There seems to be quite a lot of logic in dealing with hexagonal grids using a system of axial coordinates like the one described here.
In doing some reading about hex grids, though, I ran across the notion of the Spiral Honeycomb Mosaic (the links on that mirror are broken; the content is copied from here, about halfway down the page). It seems to be used mostly in image processing, but it's interesting from the standpoint that it assigns each hex a unique, sequential base-7 number as its identifier.
I'm not sure that it's actually useful for anything, but I'm wondering: is it possible to convert back and forth between the SHM numbering system and axial coordinates?
Both systems have methods to convert from a hex to the Cartesian coordinates of its center, so presumably one could calculate that out of one system, and then into the other. But that's messy and inefficient, and I'm convinced it's possible to do better.
Because of the fractal nature of the SHM, the conversion routine in the sample code provided relies on a loop; it seems like the general solution might involve a series?
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