Is there such a thing as "the complexity of three-dimensional space"? It is sometimes heard that the complexity of chess is $10^{120}$, or that go is more complex than chess since it has a complexity of $10^{365}$.
I’ve been thinking: Is it possible to do something similar with the three-dimensional space we live on?
To do this, let's suppose a cube made of little Planck voxels (1 Planck length by 1 Planck length by 1 Planck length):

Each of these Planck voxels can be empty or occupied. In the image, empty voxels are transparent; occupied ones are red.
So, now we have to see: How many arrangements are possible with these Planck voxels, by turning them empty of occupied?
This is what I did:
First, let's suppose a simpler case, instead of having a big cube of several Planck voxels, let's just consider squares that can also be empty (white) or occupied (black), and see all the possible arrangements that are possible:

It is possible to treat these arrangements as if we have a set $N$ which contains $A$ elements that consists of $A_1$ white elements and $A_2$ black ones, so the number of permutations with repetition is given by:
$$  P_A^{A_1,A_2} = \frac{A!}{A_1!A_2!} .    $$
If we would like to consider all the possible configurations, we have to do a summation to get the number of possible arrangements:
$$ a = \sum_{n=0}^A P_A^{A-n,n}.  $$ 
Now, we might consider these squares as just one slice of a cube, so, to compute the number of arrangements of a cube, we only need to elevate this formula by the number of slices $Z$:
$$ a = \left( \sum_{n=0}^A P_A^{A-n,n} \right)^Z $$
To put as example, let's consider a region of space with a form of a cube with an edge length of $ 6·10^{32} $ Planck lengths (roughly 1 cm).
This means that we have: 
$$ a = \left( \sum_{n=0}^{6·10^{32}} P_{6·10^{32}}^{6·10^{32}-n,n} \right)^{6·10^{32}} = \left( 2^{6·10^{32}} \right)^{6·10^{32}} $$
So, it is sometimes heard that the number of atoms in the observable universe is $10^{85}$. For some people, it is weird that chess is more complex than the number of atoms of the observable universe, since chess it is actually in the universe. But what I say is that if we take 1 cm³ of space, we are going to have a complexity much bigger than chess or even go.
So, Does it make sense to talk about such a thing as the complexity of space?
I am aware that in the universe it is possible to have things smaller than 1 Planck length, that a Planck length is the scale at which classical ideas about gravity and space-time cease to be valid, and quantum effects dominate. Also I am aware that we cannot consider space as something isolated, since it is actually part of something bigger that it is called spacetime, but for a good approximation, are my computations and my reasoning correct?
 A: The complexity estimates for chess and go serve to give you an idea of how computationally challenging these games are. For chess, the $10^{120}$ is an estimate for the number of all possible games. You can considerably prune this by excluding obviously bad moves, redundant positions, etc. I can easily devise a game with clear winning strategy that has a higher complexity than chess by this measure.

Does it make sense to talk about such a thing as the complexity of space?

I see two problems here:


*

*You cannot arbitrarily fill space with particles due to forces between particles, etc. The vast majority of ways to fill your cube would be physically highly unstable. This is somewhat analogous to including obviously bad moves in the number for chess.

*What are you going to do with the result? The result for chess at least tells you that you cannot possibly have a full catalogue of positions. It also allows you to make a rough comparison with go. For real space, there isn’t really something similar to compare it to and there is no point in computing all possible states.
However, note that your approach bears a considerable similarity the information-theoretic approach to entropy in general.
Here, you do not only consider space, but valid ways to fill it with particles.
For example, consider an amount of gas in a given box at a given temperature.
In the terms of statistical physics, this is what is called a macrostate.
Now there are plenty of ways to realise this macrostate microscopically, i.e., the actual positions and momenta of all the gas molecules.
These are called microstates.
You can often estimate the number of such microstates by simple combinatoric approaches such as yours.
Boltzmann’s formula now states that the entropy $S$ of a system or macrostate is $ S = k·\ln(W),$ where $k$ is the Boltzmann constant and $W$ is the number of microstates that realise the given macrostate.
So, the entropy of a system quantifies the complexity of its state space.
