# Fractal dimension of the universe = 2?

1. Aug 3, 2010

### sbrothy

(Maybe this should go under General Math or maybe even Topology but since it's about the dimension of the universe I'll put it here. Feel free to move it.)

I've had too much coffee and on one my frequent wiki binges - reading about life, the universe and everything - I've come up with a question, the answer to which I'm probably not going to understand. Here goes anyway:

Various sources lists the fractal dimension of the universe to be "about 2" (e.g.: http://en.wikipedia.org/wiki/Fractal_cosmology).

What is meant by "about 2" and which definition of dimension is used here? Does "about 2" mean above or below 2?

From reading a little about fractal dimensions I realize that there seems to be different definitions as to what a fractal dimension is, but the one I'm (vaguely) familiar with seems to be Hausdorff's (http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension).

Using this definition a coastline for example, can be said to be one dimensional object "aspiring" to fill out the second dimension which would give it a Hausdorff dimension somewhere between 1 and 2. Similarly, A "menger sponge" can be seen as a 2 dimensional object "apsiring" to fill out the third making it somewhere between 2D and 3D.

Following this logic I'm not entirely sure if I would expect the universe to have a dimension between 3 and 4 or 4 and 5 (?), but definitely not 2.

What gives? I'm not asking for a mathematical treatise mind you. It just puzzles me...

Regards.

2. Sep 2, 2011

### mnb96

Hi,

I accidentally found this old thread and noticed nobody had replied to the question posed.
I am not a cosmologist and I have only a basic knowledge of fractals. However it seems to me that your observations are correct, and you do have big reasons to be puzzled by the statement that the dimension of the universe is $D\approx 2$ (slightly lower than 2).
Personally, I would have rised the same questions as you did.

Loosely speaking, it turns out that the fractal dimension of the universe that was estimated in the articles cited by Wikipedia relied on the visible part of the Universe. For example, an observed galaxy might hide many other galaxies that won't be taken into account in the calculations.

"mpej.unige.ch/~eckmann/ps_files/jarvenpaa2.ps"[/URL] investigated how much influence the non visible part of the universe has when estimating its Fractal dimension. They came up with a theorem of fractal geometry that essentially states that if you have a fractal of dimension [I]>2[/I] in $\mathbb{R}^3$ and you try to estimate its dimension from the "visible part" seen from an arbitrary point, then your estimation is [U]bounded from above by 2[/U].

That essentially means that the Universe as we know it, might well have any fractal dimension in the range [2,3].

Last edited by a moderator: Apr 26, 2017