# From Tilings of Orientable Surfaces to Topological Interlocking Assemblies
## 1 Introduction  
Since the influential work of [PERSON] et al. [1], topological interlocking assemblies have been studied extensively. TIA can be applied as block systems in various architectural settings; see [2; 3; 4; 5; 6]. Moreover, there are other wide-ranging applications, from extraterrestrial construction [7] to solutions for sound insulation systems; see [8]. The common ground is that TIA, in contrast to monolithic systems, provides solutions for modular systems, where the geometry of the blocks and contact with neighbouring blocks play a key role.  
In this work, the main focus lies on the design and construction of TIA. The construction of topological interlocking assemblies (TIA) can often be formulated using the mathematical framework of tilings. The fundamental idea involves starting with a given tiling and continuously transforming it into other tilings, akin to a growth process observed in biological systems.  
In the convex case, the tiles of the initial tilings are contracted, enabling the construction of TIA with all Platonic solids, as detailed in [9; 10]. Among these, the tetrahedron is one of the most extensively studied interlocking blocks [1; 11]. Its truncated form, known as the \"Abeille Block\", named after [PERSON], was patented in 1699 for a flat-vault structure [12].  
Continuous deformation of tilings can also be achieved using Voronoi cells, as described in [13; 14]. A block in this construction has the property that its intersection with any intermediate plane is a Voronoi cell with a centre point determined by a given curve. By varying the choice of curves that are used to connect two corresponding points in the given parallel planes, one obtains assemblies of three-dimensional blocks with differentinterlocking properties. This approach yields TIA with space-filling properties that can be tailored to specific requirements [15; 16]. This paper focuses on constructing TIA based on tilings derived from the Escher trick, where regular tiles are deformed to create more complex configurations. This approach is initially discussed in [17] and further developed in [18] for tilings with wallpaper symmetry. The assemblies generated with this method are mathematically proven to possess interlocking properties, as shown in [19].  
In the literature, the construction methods for non-planar TIA are mainly focused on TIA with convex blocks, see [20; 21; 22]. The main goal of this work is the development of a general method that can be applied in order to obtain both planar and non-planar TIA with non-convex blocks. For this endeavour, the Escher-like method is generalised to create TIA based on arbitrary orientable surfaces. We demonstrate the versatility of this modified approach by constructing TIA based on the unit sphere, the truncated icosahedron, and the deltoidal hexecontahedron. Furthermore, we explore a TIA arising from a semi-regular tiling, which consists of regular polygons arranged in a repeating pattern. Semi-regular tilings provide a rich ground for constructing TIA due to their high symmetry and structural diversity. This exploration underscores the robustness of the Escher-like method in generating both planar and non-planar TIA, offering significant potential for innovative applications in various fields. The blocks of the constructed assemblies (both planar and non-planar) possess triangulated boundaries, and together with a fixed frame consisting of the most-outer blocks (in the planar case) or two distinct blocks in the spherical case, they can be shown to be TIA. For this, the infinitesimal criterion introduced in [20] can be used, which implies the regular TIA definition; see [19].