[bpbvdnby] Sprouts in higher dimensions

Consider extending Conway's game of Sprouts to three dimensions or more.

Instead of drawing lines and points on a line, we draw surfaces and lines on the surface.

As a simple intermediate example, start with a normal 2D game of sprouts, but extrude every line along the Z axis into infinitely tall cylinders or ruled surfaces.  Every new point is instead a new vertical line along a surface.  This obviously results in a 3D game isomorphic to the original.

I'm not sure what the start position of general 3D sprouts should be, but consider a middle game position with a bunch of surfaces with lines marked in them.  A move consists of drawing a new surface, not intersecting with any other, bounded by already existing lines or empty space ("a ribbon" has a free edge).  Then draw a new line on the surface.  Then (maybe?) draw a new point on the line.

The last instruction (new point) allows "sub-dimensional" moves: instead of drawing a surface between lines, you can also choose to draw a line between points, perhaps constrained to be on a connected surface, but maybe not.  If constrained, then subdimensional subgames may consist of playing 2D sprouts on weird surfaces (torus, klein bottle (which requires starting in at least 4D)).

Maybe ribbons should be forbidden, as well as subdimensional lines through empty space.  But if not, then it allows the game to begin with a bunch of free points.

Another possibility of beginning is to have free points that are infinitesimally tiny spheres with an equator marked on them. To maintain an analogue with 2D sprouts, the infinitesimal equator does not "count" in the number surfaces attached to the line. Then, perhaps "free" line segments are also permitted as infinitesimally thin cylinders between two free points.