Large-Scale Distribution of Galaxies

How big is the universe? What is our place in it?
These questions have fascinated people since the first observations of the night sky. Possible answers have been fiercely debated, occasionally at the cost of lives.
Here we are interested in a more specific question: is the universe fractal, at least over some range of distances?
First, how was the scalse of distances to the galaxies established?
Historical precedents for some aspects of a fractal galaxy distribution are found in the hierarchical cosmologies of Kant, Swedenborg, and Lambert.
The cosmologies of Newton and of Einstein are homogeneous, not hierarchical.
In the early 80s, Mandelbrot presented fractal forgeries of the distribution of galaxies.
Small-scale fractal distributions.
Large-scale fractal distributions.
De Vaucouleurs law shows hierarchical cosmologies need have no preferred observer.
One model of galaxy formation is based on percolation.
* There is general agreement about the existence of fractal structures out to about 50 million light years.
* Recent evidence suggests there is no upper cut-off of the fractal structure, and that the large-scale distribution of mass in the universe has d = 2. Both the absence of an upper cut-off and the value of the dimension are controversial.
* Isotropic fractal distributions are compatible with the equivalence of all observers.
* The Standard Cosmological Principle, that matter is distributed in a homogeneous and isotropic fashion can be replaced by the Conditional Cosmological Principle, that matter is distributed in a hierarchical and isotropic fashion.
* Observationally, hierarchical distribution implies long-range departures from Hubble's Law.
However, the distance scales to the galaxies is difficult to determine.
Thirty-five years ago, concerning the distance scale to the farthest galaxies one of my astronomy teachers remarked,"If you get the right order of magnitude in the exponent, you're doing very well."
In the time that has passed since then, observations have improved significantly.
But still, there is much uncertainty.
The range of fractality in the universe is not likely to be resolved soon to everybody's satisfaction.