  This paper examines a family of designs for magnetic cubes and counts how
many configurations are possible for each design as a function of the number of
modules.
  Magnetic modular cubes are cubes with magnets arranged on their faces. The
magnets are positioned so that each face has either magnetic south or north
pole outward. Moreover, we require that the net magnetic moment of the cube
passes through the center of opposing faces. These magnetic arrangements enable
coupling when cube faces with opposite polarity are brought in close proximity
and enable moving the cubes by controlling the orientation of a global magnetic
field. This paper investigates the 2D and 3D shapes that can be constructed by
magnetic modular cubes, and describes all possible magnet arrangements that
obey these rules. We select ten magnetic arrangements and assign a "colo"' to
each of them for ease of visualization and reference. We provide a method to
enumerate the number of unique polyominoes and polycubes that can be
constructed from a given set of colored cubes. We use this method to enumerate
all arrangements for up to 20 modules in 2D and 16 modules in 3D. We provide a
motion planner for 2D assembly and through simulations compare which
arrangements require fewer movements to generate and which arrangements are
more common. Hardware demonstrations explore the self-assembly and disassembly
of these modules in 2D and 3D.
