# Multistep Evolution Method to Generate Topological Interlocking Assemblies
## 2 Related Work
### Block-Shape Analysis  
The traditional TI shapes are the Platonic solids. [PERSON] et al. reported that the Platonic solids can maintain TI when assembled in planar sections [2]. Each solid results from the evolution of specific tile shapes. A square evolves into a tetrahedron. A hexagon evolves into an octahedron, cube, or dodecahedron. A decagon evolves into a dodecahedron or an icosahedron. A surface tessellation of decagons requires additional kite-like tiles to fill the gaps. In such a case, the generation method must ignore the kite-like tiles to avoid unexpected results. The truncated versions of the Platonic solids also have interlocking properties. [PERSON] et al. argued that \"_certain truncations of [the Platonic] solids leave the interlocking property unaffected_\" [2]. [PERSON] [12] and [PERSON] et al. [13] give examples of interlocking assemblies made of truncated tetrahedra. Furthermore, [PERSON] et al. [14] described three different arrangements for an assembly of buckyballs (i.e., truncated icosahedra) to be interlocking (associated with the symmetries of fifth and third orders). Recent advances concerning geometrical approaches come from computational architecture. [PERSON] et al. used TI assemblies to build facades, prioritizing their work on extending the catalog of resultant convex interlocking assemblies based on different types of tessellations [15]. Their conclusion is a connection between the geometry of the blocks and the structural performance of the assembly. Their work in [16] continued the search for an expanded catalog of TI assemblies. They concluded that three semi-regular tessellations are appropriate for TI purposes: 3.12\({}^{2}\), 4.6.12, and 4.8\({}^{2}\). Furthermore, the authors state that there is no limit for TI assemblies based on non-regular tessellations since their number is infinite. The authors also considered the support structure required to assemble floors made of convex interlocking blocks. The results were subjected to structural simulation for load-carrying capacity and deflection analysis. Finally, in [17], they introduced a computational method to generate planar surface tessellations aimed at TI assembly generation. Their solution is a parametric approach that generates surface tessellations by indicating the number of edges for the tiles and the angle values between the edges. Their system generates a TI configuration (TIC) using a resultant surface tessellation. [PERSON] et al. [18] studied the generation method of TI assemblies, focusing on different block geometries and their influence on structural performance. The study begins with generating two fundamental block shapes--tetrahedral and cubic--derived from regular square and hexagonal tessellations. These tessellations serve as the base for the interlocking blocks, ensuring that each block can interlock with its neighbors through their geometric configuration. A range of assemblies is created by varying the size and inclination angles of the blocks' faces using parametric modeling tools. This involves defining parameters for edge lengths, face angles, and the overall dimensions of the blocks. Based on these parameters, the blocks are arranged into assemblies, forming a grid mimicking potential real-world structures. The generated assemblies are subjected to numerical simulations using the 3 DEC software, Version 4.1 which can analyze discontinuous elements and discrete blocks. This simulation includes defining material properties and boundary conditions and applying loads to the assemblies. The software computes the structural response, including displacement and stress distribution. The blocks are often truncated or refined to ensure proper interlocking. This refinement involves modifying the initial shapes to enhance the contact area between blocks, thereby improving the interlocking and load distribution characteristics. The refinement process ensures that the blocks fit together seamlessly without gaps or overlaps.