what dimension should be assigned to objects that are not smooth and regular, but rough and irregular?
The motion of an irregular spiky index chart is certainly not two-dimensional, but nor is it simply one-dimensional. The stock index visits many more points on a flat surface than would a smooth curved or straight line. The idea of a fractional dimension was first shown to have a sound mathematical basis back in 1919 by the German mathematician Felix Hausdorff. Hausdorff used his fractional dimension to measure the amount of space occupied by objects. The reason for its development is not important here; what is important however, is that this same fractal dimension can be used as a measure of the degree of roughness or the degree of irregularity of any one-dimensional, two-dimensional, or three-dimensional object. Thus, a line with a dimension of 1.0 is a smooth straight or curved line. However a line with a dimension of 1.53 will have a rough jagged edge. The higher this dimension goes (while remaining within whole dimensional limits) the more irregular the line becomes. A line with
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- what dimension should be assigned to objects that are not smooth and regular, but rough and irregular?
