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.

*fractal dimensional*with the dimension non-integer, say between 4 and 5. (E.g. 4.2). This sounds counter-intuitive but people need to bear in mind that one of the original founders of the calculus (Leibniz) left open the possibility for a fractal operator in his derivative of a function, F(x), such that:

^{n}r 0

^{n}r 0 = (1000 ) r 0 = 10

^{3}(r 0 )

^{n}r 0 = (10) r 0 = 10

^{1}( r 0 )

^{3}/ log k

^{1}= 3

**1A**. Actually, it is a composite with two deterministic and two stochastic contributions. The deterministic fractals are the shaded and unshaded segments of the circle with radius r in

*the upper left of the sector*. The stochastic parts refer to the fully shaded circle

*and and the (open space) set intersection between it and the other shaded deterministic part. The radius R s may be thought of as an applicable scale for the whole, which also subtends an angle q in arc measure.*

**within the sector**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

*. One may define the*

**lacunarity****, L, defined:**

*lacunarity*^{-D}

^{5}. By comparison, the "objects" in Fig. 1A correcting for the spaces and density of deterministic and stochastic fractals are N ~ 10

^{2}.

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}

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^{Thus, the number of voids is about 1000 times higher. This should not be astounding if one closely inspects 1B. Going through the sector we behold innumerable tiny white spaces , many more than in 1A. Again, for lacunarity, size of the voids is not the issue so much as the frequency of voids overall within the fractal structure. Above all, what we want to see is self-similarity revealed in cosmic structures and perhaps even in the cosmos as a whole.}

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^{What does all this have to do with parsing actual cosmic structure? Only that fractal analysis may be of some use in ferreting out the dynamics of cosmic expansion, say, and in particular why interspatial topology alters at a much more rapid rate than appears to be dictated by red shift measurements. Thus, although redshift (Hubble law) calculations show a universe 13.7 billion light years across, the actual radius is more like 46 billion light years- because of the accelerating interspace dimension}

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^{See Also:}

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^{The fractal universe | Stanford News}

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^{And:}

^{Is The Universe Actually A Fractal? - Forbes}

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