[odtidrwj] Minecraft with sloped faces

Draw a closed curve on the surface of a cube.  Form the minimal surface bounded by the curve.  Divide the cube into two pieces by that surface.

Probably want constraints like non-self-intersecting on the curve to get reasonable pieces.

Let the cube be of integer edge length.  Limit the surface curve to straight segments connecting lattice points.  Under what conditions is the curve planar, i.e., the minimal surface is a plane?  Of particular interest is the hexagonal cross section of a cube of side 2.  How many planar cuts are there?

Instead of a minimal surface when not a plane, what is a "reasonable" way of forming a polyhedral surface (planar faces) bounded by the lattice-aligned curve?  Some sort of triangulation, but which one?  Introducing internal vertices is not out of the question.  How many shapes are there?

Perhaps the "faceted minimal surface" with all internal vertices also on lattice points.

If we permit multiple curves on the surface of the cube, how can we avoid the surfaces from intersecting in a weird way?  Is it merely enough for the curves not to intersect?

The original inspiration is to be able to build things in Minecraft without orthogonally aligned faces by having bricks shaped like portions of cubes.  The diagonal slope edges need to be able to line up.

The simplest shapes are the triangular wedge (half a cube), the tetrahedron and remainder after lopping off a corner.  Also cut opposite corners leaving (I think) an octahedron.  Or one corner and one edge.