Fractal objects are typically generated through some form of iterative feedback employing simple geometrical or dynamic rules. At each iteration, there may be a set of rules to choose from at random, or a parameter that can take a chosen random value. Under general conditions, continued iteration will converge on a final set that is the fractal.

The study of fractals was revolutionized and popularized by the publication in 1977 of Benoit Mandelbrot’s book, The Fractal Geometry of Nature. Since then, the field has grown immensely, with many fruitful applications and even commercialization in areas such as data/video compression, frequency-independent antennas/arrays, computer generated imagery, random process and probability modeling, image half-toning, and cluster and crack propagation analysis.

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