In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, lacunarity is a measure of how a fractal

Fractal

A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...

 fills space. It is used to further classify fractals and textures which, while sharing the same fractal dimension

Fractal dimension

In fractal geometry, the fractal dimension, D, is a statistical quantity that gives an indication of how completely a fractal appears to fill space, as one zooms down to finer and finer scales. There are many specific definitions of fractal dimension. The most important theoretical fractal...

, appear very visually different.

Dense fractals have a low lacunarity. As the coarseness of the fractal increases, so does the lacunarity. Mandelbrot introduced the term lacunarity, coming from the latin word lacuna meaning "gap", in 1983. Intuitively, more and larger gaps give a higher lacunarity. The lacunarity of a data set can be calculated using a gliding box method and is then given as a function of box size :

is the number of filled data points in the box and the normalized frequency distribution of for different box sizes.

Lacunarity is used in ecology, image processing and other scientific fields.

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