fractal dimension

(mathematics)A common type of fractal dimension is theHausdorff-Besicovich Dimension, but there are severaldifferent ways of computing fractal dimension. Fractaldimension can be calculated by taking the limit of thequotient of the log change in object size and the log changein measurement scale, as the measurement scale approacheszero. The differences come in what is exactly meant by"object size" and what is meant by "measurement scale" and howto get an average number out of many different parts of ageometrical object. Fractal dimensions quantify the static*geometry* of an object.

For example, consider a straight line. Now blow up the lineby a factor of two. The line is now twice as long as before.Log 2 / Log 2 = 1, corresponding to dimension 1. Consider asquare. Now blow up the square by a factor of two. Thesquare is now 4 times as large as before (i.e. 4 originalsquares can be placed on the original square). Log 4 / log 2= 2, corresponding to dimension 2 for the square. Consider asnowflake curve formed by repeatedly replacing ___ with _/\\_,where each of the 4 new lines is 1/3 the length of the oldline. Blowing up the snowflake curve by a factor of 3 resultsin a snowflake curve 4 times as large (one of the oldsnowflake curves can be placed on each of the 4 segments_/\\_). Log 4 / log 3 = 1.261... Since the dimension 1.261 islarger than the dimension 1 of the lines making up the curve,the snowflake curve is a fractal. [sci.fractals FAQ].

Fractal Dimension

A number that quantitatively describes how an object fills its space. In Euclidean, or Plane geometry, objects are solid and continuous. That is, they have no holes or gaps. As such, they have integer dimensions. Fractals are rough and often discontinuous, like a wiffle ball, and so have fractional, or fractal dimensions.

Fractal Dimension

A description of how a discontinuous object fills the space it occupies. For example, a basketball is not smooth; rather it has lines and grooves all over it. Because it is not a perfect sphere, it requires fractal dimensions to explain how it exists.

单词	fractal dimension
释义	fractal dimension fractal dimension (mathematics)A common type of fractal dimension is theHausdorff-Besicovich Dimension, but there are severaldifferent ways of computing fractal dimension. Fractaldimension can be calculated by taking the limit of thequotient of the log change in object size and the log changein measurement scale, as the measurement scale approacheszero. The differences come in what is exactly meant by"object size" and what is meant by "measurement scale" and howto get an average number out of many different parts of ageometrical object. Fractal dimensions quantify the staticgeometry of an object. For example, consider a straight line. Now blow up the lineby a factor of two. The line is now twice as long as before.Log 2 / Log 2 = 1, corresponding to dimension 1. Consider asquare. Now blow up the square by a factor of two. Thesquare is now 4 times as large as before (i.e. 4 originalsquares can be placed on the original square). Log 4 / log 2= 2, corresponding to dimension 2 for the square. Consider asnowflake curve formed by repeatedly replacing ___ with _/\\_,where each of the 4 new lines is 1/3 the length of the oldline. Blowing up the snowflake curve by a factor of 3 resultsin a snowflake curve 4 times as large (one of the oldsnowflake curves can be placed on each of the 4 segments_/\\_). Log 4 / log 3 = 1.261... Since the dimension 1.261 islarger than the dimension 1 of the lines making up the curve,the snowflake curve is a fractal. [sci.fractals FAQ]. Fractal Dimension Fractal Dimension A number that quantitatively describes how an object fills its space. In Euclidean, or Plane geometry, objects are solid and continuous. That is, they have no holes or gaps. As such, they have integer dimensions. Fractals are rough and often discontinuous, like a wiffle ball, and so have fractional, or fractal dimensions. Fractal Dimension A description of how a discontinuous object fills the space it occupies. For example, a basketball is not smooth; rather it has lines and grooves all over it. Because it is not a perfect sphere, it requires fractal dimensions to explain how it exists.
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