Fractals and Scale Invariance

 

Fractals are plots of non-linear equations (equations in the result is used as the next input) which can build up to astonishingly complex and beautiful designs.  Typical of fractals is their scale invariance, which means that no matter how you view them, zoom in or zoom out, you will find self-similar and repeating geometric patterns.  This distinguishes from most natural patterns (though some are fractal-like, such as mountains or the branches of a tree, stars in the sky), in which as you zoom in or out completely changes what you see -- e.g., galaxies to stars down to atoms and sub-atomic nuclei and so forth).  Nevertheless, what is most (to me) fascinating about fractals is that they allow us to simulate reality in so many ways.

The Mandelbrot set shown above is the most famous example of fractal design known. 
The Mandelbrot set is a mathematical set of points whose boundary is a distinctive and easily recognizable two-dimensional fractal shape. The set is closely related to Julia sets (which include similarly complex shapes), and is named after the mathematician Benoit Mandelbrot, who studied and popularized it.

Mandelbrot set images are made by sampling complex numbers and determining for each whether the result tends towards infinity when a particular mathematical operation is iterated on it. Treating the real and imaginary parts of each number as image coordinates, pixels are colored according to how rapidly the sequence diverges, if at all.

More precisely, the Mandelbrot set is the set of values of c in the complex plane for which the orbit of 0 under iteration of the complex quadratic polynomial

z n+1 =z n squared +c

remains bounded.^[1] That is, a complex number c is part of the Mandelbrot set if, when starting with z₀ = 0 and applying the iteration repeatedly, the absolute value of z_n remains bounded however large n gets.

In general, a fractal is a mathematical set that typically displays self-similar patterns, which means they are "the same from near as from far".^[1] Often, they have an "irregular" and "fractured" appearance, but not always. Fractals may be exactly the same at every scale, or they may be nearly the same at different scales.^[2][3][4][5] The definition of fractal goes beyond self-similarity per se to exclude trivial self-similarity and include the idea of a detailed pattern repeating itself.^{[2]:166; 18[3][6]}

One feature of fractals that distinguishes them from "regular" shapes is the amount their spatial content scales, which is the concept of fractal dimension. If the edge lengths of a square are all doubled, the area is scaled by four because squares are two dimensional, similarly if the radius of a sphere is doubled, its volume scales by eight because spheres are three dimensional. In the case of fractals, if all one-dimensional lengths are doubled, the spatial content of the fractal scales by a number which is not an integer. A fractal has a fractal dimension that usually exceeds its topological dimension^[7] and may fall between the integers.^[2]

As mathematical equations, fractals are usually nowhere differentiable.^[2][5][8] An infinite fractal curve can be perceived of as winding through space differently from an ordinary line, still being a 1-dimensional line yet having a fractal dimension indicating it also resembles a surface.^[7]:48[2]:15

The mathematical roots of the idea of fractals have been traced through a formal path of published works, starting in the 17th century with notions of recursion, then moving through increasingly rigorous mathematical treatment of the concept to the study of continuous but not differentiable functions in the 19th century, and on to the coining of the word fractal in the 20th century with a subsequent burgeoning of interest in fractals and computer-based modelling in the 21st century.^[9][10] The term "fractal" was first used by mathematician Benoît Mandelbrot in 1975. Mandelbrot based it on the Latin frāctus meaning "broken" or "fractured", and used it to extend the concept of theoretical fractional dimensions to geometric patterns in nature.^[2]:405[6]

There is some disagreement amongst authorities about how the concept of a fractal should be formally defined. Mandelbrot himself summarized it as "beautiful, damn hard, increasingly useful. That's fractals."^[11] The general consensus is that theoretical fractals are infinitely self-similar, iterated, and detailed mathematical constructs having fractal dimensions, of which many examples have been formulated and studied in great depth.^[2][3][4] Fractals are not limited to geometric patterns, but can also describe processes in time.^[1][5][12] Fractal patterns with various degrees of self-similarity have been rendered or studied in images, structures and sounds^[13] and found in nature, technology, art, and law.

Much of this was taken from Wikipedia Mandelbrot Set and Fractal.