  A method is proposed for generating compact fractal disordered media, by
generalizing the random midpoint displacement algorithm. The obtained
structures are invasive stochastic fractals, with the Hurst exponent varying as
a continuous parameter, as opposed to lacunar deterministic fractals, such as
the Menger sponge. By employing the Detrending Moving Average algorithm [Phys.
Rev. E 76, 056703 (2007)], the Hurst exponent of the generated structure can be
subsequently checked. The fractality of such a structure is referred to a
property defined over a three dimensional topology rather than to the topology
itself. Consequently, in this framework, the Hurst exponent should be intended
as an estimator of compactness rather than of roughness. Applications can be
envisaged for simulating and quantifying complex systems characterized by
self-similar heterogeneity across space. For example, exploitation areas range
from the design and control of multifunctional self-assembled artificial nano
and micro structures, to the analysis and modelling of complex pattern
formation in biology, environmental sciences, geomorphological sciences, etc.
