A generalized framework for designing topological interlocking configurations

A topological interlocking configuration is an arrangement of pieces shaped in such a way that the motion of any piece is blocked by its neighbors. A variety of interlocking configurations have been proposed for convex pieces that are arranged in a planar space. Published algorithms for creating a topological interlocking configuration start from a tessellation of the plane (e.g. squares colored as a checkerboard). For each square S of one color, a plane P through each edge E is considered, tilted by a given angle θ against the tessellated plane. This induces a face F supported by P and limited by other such planes nearby. Note that E is interior to the face. By adjacency, the squares of the other color have similarly delimiting faces. This algorithm generates a topological interlocking configuration of tetrahedra or antiprisms. When checked for correctness (i.e. for no overlap), it rests on the tessellation to be of squares. If the tessellation consists of rectangles, then the algorithm fails. If the tessellation is irregular, then the tilting angle is not uniform for each edge and must be determined, in the worst case, by trial and error. In this article, we propose a method for generating topological interlocking configurations in one single iteration over the tessellation or mesh using a height value and a center point type for each tile as parameters. The required angles are a function of the given height and selected center; therefore, angle choices are not required as an initial input. The configurations generated using our method are compared against the configurations generated using the angle-choice approach. The results show that the proposed method maintains the alignment of the pieces and preserves the co-planarity of the equatorial sections of the pieces. Furthermore, the proposed method opens a path of geometric analysis for topological interlocking configurations based on non-planar tessellations.