Turns out that what I was visualizing is an "alternated cubic honeycomb" which involves octahedra and two different orientations of tetrahedra. I still think this is probably the most organic way to approach a voxel system from the perspective of triangles and resolution.

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turns out there's a scheme for spacial subdivision in the alternated cubic honeycomb. tetrahedra subdivide into four tetrahedra and an octahedron; octahedra subdivide into six octahedra and eight tetrahedra. At any level of subdivision, all faces are exactly the same size equilateral triangles.
I think this is what matters for a relatively organic voxel system (relative to cube voxels).

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