How to generally describe all possible quasi-crystal structures in $\mathbb{R}^3$? According to what I found on Wikipedia[1,2], you can represent any quasi-crystal structure in $\mathbb{R}^n$ by cutting a space $\mathbb{R}^N, N>n$ at an angle with the $\mathbb{R}^n$ space and then look at a periodic lattice in $\mathbb{R}^N$ and see where its points land in  $\mathbb{R}^n$.
It also vaguely mentions cutting through $\mathbb{R}^6$ to result in certain quasi-crystal structures in $\mathbb{R}^3$. There is also a linked paper which, sadly, is behind a paywall.
Is that the highest-dimensional space you need to get all the quasi-periodic tilings of $\mathbb{R}^3$? If not, where is this limit if there is one?
Also, does it suffice to take a simple cubical lattice (i.e. just look at all Integer vectors $\mathbf{x}=\left(x_i\right), x_i \in \mathbb{Z}$, or do I need to use more generic lattices? - My guess on that is that, eventually, a cubical lattice would suffice since cutting through it with some lowerdimensional space would result in a different lattice which you could then cut through at even lower dimensions to get general quasi-lattices.
So in short, is there a unified, finitely representable way to represent all quasi-crystal-structures for $\mathbb{R}^3$? And I suppose, if that isn't too much extra information, the generic version for $\mathbb{R}^n$ would be interesting too, though I do not need that right now if it's too complicated.
Lastly and perhaps obviously, what I do not care about are rotations or translations of the quasi-crystal-structure in $\mathbb{R}^3$. So I assume to achieve that I must not rotate my $\mathbb{R}^N$-space around any vector spanned by my $\mathbb{R}^3$ subspace. Is this indeed a necessary condition? Is it sufficient?

Somewhat related but different question I found:
Projection of a lattice onto a subspace
 A: There are quasi-crystals arising from cut and project schemes from $\mathbb{R}^N$ to $\mathbb{R}^3$ for all $N>3$. All that is required to force aperiodicity is that the subspace being projected onto is spanned by 'rationally independent' basis vectors (the generic case in terms of measure). The integer $N$ determines how many 'tile types' there are (a tile type would be the shape of possible tiles appearing in the Voronoi tiling of the point set).
Essentially, in the generic case with canonical window, the number of tile types will be the number of $3$-faces on the unit $N$-cube which is given by ${{N}\choose{3}} 2^{N-3}$. Apart from the codimension ($N-3$) of the tiling , you can also change the direction of the subspace and the size and shape of the 'acceptance domain' (or 'window') these all affect the properties of the quasi-crystal that you can end up with. This can all be formalised, but I'm guessing that for the moment you only want a general idea of what kinds of forms these quasi-crystals can take. Even for this rather tame collection, the answer is essentially that there are an uncountable number of quasi-crystals, even up to rather weak notions of equivalence of crystals (say, up to dynamical equivalence or topological).
Things get almost impossibly more complicated if you decide you want to look at the much larger class of all FLC (finite local complexity) Delone sets or FLC Mayer sets.
There are various other methods of generating classes of quasi-crystals which you may be interested in, such that the self-similar quasicrystals related to the so-called "substitution tilings", of which there are many variants and generalisations. In-situ, these kinds of quasi-crystals are rather well understood, but one should again be careful to understand that they are only a small subcollection of all possible quasi-crystals.
