Negative fractal Dimension

Towers. There are many 0 Dimensions here. From http://www.arteyfotografia.com.ar/contenido/objetos

There is a paper by B. Mandelbrot that interested me in the theoretical aspects. I don´t know if this paper is famous in the world of Physics, but I can tell you that negative fractal dimension concepts is not a common one. This paper is called  Negative fractal dimensions and multifractals. Published in Physica A 163, p. 306-315, North Holland. Also available on line.

I´ll reproduce some paragraphs here.

¨A new notion of fractal dimension is defined. When it is positive, it effectively falls back on known definitions. But its motivating virtue is that it can take negative values, which measure usefully the degree of emptiness of empty sets. The main use concerns random multifractals¨.

Mandelbrot says the applications are to turbulence and DLA. (Diffusion Limited Aggregation)

¨...negative fractal dimensions is briefly announced ... as one of two separate aspects of dimension, that are latent (hidden, but present). Lately, many authors have added much to the topic of multifractals, and it has greately changed (though our early papers may not yet be exhausted). Despite these advances, however, even the most basic aspects of multifractality continue to present features that deserve further research¨.

¨The link between the two topics in the title came to focus recently, and it is elementary, that is, should be widely used. First, we develop negative dimension as a new notion, and introduce those physicists who have already become used to life in fractional dimension to the charms of negative dimension, and to its inevitability¨…

Then, Mandelbrot writes about the generic rule of intersection for dimensions, BUT there is a major exception to this rule: its value does not matter, the intersection S is generically empty.

¨A way to redefine dimension, which avoids this exception, and simplifies but enriches the intersection rule. The example of points, lines, planes and the like. Compare the intersection of two lines and the intersection of a line by a plane. Both sets are ¨generically¨ of dimension 0, in agreement  with the intersection rule and its exception. Yet, one would like to discriminate more finely between these various ways of being of dimension 0, by expressing numerically the idea that the intersection of two lines is ¨emptier¨ than the intersection of a line by a plane. If one could get ride of the exception to the intersection rule, one may perhaps be allowed to say that these two sets have the dimensions –1 and 0¨.

Needless to say I couldn´t understand the rest of his findings, because the paper is full of formulae that are very obscure for an architect.

Nevertheless, it certainly reminded me my concern about the dimension 0 and the paradox of a building that constitutes a city. I cited the example in this blog before. If we are far enough, suppose in the sky, and we see the building as a point, its dimension would be 0, but if I look on the side, suppose we have a huge rock, the dimension of the rock would also be 0. In theory, both dimensions wouldn´t be the same. One should be ¨less empty¨, the building of course.

Another thought, suppose we have two buildings now. And from the satellite scale, I can consider both of them of Dimension 0. But one of them has constructions underground. Again, one of those buildings dimension is emptier.

Further readings.
http://myriammahiques.blogspot.com/2009/11/urban-exercises-with-fractal-spectrum.html