Fractals

Fractals are  mathematical sets that exhibits  repeating patterns displayed at every scale. 

They are also known as expanding symmetry or evolving symmetries. If the replication is exactly the same at every scale, it is called a self-similar pattern. An example of this is the Menger Sponge. 

Mengersponge

A Menger Sponge. image: Mattcomm/ wikipedia

Fractals can also be nearly the same at different levels. This latter pattern is illustrated in small magnifications of the Mandelbrot set. 

Mandelbrot-similar-x1

A mandelbrot set is fractal pattern

Mandelbrot-similar-x6

that shows self-similarity at smaller scales

Fractals also include the idea of a detailed pattern that repeats itself.

Von_Koch_curve

A koch 'curve' shows self-similarity as each side is broken evenly. image: wikipedia

How they are different from other geometries

Fractals are different from other geometric figures because of the way in which they scale. Doubling the edge lengths of a polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the dimension of the space the polygon resides in). 

Likewise, if the radius of a sphere is doubled, its volume scales by eight, which is two (the ratio of the new to the old radius) to the power of three (the dimension that the sphere resides in). 

But if a fractal's one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily an integer.  This power is called the fractal dimension of the fractal, and it usually exceeds the fractal's topological dimension. 

As mathematical equations, fractals are usually nowhere differentiable.  An infinite fractal curve can be conceived 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. 


Historical origins of fractals

The mathematical roots of the idea of fractals have been traced throughout the years as a formal path of published works, starting in the 17th century. 

This started with 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 by the seminal work of Bernard Bolzano, Bernhard Riemann, and Karl Weierstrass. 

Computers become critical in visualisations of fractals

This continued 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 20th century.

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.

Experts don’t have a formal definition 

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.” 

 The general consensus is that theoretical fractals are infinitely self-similar, iterated, and detailed mathematical constructs having fractal dimensions. 

Other processes where fractals form

Fractals are not limited to geometric patterns, but can also describe processes in time. Fractal patterns with various degrees of self-similarity have been rendered or studied in images, structures and sounds and found in nature, technology,  art, and law. 

KarperienFractalBranch

Self-similar branching pattern modeled in silico using L-systems principles. image: wikipedia

 Fractals are of particular relevance in the field of chaos theory, since the graphs of most chaotic processes are fractal.

Fractal_fern_explained

Barnsley's fern, an early IFS(iterated function systems) shows natural looking  self similarity image: wikipedia

Natural phenomena with fractal features

Here are a few examples of fractal like structures in nature

Fractal_Broccoli

Romanesco broccoli, showing self-similar form approximating a natural fractal. image: Jon Sullivan/wikipedia

Animated_fractal_mountain

A fractal that models the surface of a mountain (animation). image:António Miguel de Campos/wikipedia 


Source adapted from: Fractal. (2017, January 3). In Wikipedia, The Free Encyclopedia. Retrieved 18:51, January 3, 2017, from https://en.wikipedia.org/w/index.php?title=Fractal&oldid=758143234