Bingo. The trick, as it turns out, is to start with an octahedron (although any uniform polyhedra should work, including a cube, but an octahedron just requires a point on each axis, so it's easiest to spawn). Since you want a uniform, edgeless surface, you need to start with one. Can't build it outwards like you can in a bounded 2D volume, so you have to build it inwards.
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