This is related to my previous question here: Finding a way to describe the position of vertices of $n$ nested hexagons
In the setting described in my previous question, I wanted to find a set of points forming regular hexagons such that the pink lines are parallel with a sequence of nested regular hexagons. In this case the nested hexagons are of equal distance $s$ where $s=\frac{r}{6}(-1+\sqrt{13})$ and $r$ is the radius of the innermost hexagon.
Now, in this question, I am trying to find a hexagonal (or similar, does not absolutely necessarily need to be regular hexagons, but this would be preferred if such construction exists) construction where all the lines of the same color are parallel with one another. Note that these lines also must be ordinary lines i.e. they can only contain two points per line.
Without the loss of generality, I am setting the origin $O$ to be $O=(0,0)$ in $\mathbb{R}^2$ and the points of the innermost hexagon to be $(-2,0), (2,0),(1,\sqrt{3}),(-1,-\sqrt{3}),(-1,\sqrt{3}),(1,-\sqrt{3})$. Again, if this condition absolutely needs to change for the construction to work, that may be done.
Now, since the pink lines must be parallel, we can somewhat employ the result from my previous question: the distance between points $R$ and $R'$ remains as $s=\frac{r}{6}(-1+\sqrt{13})$. However, this construction given for my previous question otherwise does not work: this is because, for example, not all the green lines can be parallel. This means that the distances between hexagons needs to be different, and we may not assume the distances to be equal.
I am however unable to develop this idea further, because I have a tough time coming up with a general rule / overall general conditions that would be entirely fulfilled by a construction. Certainly the rules will be a lot more complicated than the ones in my previous question, but I want to be able to somehow express the placements of all the vertices in the construction such that all the lines of same color would be parallel.
