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    \$\begingroup\$ So the gist of the video is that we'll ever have only regular 3/4/6-gons made of 3D grid points, however large the grid is, right? \$\endgroup\$ Commented Oct 23, 2020 at 0:34
  • \$\begingroup\$ @Bubbler—that's right! \$\endgroup\$ Commented Oct 23, 2020 at 0:35
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    \$\begingroup\$ I conjecture that the squares only appear on the planes parallel to the cube's faces, and the triangles and hexagons only appear on the (1,1,1)-lattice equivalents. Is this true? \$\endgroup\$ Commented Oct 23, 2020 at 0:57
  • \$\begingroup\$ @Bubbler, I've edited the example of the square to provide a square that is not parallel to a face. \$\endgroup\$ Commented Oct 23, 2020 at 1:06
  • \$\begingroup\$ @Bubbler, for a triangle, see \$(0,4,0),(1,0,1),(4,3,1)\$. \$\endgroup\$ Commented Oct 23, 2020 at 1:11