Fractal Dimension - Role of Scaling

Role of Scaling

The concept of a fractal dimension rests in unconventional views of scaling and dimension. As Fig. 4 illustrates, traditional notions of geometry dictate that shapes scale predictably according to intuitive and familiar ideas about the space they are contained within, such that, for instance, measuring a line using first one measuring stick then another 1/3 its size, will give for the second stick a total length 3 times as many sticks long as with the first. This holds in 2 dimensions, as well. If one measures the area of a square then measures again with a box of side length 1/3 the size of the original, one will find 9 times as many squares as with the first measure. Such familiar scaling relationships can be defined mathematically by the general scaling rule in Equation 1, where the variable stands for the number of new sticks, for the scaling factor, and for the fractal dimension:

(1)

This scaling rule typifies conventional rules about geometry and dimension - for lines, it quantifies that, because =3 when =1/3 as in the example above, =1, and for squares, because =9 when =1/3, =2.

The same rule applies to fractal geometry but less intuitively. To elaborate, a fractal line measured at first to be one length, when remeasured using a new stick scaled by 1/3 of the old may not be the expected 3 but instead 4 times as many scaled sticks long. In this case, =4 when =1/3, and the value of can be found by rearranging Equation 1:

(2)

That is, for a fractal described by =4 when =1/3, =1.2619, a non-integer dimension that suggests the fractal has a dimension not equal to the space it resides in. The scaling used in this example is the same scaling of the Koch curve and snowflake. Of note, these images themselves are not true fractals because the scaling described by the value of cannot continue infinitely for the simple reason that the images only exist to the point of their smallest component, a pixel. The theoretical pattern that the digital images represent, however, has no discrete pixel-like pieces, but rather is composed of an infinite number of infinitely scaled segments joined at different angles and does indeed have a fractal dimension of 1.2619.