Fractal Dimension

Fractal dimensions appear quite often in the technical literature. This popular member of the dimension family isn't really a separate dimension, however. Instead, "fractal dimension" is a general or basket term that might refer to any of the exponent dimensions mentioned above or even to other dimensions as well.
There are at least two possible reasons for attaching "fractal" to the various dimensions. First, if the estimated lengths of such objects follow a power law when plotted against scale size, the objects are often fractal. In other words, the same geometric rule or relation holds at lower and lower measuring scales. Secondly, the dimension value for the analytical types of dimension usually is noninteger, and people tend to associate a noninteger dimension with a fractal. Because of that noninteger quality, some people unintentionally introduce further confusion by using not only the term "fractal dimension" but also fractional dimension. In general, "fractional dimension" and ''fractal dimension" are the same thing.
Early (c. 1980) usage of the term "fractal dimension" usually meant one or more main types prevalent at that time—similarity dimension, Hausdorff dimension, and/or capacity. Grouping those three dimensions together tacitly assumes or implies that a straight-line relation between number of increments and scale size can legitimately be extrapolated to a scale size of zero. If that assumption is correct (and there's no way to prove or disprove it), then at a scale size of zero the similarity dimension, capacity, and Hausdorff dimension all have the same value, since they all derive from the same basic formula. That's why some people think of them as one family.
When other measures of dimension were developed or refined, people began applying the term "fractal dimension" to the other types as well. Hence, "fractal dimension" and "fractional dimension" might mean any or all of similarity dimension, capacity, Hausdorff, pointwise, correlation, information or nearest-neighbor dimension, and perhaps others. Bergé et al. (1984), Holzfuss & Mayer-Kress (1986), and Rasband (1990), among others, point out the ambiguity. Whenever you see the terms "fractal dimension" or "fractional dimension," you've got to choose between two possible interpretations:
• The author is using the term in a general way to refer to most or all exponent dimensions as a group (and with luck this will be evident from the context). Example: "The fractal dimension usually is noninteger."
• The author really means a particular exponent dimension. In that case you have to examine the fine print and try to figure out which one.