Having dealt at some length with basic cosmic fractal features, then basic fractals and how to obtain their dimension and density, we now go back to cosmic fractals 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) 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.
In the figure, the basic template fractal is shown in 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 within the sector 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 theta in arc measure.
In Fig. 1B, the reader will see a sector subtending about five hours (~ 75 deg) of Right Ascension. This is a cosmic snapshot of galaxy distribution obtained using the Two Degree (2dF) sky survey. One comparative aspect immediately visible is the difference in void distribution between the ideal and actual cosmic case. To fix ideas, the void in the ideal fractal sector is more or less uniformly distributed, at least around the circular stochastic element. In the real 2dF sector, the void is distributed toward the periphery of the field. There is much greater fractal density and complexity toward the center of 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 lacunarity. One may define the lacunarity, L, to be:
L = Nr(k> R_s)/ R_s^-D
Here, I have taken the liberty of using R_s in the abstract case (1A) to double as the scale size for the radius of the actual cosmic sector in 1B. There is no loss of generality here, though Mandelbrot used the capital Greek letter LAMBDA to denote it. Readers may substitute LAMBDA for R_s if they are more comfortable doing so. Nr is a product of objects and linear space but which can be generalized into a sector volume, in 3D. If one wants, he may think of it as a kind of proxy for fractal density.
Okay, let's examine the two figures-fractal sectors, ideal and actual - and see if the comparative lacunarity can be assessed, if only roughly. First, we are already asserting that R_s = Lambda in both cases. Thus, R_s is the same for each, and for simplicity, call it 1. The dimension, D, is more difficult, but if we understand that we are projecting (essentially) 3D objects into (or onto) a 2D space then we also understand if a fractal dimension is projected from a d=3 to d=2 subspace the result will be close to D' = 2.5. So, in this case, we call D= D' = 2.5. In terms of 'N', the actual cosmic sector holds about 100,000 galaxies so N = 10^5. 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
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.
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 dimensions.
One of the projects I am currently pursuing is how to make this interspace difference more understandable from a fractal perspective.
In terms of dimensionality, it may be necessary to imprt different fractal dimensions to associate with physical dimensions. For example, assign a galaxy point status so it is 0D. (Zero dimensional). Then assign long cosmic filaments or "ripples" denoting the COBE (density perturbation) structures as 1D, and immense plasma current sheets in intergalactic space as 2D, while the spherical galaxy clusters are 3D. This all adds lots of hierarchical complexity, especially to the calculations.
Will it be worthwhile? I hope to soon find out, and then do a later blog entry on it!
In the figure, the basic template fractal is shown in 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 within the sector 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 theta in arc measure.
In Fig. 1B, the reader will see a sector subtending about five hours (~ 75 deg) of Right Ascension. This is a cosmic snapshot of galaxy distribution obtained using the Two Degree (2dF) sky survey. One comparative aspect immediately visible is the difference in void distribution between the ideal and actual cosmic case. To fix ideas, the void in the ideal fractal sector is more or less uniformly distributed, at least around the circular stochastic element. In the real 2dF sector, the void is distributed toward the periphery of the field. There is much greater fractal density and complexity toward the center of 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 lacunarity. One may define the lacunarity, L, to be:
L = Nr(k> R_s)/ R_s^-D
Here, I have taken the liberty of using R_s in the abstract case (1A) to double as the scale size for the radius of the actual cosmic sector in 1B. There is no loss of generality here, though Mandelbrot used the capital Greek letter LAMBDA to denote it. Readers may substitute LAMBDA for R_s if they are more comfortable doing so. Nr is a product of objects and linear space but which can be generalized into a sector volume, in 3D. If one wants, he may think of it as a kind of proxy for fractal density.
Okay, let's examine the two figures-fractal sectors, ideal and actual - and see if the comparative lacunarity can be assessed, if only roughly. First, we are already asserting that R_s = Lambda in both cases. Thus, R_s is the same for each, and for simplicity, call it 1. The dimension, D, is more difficult, but if we understand that we are projecting (essentially) 3D objects into (or onto) a 2D space then we also understand if a fractal dimension is projected from a d=3 to d=2 subspace the result will be close to D' = 2.5. So, in this case, we call D= D' = 2.5. In terms of 'N', the actual cosmic sector holds about 100,000 galaxies so N = 10^5. 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
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.
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 dimensions.
One of the projects I am currently pursuing is how to make this interspace difference more understandable from a fractal perspective.
In terms of dimensionality, it may be necessary to imprt different fractal dimensions to associate with physical dimensions. For example, assign a galaxy point status so it is 0D. (Zero dimensional). Then assign long cosmic filaments or "ripples" denoting the COBE (density perturbation) structures as 1D, and immense plasma current sheets in intergalactic space as 2D, while the spherical galaxy clusters are 3D. This all adds lots of hierarchical complexity, especially to the calculations.
Will it be worthwhile? I hope to soon find out, and then do a later blog entry on it!
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