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These are animations of Julia set fractals. These objects look complex and intricate, but in fact they are defined through the simplest mathematical operations: repeated squaring and addition.

A fractal is a geometric figure with two special properties. First, it is irregular, fractured, fragmented, or loosely connected in appearance. Second, it is self-similar; that is, the figure looks much the same no matter how far away or how close up it is viewed.

One of the natural objects most often used to explain fractals is a coastline. A coastline has the three properties typical for any fractal figure. First, a coastline is irregular, consisting of bays, harbors, and peninsulas. By definition, any fractal must be irregular in shape.

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. This property is the self-similar property that also is basic to all fractals.

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 give only an estimate of its length. 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. Thus, 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.

http://www.encyclopedia.com/topic/fractal_geometry.aspx

A fractal is a geometric figure with two special properties. First, it is irregular, fractured, fragmented, or loosely connected in appearance. Second, it is self-similar; that is, the figure looks much the same no matter how far away or how close up it is viewed.

One of the natural objects most often used to explain fractals is a coastline. A coastline has the three properties typical for any fractal figure. First, a coastline is irregular, consisting of bays, harbors, and peninsulas. By definition, any fractal must be irregular in shape.

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. This property is the self-similar property that also is basic to all fractals.

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 give only an estimate of its length. 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. Thus, 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.

http://www.encyclopedia.com/topic/fractal_geometry.aspx

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2014-05-23

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