Generalized Balanced Ternary
GBT remains my happiest and most grandiose programming achievement so far. This is the case because it is more complicated than anything else I have ever completed.
Generalized Balanced Ternary is the invention of Dean Lucas and Laurie Gibson, who applied the idea to Geographical Information Systems during the 1970s and 1980s. In the practical world, the main advantage of two-dimensional GBT is that it divides areas into congruent regular hexagons, and such hexagons fit together very neatly, according to certain criteria.
Any regular hexagon shares an edge with six other identical hexagons in the tessellation of the plane by regular hexagons.
When squares tessellate the plane, each square shares an edge with four other squares, and at the same time it shares a corner with yet four other squares. This makes dealing with squares cumbersome and messy.
For me, GBT opened vast new worlds of exotic geometry and also of combinatorics. It is based on rosettes of hexagons, of "septrees" of sevens within sevens within sevens.
In three dimensions, the GBT units are truncated octahedra, and a 3-D rosette consists of a cluster of 15 of these polyhedra. These clusters can be packed hierarchically, just as the rosettes of 7 hexagons can be packed in two dimensions.
So far, I have only worked in two dimensions with rosettes of regular hexagons. My method of making abstract designs out GBT and its hexagons has been to put a large number of hexagonal cells on the computer screen, and then consider the GBT labels for each cell. From those labels, color numbers are derived.
Figuring out how to turn GBT labels into unique color designations was a daunting task. To solve it, I had to turn to "Greep Theory."