Fractal dimension

How can be measured a length or an area?

It can also be said that, for a ruler small enough (infinitely small), if I divide by n its length, I multiply by n the number of times that I use it to measure the line. This leads to a ratio of n/n, that is to say 1 (this is true also if I write log n/log n, a remark which will be useful soon); 1 is the dimension of linear figures (Euclidean or topological dimension... yes I know that these two terms are not exactly equivalent, but this doesn't matter here).

Now we will use the same argument for a surface. Let us consider a square with a side length L. If I use a unit square with a side length l = L/2, 4 small squares are needed to pave the initial square. For a side l = L/4, 16 of them are needed, etc. If the side is divided by a factor n, the number of small squares is multiplied by n².

The ratio n²/n cannot be used to measure the dimension of a surface (we know since Euclid that it is 2) because 4/2 = 2, but 16/4 = 4 etc.
On the contrary log n²/log n = 2 in all cases. You can verify it with a calculator but, if you still remember your maths, you know that log n²/log n = 2 (log n/log n). The same approach can be applied to volumes where we have log n³/log n = 3.
Let us consider now any surface, an arbitrary unit square that we will divide by an arbitrary factor n. If the size of the square is not negligible I cannot pave the surface in a satisfying way and I will underestimate its value. Let N be the number of squares included in the area. The smaller the square is, the better is the paving. It is obvious that the estimation of the area will be much nearer the actual area since the unit square is smaller. Here again it can be proved that log N/log n tends towards 2 if the size of the square tends towards 0 (in other words if N tends towards infinity).

Why such complications since the result gives the value of the well known Euclidean dimension? Because, if it is true for the figures of classical geometry, this is not true in other cases.

Look indeed at the von Koch snowflake that I have described in another page. If I use a ruler the length of which is equal to the side of the initial triangle and if I apply it on the snowflake, I will find a length equal to 3 L (the ruler is contained 3 times because there are 3 sides in the initial triangle). Now if I use a ruler the length of which is L/3 I can cover more details of the snowflake, and I must apply the ruler 12 times (3*4) to go round the snowflake. If I repeat the experiment by dividing 3 times more the length of the ruler, I can apply it on more and more short segments (don't forget that the snowflake has an infinite number of tips which are smaller and smaller). The ruler will be applied 48 times (3*4*4) on the perimeter. In other words each time I divide the ruler by 3 I multiply by 4 the number of times I apply it to go around the snowflake. The argument can be repeated endlessly.

Green: a ruler with a length L Yellow: a ruler with a length L/3 Blue: a ruler with a length L/9

It can be seen that the dimension of this strange figure is not 1 like for all the figures made of lines of the classic geometry because we have a ratio of 4/3. The fractal dimension is given by the ratio log 4/log 3 and, surprise, this dimension has for value 1,26... This is a figure the Euclidean dimension of which is 1 (it is a broken line) and the fractal dimension of which is greater than 1 and, moreover, is not a whole number.
Mandelbrot comes to the same result by using a slightly different process based on the ratio of homothety.

The example of the von Koch snowflake is easy to understand because it is simple and because the ratio remains constant in the peculiar case I have chosen. This might be slightly more complex if the initial length of the ruler was different from the length of the side of the triangle and if it was divided by a divisor different from 3. But in all the cases the ratio tends towards log 4/log 3 when the ruler becomes smaller and smaller.

This leads to the general law:

The fractal dimension D, for a linear figure, is defined as

         log (L2/L1)
     D = -----------    when S tends towards 0
         log (S1/S2)

where L1, L2 are the lengths measured on the curve (expressed in number of units), and S1, S2 are the sizes of the unit (ie. the scale) used for the measurements.
or again, using the ratio of homothety

            log N
     D = ---------    where r is the ratio of homothety and N (I simplify)
         log (1/r)    the number of "elements" created by the operation of homothety.

For the usual figures of classic geometry, drawn with straight or curved lines, this dimension is equal to 1, like their topological dimension, but for fractal curves the topological dimension is really 1 whereas the fractal dimension is greater than 1 and lower or equal to 2 (2 cannot be exceeded because it is the topological dimension of surfaces).

This result can be generalized to fractal surfaces which have a topological dimension of 2 and a fractal dimension greater than 2.

This dimension is often said to be the dimension of Hausdorff-Besicovitch or, at least, it is suggested by the context. As a matter of fact this is false and true at the same time. False because the general expression of the dimension of Hausdorff-Besicovitch is very abstract (it is often too difficult to compute to be used). True because for the fractal made of lines which are simple, as is the von Koch curve, the dimensions of homothety and of  Hausdorff-Besicovitch are equal.
If I have correctly understood the explanations of Mandelbrot in the mathematical appendix of his book "Les objets fractals" (I suppose that there might be something similar in "The fractal geometry of Nature"), this dimension seems to be the covering dimension of Pontrjagin and Schnirelman :
suppose an object in a space with n dimensions which is covered by the smallest possible number N of "balls" the radius of which is r. Its covering dimension is log N/log (1/r) for r tending towards 0 (I have slightly modified the formula to avoid a ro because its rendering on the screen is problematical with some character sets).

The conclusion is that the explanations given in this page (if you except the previous paragraph) give a false idea of simplicity on a question which is in fact very complex. Moreover there are other approaches of the notion of dimension which are not all equivalent, and the works of Kolmogorov and Tihomirov show a connection between the covering dimension and the notion of entropy (this may bring back some vague memories, for some of you!).