It is possible to cover the points of a size 4 cubic lattice with a closed loop of 28 turns, as seen in the images below.
In these solutions, the turns only occur at the given points, and no point is revisited. Is 28 turns minimal for order 4, with these restrictions? What are the minimal number of turns for orders 5 through 8?
For the 2D problem, here are some connect-the dots results. In most of these, there is no restriction on where you can turn. For the 3D case, what is the minimal number of turns if one can turn anywhere, with the restriction that no points can be revisited?