Fractal
A fractal is a manifold which exhibits a kind of self-similarity; pieces of the manifold bear a resemblance to the whole, down to arbitrarily small scales. Generally manifolds which are smooth and continuous are excluded from consideration, therefore ruling out trivial cases such as places. Some fractals are precisely identical to scaled-up versions of their parts, but such exact correspondence is not a requirement; other fractals bear a close resemblance to their parts, but not precise conformity. Nor is it necessarily the case that any arbitrary piece of the whole bears it a resemblance; the self-similarity may be visible only by choosing certain parts.
Examples
The self-similarity inherent in fractals often arises from recursive and iterative equations and algorithms. Such an equation can be used to construct one of the best-known fractals of all, the Mandelbrot set, named after Benoit Mandelbrot, the mathematician primarily responsible, on True Earth, for developing the concept of the fractal (as well as coining the word "fractal"). The Mandelbrot set is the set of all points in the complex plane for which the recursive equation [math]\displaystyle{ z_{n+1}=z_n^2+c }[/math] remains bounded. Related to the Mandelbrot set are Julia sets, defined as the boundary of the set of points that remain bounded after repeated application of some rational function—though not all rational functions give rise to a fractal. The bifurcation diagram of the logistic map, one of the best known cornerstones of chaos theory, is also such a fractal, arising from the stable solutions to the simple recursive equation [math]\displaystyle{ x_{n+1}=r x_n(1-x_n) }[/math]. In fact, chaos theory is full of strange attractors with similarly fractal properties.
Recursive equations are not the only way to produce a fractal mathematically, however; some relatively simple non-recursive functions also yield fractal graphs, such as the cathedral function, the Weierstrass function, and the question mark function (though the last can be defined recursively for rational arguments between zero and one).
Many fractals are defined by recursive geometric procedures... though equivalent mathematical operations could be defined to produce them, the geometric explanation is generally easier to visualize (or at least more common). The procedure can be formalized by a grammar called a Lindenmayer system. One of the simplest fractals formulated this way is the Cantor set, generated by starting with a finite line segment, then removing the middle third, then removing the middle third of each remaining segment, and so forth; what is left after infinitely many iterations is a disconnected set of points with a self-similar distribution. Performing a similar operation with a square in two dimensions produces a fractal called the Sierpiński carpet, and in three dimensions the Menger sponge. Similar to the Sierpiński carpet but using triangles instead of squares is the Sierpiński gasket. Another fractal similar to the Sierpiński carpet, but produced by removing at each step four "edge" squares instead of the center square, is the box fractal. Some fractals are similarly produced but by adding pieces rather than subtracting them, such as the Koch snowflake and the Pythagoras tree. (Naturally, many fractals can be produced by several different but equivalent algorithms.) One fractal produced geometrically using circles instead of polygons is the Apollonian gasket.
A large number of familiar fractals are produced by taking a particular curve (usually one composed of straight line segments) and replacing parts of it with smaller, similar curves, then iteratively repeating the process. Some of the best known fractals produced this way include the Cesàro fractal, the Koch snowflake, the Lévy curve, and the Minkowski sausage. Some fractals produced this way at their limit occupy every point in space (or at least every point within a particular area), and are accordingly called space-filling curves; examples include the Gosper curve, the H tree, the Hilbert curve, the Heighway dragon, and the Peano curve.
Many aperiodic tilings contain properties of fractals as well, particularly if each tile at one scale is regarded as a pattern of tiles at a smaller scale (as is possible with many such tilings).
Fractal sequences also exist consisting entirely of integers. According to a definition formulated by the mathematician Clark Kimberling, a sequence is a fractal sequence if every natural number appears in it infinitely many times, smaller numbers have earlier first appearances, and each number appears exactly once in the sequence between any two consecutive appearances of each larger number. Despite this seemingly restrictive definition, many examples of such sequences exist, including the Kimberling triangle sequences, the signature sequences of positive irrational numbers, and the sequences comprising the rows in which each successive natural number appears in an interspersion. There are also, however, integer sequences that have fractal properties but do not meet all Kimberling's criteria, such as the Thue-Morse sequence and skyline sequences. Furthermore, cosmic partitions form an example of arrays of numbers with fractal properties although no single row or column in the array is fractal in any meaningful sense.
Most of the fractals mentioned here can be varied in numerous ways, by altering certain parameters or by generalizing to higher dimensions. In addition, many other fractals completely different from these have been described, and countless more possibilities exist that have not been.
Fractal dimension
While most familiar shapes and objects have an integral number of dimensions, many fractals are an exception. Even fractals that still have an integral number of dimensions in the topological sense may have a different number of dimensions according to some definitions. A line unambiguously has one dimension, a plane two dimensions, but, for example, the boundary of the Gosper curve has in a sense a dimension of about 1.13, the Apollonian gasket has a dimension of about 1.3, and the Sierpiński carpet has a dimension of about 1.9. Dimensions less than one or greater than two are also possible.
There are a number of different ways to measure the fractal dimension, which in some cases may not be completely equivalent. The numbers above correspond to the Hausdorff dimension, which is defined, speaking roughly, as the additive inverse of the limit, as r goes to infinity, of the ratio of the minimum number of spheres of radius r it will take to completely cover the shape to r. For self-similar fractals, the Hausdorff dimension can be more easily estimated by the following equivalent procedure: if you can duplicate a shape by putting together N copies of it, each scaled down by a linear factor of r, then the Hausdorff dimension is the additive inverse of the ratio of the logarithm of N to the logarithm of r. Smooth and continuous shapes have Hausdorff dimensions equal to their topological dimensions, but for fractals this dimension may not be an integer. For instance, the Cantor set can be reproduced by putting together two copies of it, each scaled down by a factor of 1/3, so its Hausdorff dimension is equal to [math]\displaystyle{ log (2) / log (3) \approx 0.63 }[/math].
Not all fractals, however, have fractional Hausdorff dimensions. Most familiar fractals do, but the blancmange curve, for instance, has both a fractal and a topological dimension of one, and the Mandelbrot set has both fractal and topological dimensions of two. There are some fractal shapes that have Hausdorff dimensions and topological dimensions that are both integers, but different integers. The tetrix, a three-dimensional extrapolation of the Sierpiński gasket, has a topological dimension of three and a fractal dimension of two. The space-filling curves have a topological dimension of one and a fractal dimension of two (or more, if extended into higher dimensions); the same is true of the boundary of the Mandelbrot set.
Fractals in real life
Many objects in real life have an approximately fractal form. Many coastlines, with their meandering contours, are self-similar at many scales, and have fractal properties. Many trees and other branching structures likewise have fractal characteristics. Mountains, clouds, and lightning all have fractal qualities. It's possible to estimate the fractal dimensions of these objects and phenomena. The fractal dimension of the coastline of Taiwan has been estimated as 1.04, that of Australia as 1.13, that of Great Britain as 1.24. The eastern white pine has been estimated to have a fractal dimension of 2.24, the Japanese yew of 2.45[1].
There are limits to the scales at which fractals are good fits to such real-life phenomena, of course. For coastlines, for instance, the approximation breaks down at the level of individual grains of sand, if not before, and has an obvious upper limit at the perimeter of the landmass in question. For trees and other branching plants, there is a clear limit as to just how small their twigs and branches can get. Still, despite its limitations, the fractal model can be a fairly good fit in some cases for several orders of magnitude.
Applications
The fact that fractals are good approximations to real-life phenomena (at least at certain scales) makes them useful for modeling and simulating such phenomena. Fractals can generate passably realistic models and images of coastlines, clouds, and so on. Fractal landscapes can be randomly generated for computer games. One particularly useful kind of fractal for these purposes is the plasma fractal, which incorporates a random element that avoids the obvious regularity of many of the classic types of fractal. These fractals can also be "tuned" by various parameters to affect such matters as the irregularity of generated coastlines, the height and jaggedness of generated mountains, et cetera.
Fractals have also found use in art. Abstract fractals can have an aesthetic appeal of their own, but the self-similarity algorithms, with or without randomizing elements, can also be applied to images and three-dimensional models to produce complex and fantastic structures and landscapes and other designs. Fractal art has been compared to photography; both could be thought of, in a sense, as the skilled capturing (and perhaps manipulation) of an image that already exists, though in the case of fractals this existence occurs in an abstract mathematical realm rather than in physical reality.