Enticed by this answer, I wrote some python code to generate lattice projections from 7-dimensional cubic lattices, as I was interested to see if a heptagrid tiling also exist.
Basically I generated a 7-dimensional orthonormal basis that had irrational components, then choose 5 of the base vectors as the orthogonal space of my projection, and projected grid points that were at a cutoff distance from the 2-dimensional complement that intersected the $(0,0,0,0,0,0,0)$ lattice point at the origin. The best looking example I was able to get was this: 
However, for the most part the results seem to require lots more than 2 shapes, and here comes my question:
Question: Do criteria exist that should be satisfied the projection basis in order for getting proper Penrose tilings?
I imagine there are some special symmetries that the basis choice should fulfill in order to result in proper tilings, but I wonder what those might be, or how to compute such basis. My ultimate goal was to build Penrose tilings for 3-dimensional space, hopefully the criteria I am looking for will not be specific to a number of dimensions
According to this reference, a possible criteria is that the projections of the cubic lattice basis add up to zero:
$$ \sum_{i=0}^6 { \gamma_i} = 0 $$