Is the Cosmos Fractal in Nature?

Among the recurring questions which have arisen to do with our universe, is whether it is fractal in nature. Fractals, for those unaware, are generated via a self-similarity property that enables us to compute them via repeated iterations. Fig. 1 shows the outcome of one such series of computations. One can easily see from this that the self-similarity replicating property repeats at smaller and smaller scales.

The most basic way for the cosmos to exhibit a fractal nature is in terms of its dimension. Thus, the universe may well not be four dimensional, or five dimensional, but 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:

Given: (d/dx)^n F(x)

the value of n need not be an integer.

In the case of the cosmos at large, one will wish to examine concentric spheres of radii: R1, R2, R3....R_N and assess using these radii the density of objects within - assuming a hierarchical configuration. In the most abstract sense then, the cosmos' fractal dimension will depend on regularity between successive expansions factors k, k', such that the dimension will be:

D = (log k')/ (log k)

and one uses radii: r_n = k^n r_o

where cosmic radius r_o contains N_o objects, and r_1 contains N_1 objects, r_2 contains N_2 objects and so forth. In each r_n we assign what we call "strata" and "sub-strata" for both particles-objects and cosmic space.It's important to understand at least in a general way how the boundaries apply between sphere radii. We take a prosaic example to try to illustrate the separation of substrata-strata by radii and also by fractal dimension.

'Fluids of information' (of different information capacity) are subject to being fractionated. A nice example is after cooking a roast, say, and pouring the residual fluid into a small container. This is then put into a cold refrigerator for a day or two, then taken out. On observation, a top layer of whitish fat is seen resting atop the less dense supporting fluid - which is now a gel. In the fluid analogy model, a phase transition has occurred.

Let's carry this further. The fat has a viscosity y1, and the gravy -y1. Now, let Z denote a scalar[1] function of position through the two dimensional region separating the fluids, we have three regions in all:

--------------------------------------------------------
Z (f) = y1 (fat) FAT
Z (c) = 0 (boundary) -------------------------------
Z (g) = -y1 (gravy) GRAVY
---------------------------------------------------------

where Z (c) represents the center of the region, separating the fat from the (fatless) gravy. This two dimensional boundary is a domain wall. Not surprisingly, information and energy are interchangeable. Separations of energy, therefore indicate separations of information domains. What might these represent? In a crude way, the two information fluids are two parallel cosmi - maybe one matter, the other anti-matter. One domain might be called 'Aleph-nought', and the other (with greater information density) 'Aleph-One'. From observations of our universe, it seems that the material cosmos (the one we inhabit) is 'Aleph-One' since so little anti-matter is discernible. Aleph-nought and one are separated by fractional dimensionality.

Let us see how a general cosmic fractal dimension may be ascertained using some basic observations. Let the zeroth radius r_o = 1 pc or 1 parsec (3.26 light years). Let it possess N_o = 1 (1 object, say a star). Then we go to r_1 = 10 pc and N_1 = 100. Then, r_2 = 1000 pc, and N_2 = 10,000. If one truncates the data right here, then we can establish the expansion factor using the last two r-values:

r_2 = k^n x r_o = (1000)r_o = 10^3 (r_o)

r_1 = k^n x r_o = (10)r_o = 10^1 (r_o)

From this we determine: D = log k^3/ log k^1 = 3

Note, however, that if r_2 = 1500 pc (same objects interior) then D = 3.17 and we have a true fractal dimension. In practice, things to determine an overall cosmic density are nowhere this simple. The main problem is the ability to correctly assess the relevant objects (in this case galaxies, or galaxy clusters) interior to the respective radii. One can use a more simplified number here, which we call the average density.

For example, based on the configuration that yields D = 3.17, we work out the average density using:

= N(R_2)/ V(R_2) = (7 x 10^-7) = [3/4pi] AR^-y

Now, A is the normalizing constant = N_o/ r_o^D, and for the quantities already defined we have: A = 1

Here, (y) is the fractal density index.

Using logs we can easily calculate it to be: y = 1.747

Again, this is only intended to be instructive, not to be based on actual cosmic data. However, for comparison, we may compare it to an early estimate obtained by Vaucouleurs (SCIENCE, Vol. 167, 1970, p. 1203) for the cosmos, which was y = 1.8. It is incredible we could come so close to Vaucouleurs' estimate using basically numbers and sphere radii pulled out of thin air. This appears to show perhaps an insensitivity of the cosmic fractal dimension to any particular choice of objects N within radii r_N. Or does it? We will pursue this further in a subsequent entry.

[1] Scalar means it depends only on magnitude, not direction.