Having dealt at some length in previous posts with basic fractal features, then cosmic fractals and how to obtain their dimension and density, we now look again at the latter in more detail. What I will do is examine an ideal example of a type of simple topological fractal which, with suitable adaptation and modification (mostly by including many self-similar, complex parts) and which could double as a basic "cosmic fractal". I will then show an actual cosmic fractal (or perhaps more accurately what might be considered as such ) and how it compares to the ideal case. These fractal applications are important because they serve to enable us to perceive the cosmos in a new and more practical light, with centers of self-similarity and non-linear complexity that may explain some large scale cosmic attributes, up to an including (maybe) cosmic accelerated expansion.
Early on, fractal researchers and investigators like Benoit Mandelbrot realized that fractal dimension, D, was not adequate to to accurately assess or determine the topology. A new measure was needed that reckoned in void extent, and that was defined by Mandelbrot as the lacunarity. One may define the lacunarity, L, defined:
Now, the denominators for both are respectively, R s - D = (1) 2.5 = 1
Thus, the lacunarity is solely dependent on N, and for the real cosmic case, the ratio of it lacunarity to that of the abstract or ideal example is:
L c / L = (10 5 )/ 10 2 = 10 3