In mathematics, a fractal is a subset of a Euclidean space for which the Hausdorff dimension strictly exceeds the topological dimension. Fractals are encountered ubiquitously in nature due to their tendency to appear nearly the same at different levels, as is illustrated here in the successively small magnifications of the Mandelbrot set. Fractals exhibit similar patterns at increasingly small scales, also known as expanding symmetry or unfolding symmetry; If this replication is exactly the same at every scale, as in the Menger sponge, it is called affine self-similar.
One way that fractals are different from finite geometric figures is 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). However, 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 – although it is still 1-dimensional, its fractal dimension indicates that it also resembles a surface.
The mathematical roots of fractals have been traced throughout the years as 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 by the seminal work of Bernard Bolzano, Bernhard Riemann, and Karl Weierstrass, 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 20th century. The term "fractal" was first used by mathematician Benoit 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.
There is some disagreement among mathematicians about how the concept of a fractal should be formally defined. Mandelbrot himself summarized it as "beautiful, damn hard, increasingly useful. That's fractals." More formally, in 1982 Mandelbrot stated that "A fractal is by definition a set for which the Hausdorff–Besicovitch dimension strictly exceeds the topological dimension." Later, seeing this as too restrictive, he simplified and expanded the definition to: "A fractal is a shape made of parts similar to the whole in some way." Still later, Mandelbrot settled on this use of the language: "...to use fractal without a pedantic definition, to use fractal dimension as a generic term applicable to all the variants".
The 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. 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, architecture and law. Fractals are of particular relevance in the field of chaos theory, since the graphs of most chaotic processes are fractals.
XaoS is an interactive fractal zoomer program. It allows the user to continuously zoom in or out of a fractal in real-time.
XaoS is licensed under GPL. The program is cross-platform, and is available for a variety of operating systems, including GNU/Linux, Windows, Mac OS X, BeOS and others.
XaoS can show the Mandelbrot set (power 2, 3, 4, 5 and 6), the Octo fractal, three types of Barnsley's fractals, the Newton fractal (order 3 and 4), Phoenix fractal and Magnet (1 and 2). XaoS can display Julia sets from selected fractal parts. Xaos also allows users to enter custom formulas.
XaoS is capable of displaying fractals as ASCII art using AAlib, which, in combination with being built on freely available GNU tools, allows it to run almost anywhere.
An interactive help and an animated introduction to fractals are included. The introduction deals in ten chapters with different formulas presented in the software and their features.
FractalNow provides users with tools to generate pictures of various types of fractals quickly and easily.
It is made of both a command line tool, FractalNow, and a graphical tool, QFractalNow.
The graphical tool, based on Qt library, allows users to explore fractals intuitively and generate pictures.
Both tools are entirely multi-threaded and implement advanced algorithms and heuristics that make computation very fast compared to most existing free fractal generators.
Mandelbulber is an easy to use, handy application designed to help you render 3D Mandelbrot fractals called Mandelbulb and some other kind of 3D fractals like Mandelbox, Bulbbox, Juliabulb, Menger Sponge
Mandelbulber v2 is new line of this application. It was rewritten from scratch with new Qt based interface. Functionality is still limited, but this is only matter of time.
Gnofract 4D is a free, open source program which allows anyone to create beautiful images called fractals. The images are automatically created by the computer based on mathematical principles. These include the Mandelbrot and Julia sets and many more. You don’t need to do any math: you can explore a universe of images just using a mouse. It runs on Unix-based systems such as Linux and FreeBSD and can also be run on Mac OS X.
Fracplanet is an interactive tool for creating random fractal planets and terrain areas with oceans, rivers, lakes and icecaps. The results can be exported as models to POV-Ray and to Blender, or as texture maps for more general usage. The code is licensed under the GPL. It uses Qt and OpenGL.
Flames are algorithmically generated images and animations. The software was originally written in 1992 and released as open source, aka free software. Over the years it has been greatly expanded, and is now widely used to create art and special effects. The shape and color of each image is specified by a long string of numbers – a genetic code of sorts.