the measurement of the deformation of a contrast
the trap in the vectorial representation of the forces
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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.
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.
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.|
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.
|So, we have made a little progress
to define a deformation dimension.
we saw how a deformation could have a value, without any notion of coordinates.
|next : the trap in the vectorial representation of the forces||