Enumeration of Polyominoes & Polycubes Composed of Magnetic Cubes

TL;DR

This paper explores the design, enumeration, and assembly of magnetic modular cubes in 2D and 3D, providing methods to count configurations, visualize arrangements, and demonstrate self-assembly through simulations and hardware experiments.

Contribution

It introduces a novel enumeration method for magnetic cube configurations and analyzes their assembly properties in 2D and 3D.

Findings

Enumerated all magnetic arrangements up to 20 modules in 2D and 16 in 3D.

Developed a motion planner for efficient assembly.

Demonstrated hardware self-assembly and disassembly of magnetic cubes.

Abstract

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.