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'In theory' the dimension of a contrast can be self-similar
 
 

The example of the spotted sheet has a limit: you do not always measure the same contrast dimension, even on the same surface, even with the same distribution of spots. As a matter of fact, the result will depend on the portion of the surface we measure: shall we calculate on the whole surface, or only on a part? If we take only a part and fall in the space between two spots, we shall find a contrast randomly varying between 0 and 1. It would as well be 0.3 in a measurement, and 0.7 in another. If the contrast is homogeneous on the sheet, that is, if the spots are regularly distributed, nothing will change. The bigger the surface, the more constant the result, but the smaller the surface we shall take, the more unpredictable and the more various the result will be. There is no 'trick' to correct the result according to the scale of the measurement, and to always pick the same result.

 

It's not the same thing with the coordinate dimensions: you can look from far away or from nearby, they always measure the same dimensions. We only have to adapt the measurement scale to the reading scale.

The same thing fits to the deformation dimensions, only if they have a special characteristic: they have to be self-similar, that is, similar to themselves in all scales.
To take again the example of the spots, the size and the organization of the spots should be coordinated, so that the spots we see from far away, appear when seen more closely, as made of smaller spots, which themselves when seen even more closely, etc., from the infinitely great to the infinitesimal. Of course, the proportion between black and white should also be the same in every 'spot level'.

In practice, it cannot be so, for we cannot divide a spot infinitely.
But theoretically, that's possible. And in fact, it does exist.
     - a 'Cantor-dust' fits with this definition. That's a line whose central 1/3 is removed, and on each remaining 1/3, the central 1/3 is removed, and so on infinitely.
     - a 'Sierpinski-carpet' as well. That's a square whose central square is removed, then on every square in what the remain can be split up, we remove the central square, etc.
     - a 'Menger-sponge' too. That's the same thing than the 'carpet', but with cubes instead of squares.

 

a 'Cantor-dust'

a 'Sierpinski-carpet'

a 'Menger-sponge'

[the drawings are from the book of B. Mendelbrot : les objets fractals - Editions Flammarion]
 
 
 

So, we have made a little progress to define a deformation dimension.      - first, we saw how a deformation could have a value, without any notion of coordinates.      - then we saw an important difference between deformation dimensions and coordinate dimensions: the firsts oscillate between 0 and 1, the seconds go from - infinity to + infinity.      - finally, we saw that a deformation dimension does not always have the self-similar character of a system of coordinates, but could have this quality, in some cases.

Now we are rady to study a second type of deformation dimension, that which corresponds to the making of curves.
 
 
 

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