The maths is what interests me...
and like I've said I think your concept is a brilliant explanation.
Well the math is quite complex and like most math, very generalized.
I think you are more interested in the implementation process.
So, that I can explain. I happened to use a program called nbos astrosynthsis 3, but a spreadsheet can be used as well.
Step 1. Gather together the x,y position of stars in light years. No other data is needed. They can be relative to the galactic center or earth or any arbitrary xyz frame of reference.
Step 2. Calculate the distance between each and every star.
If the distance is 1.8 ly or less, tag those star together into a system. (This is a sphere that covers the current cubic pc and overlaps the neighbors by a little bit)
Now that you have worked out the "systems", calculate the center of the system ie the average midpoint for each of them.
This is your new xyz grid.
Step 3. Pick a central system. That system is a cubic parsec. Since you know it's xyz center point, you can work out if any systems are in one of the neighbouring 26 cubes around that central system. The distance is between 3.26 and 10ly based upon angle. (Refer to diagrams for how this works)
Place them as per the diagram in the article.
Starting with the inner ring, then the middle ring.
For the first system, use the placement in the diagram as shown, but as you add systems beyond the first, you will find that the placement on the outer ring will need to be adjusted.
Step 4. Pick a new system that is not already mapped, but which shares some of the systems mapped to your first system. Map up this new system, overlapping any shared relationships with the first system. Remember that outer ring system placement can rotate but keep in mind real world xyz distance when placing the outer ring systems.
Step 5. Pick a new system and repeat the work of step 4 until no more systems remain unmapped.
You have effectively separated space into 20ly cubes, that overlap by about 3.5ly to a side.
(Central system, 10ly radius, trimmed as a slight ly deformed cube)
My first pass at this was using nbos astro 2 and a diagramming program called yED.
Second pass was a series of macros that would process the systems and change a text value on the system that held its position within hexagonal xy space.
As each new system was added, it would nudge these values depending upon overall placement.
Finally I showed my work to one of my wife's friends who is a math teacher and she immediately saw what I was up to and put me in touch with a professor who showed me what I was doing wrong. That led to my further breaking down a single system into 27 cubes, a 3x3x3 grid, each slightly larger than a ly to a side (3.26/3). This held the original xyz position within the system so that it could be extrapolated back out to 3d.
So the whole thing is a series of cubes within cubes, spread out onto a plane. The computer version that I ended up with was based on Einstein's work (he did alot more than just physics, he was a mathematician first and foremost). That was a period of months of difficult work studying to understand how it works. I hope that I may reach the point of understanding that I can explain it simply (a task I am not yet up to). But at least I can do it manually or program it up in Pascal.
Hopefully this rambling made some sense to you.
Best regards,