“fractal dimension”的概念、定义、翻译、参考文献-科学参考

fractal dimension

Mathematics

One of the many extensions (see Hausdorff dimension) of the notion of dimension to objects for which the traditional concept of dimension is not appropriate. The fractal dimension may have a non‐integer value. The Koch curve has dimension log₃4 ≈ 1.26. The Cantor set has dimension log₃2 ≈ 0.63. Fractal dimension has found applications analysing chaotic or noisy processes (see chaos).

Statistics

A generalization of the usual idea of dimension that permits non-integer values. For a plane set of points, the fractal dimension is estimated as follows. A lattice of square boxes of side s is superimposed over the set. The number, N(s), of these boxes that contain a portion of the set is determined. This is repeated for a range of values of s. When ln{N(s)} is plotted against −ln(s) the result will be an approximate straight line. The fractal dimension is the slope of this line. An ordinary curve has a fractal dimension equal to 1. Any set of points in a plane has a fractal dimension not greater than 2.

单词	fractal dimension
释义	fractal dimension Mathematics One of the many extensions (see Hausdorff dimension) of the notion of dimension to objects for which the traditional concept of dimension is not appropriate. The fractal dimension may have a non‐integer value. The Koch curve has dimension log₃4 ≈ 1.26. The Cantor set has dimension log₃2 ≈ 0.63. Fractal dimension has found applications analysing chaotic or noisy processes (see chaos). Statistics A generalization of the usual idea of dimension that permits non-integer values. For a plane set of points, the fractal dimension is estimated as follows. A lattice of square boxes of side s is superimposed over the set. The number, N(s), of these boxes that contain a portion of the set is determined. This is repeated for a range of values of s. When ln{N(s)} is plotted against −ln(s) the result will be an approximate straight line. The fractal dimension is the slope of this line. An ordinary curve has a fractal dimension equal to 1. Any set of points in a plane has a fractal dimension not greater than 2.
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