Fractals
From NECSIWiki
Fractals are entities that look the same under magnification, they are "self-similar." More specifically, a geometric fractal is formed of parts, which, when magnified, are the same as the original shape. Fractals can also involve randomness, so that the similarity of parts to the whole can be of a statistical or average property.
An example of a geometric fractal is the Koch curve (an approximate version is shown below) which can be made using a simple algorithm: Take away the middle third of a line segment and insert two segments that are the same length as the one that was removed. Place them so they would make an equilateral triangle with the removed segment. Repeat this procedure with each of the line segments that now make up the curve.
Fractals get their name because they can be seen to have fractional dimension. Length, area and volume measure the size of one, two and three dimensional shapes respectively. However, a Koch curve can be shown to be infinite in length and to cover no area. It has a dimension between one and two (strictly its dimension is d=ln(4)/ln(3)~1.2619). This means that if we use a ruler to measure the length of the curve, the length we find depends on how long the ruler is. The shorter the ruler, the longer is the length we measure. This is because a longer ruler cannot "get inside" smaller bumps.
An example of a fractal-like shape found in nature is a coastline. How long is the coastline? Just as with the Koch curve, the length of the coastline depends on how long the ruler is used to measure it because there are inlets and peninsulas of many different sizes. A picture of the coastline of Cape Cod is shown below.
Other shapes that have fractal-like properties include trees or other branching structures, hierarchical organizational structures, and time series like stock market averages. In each case amplifying or enlarging a part makes it look similar to the whole because the internal structure (or time dependence) of each part is similar to the structure of the entirety.
Fractal shapes are important in modern science (since the 1970s) because much of traditional scientific thought is based on calculus. Calculus assumes that everything becomes smooth when you look at a fine enough scale. The recognition that there are characteristic shapes that do not become smooth was important. That these shapes can be studied and analyzed mathematically and observed in nature has led to great changes in science.
[edit] Related concepts:
scale, separation of scales, chaos, chaos and fractals, complexity at different scales, complexity profile
[edit] References
Mandelbrot, Benoît B., The Fractal Geometry of Nature. (W. H. Freeman and Co., New York, 1982)
