# Fractal

### dimension length coastline factor

A fractal is a geometric figure, often characterized as being self-similar; that is, irregular, fractured, fragmented, or loosely connected in appearance. Benoit Mandelbrot coined the term fractal to describe such figures, deriving the word from the Latin "fractus" meaning broken, fragmented, or irregular. He also pointed out amazing similarities in appearance between some fractal sets and many natural geometric patterns. Thus, the term "natural fractal" refers to natural phenomena that are similar to fractal sets, such as the path followed by a dust particle as it bounces about in the air.

Another good example of a natural phenomenon that is similar to a fractal is a coastline, because it exhibits three important properties that are typical of fractals. First, a coastline is irregular, consisting of bays, harbors, and peninsulas. Second, the irregularity is basically the same at all levels of magnification. Whether viewed from **orbit** high above **Earth**, from a helicopter, or from land,

whether viewed with the naked **eye**, or a magnifying **glass**, every coastline is similar to itself. While the patterns are not precisely the same at each level of magnification, the essential features of a coastline are observed at each level. Third, the length of a coastline depends on the magnification at which it is measured. Measuring the length of a coastline on a photograph taken from space will only give an estimate of the length, because many small bays and peninsulas will not appear, and the lengths of their perimeters will be excluded from the estimate. A better estimate can be obtained using a photograph taken from a helicopter. Some detail will still be missing, but many of the features missing in the space photo will be included, so the estimate will be longer and closer to what might be termed the "actual" length of the coastline. This estimate can be improved further by walking the coastline wearing a pedometer. Again, a longer measurement will result, perhaps more nearly equal to the "actual" length, but still an estimate, because many parts of a coastline are made up of **rocks** and pebbles that are smaller than the length of an average stride. Successively better estimates can be made by increasing the level of magnification, and each successive measurement will find the coastline longer. Eventually, the level of magnification must achieve atomic or even nuclear resolution to allow measurement of the irregularities in each grain of **sand**, each clump of dirt, and each tiny pebble, until the length appears to become infinite. This problematic result suggests the length of every coastline is the same.

The resolution of the problem lies in the fact that fractals are properly characterized in terms of their dimension, rather than their length, area, or **volume**, with typical fractals described as having a dimension that is not an integer. To explain how this can happen, it is necessary to consider the meaning of dimension. The notion of dimension dates from the ancient Greeks, perhaps as early as Pythagoras (582-507 B.C.) but at least from Euclid (c. 300 B.C.) and his books on **geometry**. Intuitively, we think of dimension as being equal to the number of coordinates required to describe an object. For instance, a line has dimension 1, a **square** has dimension 2, and a cube has dimension 3. This is called the topological dimension. However, between the years 1875 and 1925, mathematicians realized that a more rigorous definition of dimension was needed in order to understand extremely irregular and fragmented sets. They found that no single definition of dimension was complete and useful under all circumstances. Thus, several definitions of dimension remain today. Among them, the Hausdorf dimension, proposed by Felix Hausdorf, results in fractional dimensions when an object is a fractal, but is the same as the topological value of dimension for regular geometric shapes. It is based on the increase in length, area, or volume that is measured when a fractal object is magnified by a fixed scale **factor**. For example, the Hausdorf dimension of a coastline is defined as D = log(Length Increase)/log(scale factor). If the length of a coastline increases by a factor of four whenever it is magnified by a factor of three, then its Hausdorf dimension is given by log(Length Increase)/log(scale factor) = log(4)/log(3) = 1.26. Thus, it is not the length that properly characterizes a coastline but its Hausdorf dimension. Finally, then, a fractal set is defined as a set of points on a line, in a **plane**, or in space, having a fragmented or irregular appearance at all levels of magnification, with a Hausdorf dimension that is strictly greater than its topological dimension.

Great interest in fractal sets stems from the fact that most natural objects look more like fractals than they do like regular geometric figures. For example, **clouds**, trees, and **mountains** look more like fractal figures than they do like circles, triangles, or pyramids. Thus, fractal sets are used by geologists to model the meandering paths of **rivers** and the rock formations of mountains, by botanists to model the branching patterns of trees and shrubs, by astronomers to model the distribution of **mass** in the universe, by physiologists to model the human **circulatory system**, by physicists and engineers to model turbulence in fluids, and by economists to model the stock market and world economics. Often times, fractal sets can be generated by rather simple rules. For instance, a fractal dust is obtained by starting with a line segment and removing the center one-third, then removing the center one-third of the remaining two segments, then the center one-third of those remaining segments and so on.

Rules of generation such as this are easily implemented and displayed graphically on computers. Because some fractal sets resemble mountains, islands, or coastlines, while others appear to be clouds or snowflakes, fractals have become important in graphic art and the production of special effects. For example, "fake" worlds, generated by computer, are used in science fiction movies and **television** series, on CD-ROMs, and in video games, because they are easily generated from a set of instructions that occupy relatively little computer memory.

## Resources

### Books

Peterson, Ivars. *Islands of Truth, A Mathematical Mystery* *Cruise.* New York: W. H. Freeman, 1990.

J. R. Maddocks

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