Construction Methods for Space-Filling Heterogeneous Topological Interlocking Assemblies

Deforming fundamental domains of wallpaper groups provides a systematic way to generate non-convex blocks which admit topological interlocking assemblies (TIAs). We use this approach to construct TIAs that fully occupy the space between two parallel planes and incorporate multiple block types. In addition to wallpaper groups, semiregular tessellations are employed in the construction of such TIAs. These construction methods open up an extensive design space for TIAs, expanding the possibilities of feasible interlocking systems and creating new opportunities for architectural and material design. Several resulting block families can be interpreted as geometric realizations of generalized Truchet tiles or decorated lozenge tilings and, with suitable colouring rules, we establish a one-to-one correspondence between these tilings and specific TIAs. This framework enables a systematic investigation of symmetric and asymmetric assemblies derived from diverse block types.