click for Postscript
Adapted from Benoit B Mandelbrot
and R Pastor-Satorras, Antipodal Correlations and the Texture (Fractal Lacunarity) in Critical Ising Clusters; figure by Erik Rauch
click for Postscript
The properties and characteristics of a fractal set are not
completely determined by its fractal dimension D. Indeed, it is easy to
construct a family of fractals that share the same D but differ sharply.
As an example of such a construction, the above
depicts a stack of Cantor sets made up by a generator of 
intervals and
a reduction factor 
. By construction, all have the same fractal
dimension 
. Starting from the
upper middle set, the index k varies from 1 to 6 when moving up and 1 to 5 when moving down.
Below the middle, the 
segments are uniformly distributed; above, they
are closely packed at the ends of the set. The extremes do not look ``like
fractals'': the one in the bottom
(
uniformly distributed segments)
begins to resemble a filled interval, while the one on the top (
packed
segments) looks like two isolated points at the ends of the set. Only the
central Cantor set looks like a true fractal.
These sets have different ``texture'', more specifically, different
lacunarity. Lacunarity is a notion distinct and
independent from D; it is not related with the topology of the fractal and
needs more than one numerical variable to be fully determined. Lacunarity is
strongly related with the size distribution of the holes on the fractal and
with its deviation from translational invariance; roughly speaking, a
fractal is very lacunar if its holes tend to be large, in the sense that
they include large regions of space.