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Great puzzle @nedbat.com @wang.social. The trick was to find enough minus signs! Here's the answer: Replace each diagonal with a special point to its top left, and associate ALL edges below with a minus sign. The first three columns are X,Y,Z. This gives coordinates of each cube in the octahedron.

The third picture shows the ninth generation of three crystal seeds growing inside a 32×32×32 cube with transparent faces, revealing the crystals' internal structure.