A cube has 24 orientations. By rolling the cube on its edge within the perimeter of a $2\times4$ rectangle 3 times, all 24 orientations are reached and the next roll returns the cube to both the starting position and starting orientation.
A rolling icosahedron has 120 orientations. The top face can point up or down in each of 60 orientations. What is the smallest triangular grid for which a rolling icosahedron can roll through a complete Hamiltonian cycle of all 120 orientations? What are the properties of the 120-vertex cubic graph?
Similar question for the other 7 deltahedra. What is the smallest triangular grid allowing a complete cycle of all orientations?
For other polyhedra that can be rolled through all possible orientations on a simple 2D grid of polygons, what is the smallest grid that supports a Hamiltonian cycle?
For the 1x1x2 cuboid, here's a grid that allows a Hamiltonian path through all 24 orientations. Is there a grid with fewer cells?