Also on fractals: A Fractal Brownian Archipelago
Lacunarity is a counterpart to the fractal dimension that describes the texture of a fractal. It has to do with the size distribution of the holes. Roughly speaking, if a fractal has large gaps or holes, it has high lacunarity; on the other hand, if a fractal is almost translationally invariant, it has low lacunarity. Different fractals can be constructed that have the same dimension but that look widely different because they have different lacunarity. There are applications of lacunarity in image processing, ecology, medicine, and other fields.
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 bottombegins to resemble a filled interval, while the one on the top 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.
An expanded version more appropriate for physicist is found in
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