# From Tilings of Orientable Surfaces to Topological Interlocking Assemblies
## 4 Constructing Spherical Topological Interlocking Assemblies
### The Modified Escher-like Approach  
The main aim of this work is to modify the Escher-like approach in such a way that it is possible to construct TIA based on tessellations of arbitrary orientable surfaces. So by introducing this modification, we establish the construction of TIA that result from  
Figure 7: (**a**) The square block that results from interpolating between three copies of modified square tiles; (**b**) The octagon block that results from interpolating between three modified octagon tiles; (**c**) TIA of the square and the octagon block in the fashion of the truncated square tiling where the frame consists of the 20 most outer octagon blocks. structures such as curved surfaces, convex, and non-convex polyhedra, the unit sphere and the torus. First examples of TIA arising from a slight modification of the Escher-like approach have been illustrated in [33], where [PERSON] et al. formulate a construction method that produces TIA realising the surface of given convex polyhedra whose vertices all lie on the unit sphere. Based on this formulation, [PERSON] et al. provide different examples of TIA constructed by exploiting the structure of the Platonic solids and (angular) tubes. The approach proposed here can be viewed as a generalisation of the method introduced in [33], expanding the design space of TIA and additionally enabling the customisation and optimisation of interlocking structures for specific applications. The _modified Escher-Like approach_ to construct TIA realising arbitrary orientable surfaces can be described as follows: For simplicity, the orientable surfaces in this paper are considered to be surfaces without boundaries. First, let \(\mathcal{M}:=\{\mathcal{S}_{t}\mid t\in[0,1]\}\) be a set of orientable surfaces such that  
* for two real numbers \(0\leq t_{1}<t_{2}\leq 1\) the surfaces \(\mathcal{S}_{t_{1}}\) and \(\mathcal{S}_{t_{2}}\) are disjoint, i.e., the intersection \(\mathcal{S}_{t_{1}}\cap\mathcal{S}_{t_{2}}\) is empty,
* the set \(\mathcal{L}\) defined as \(\mathcal{L}:=\bigcup_{S\in\mathcal{M}}\mathcal{S}\) is a connected set in \(\mathbb{R}^{3}\). As \(\mathcal{S}_{0}\) is a surface without boundary, the set \(\mathbb{R}^{3}\setminus\mathcal{S}_{0}\) consists of exactly two connected components, \(C_{1}\) and \(C_{2}\). In order to formulate the desired construction, assume that for all \(t\in(0,1]\) the surface \(\mathcal{S}_{t}\) is contained either in \(C_{1}\) or in \(C_{2}\). Further, let \(n\in\mathbb{N}\) be a natural number and let \(\mathcal{T}=\{\tau_{1},\ldots,\tau_{n}\}\) be a tiling of \(\mathcal{S}:=\mathcal{S}_{0}\). Lastly, let \(f:[0,1]\times\mathcal{L}\to\mathbb{R}^{3}\) be a continuous function such that for all \(t\in[0,1]\) the set \(\mathcal{T}^{\prime}:=\{f(t,\tau_{1}),\ldots,f(t,\tau_{n})\}\) is a tiling of the surface \(\mathcal{S}_{t}\). With this information, the construction of the desired interlocking blocks can be achieved. In particular, for all \(1\leq i\leq n\), the block \(B_{i}\) is constructed by defining \(B_{i}:=\bigcup_{t\in[0,1]}f(t,\tau_{i})\). Since two blocks \(B_{i}\) and \(B_{j}\) with \(i\
eq j\) satisfy the equality \(B_{i}\cap B_{j}=\partial B_{i}\cap\partial B_{j}\), the resulting set \(\{B_{1},\ldots,B_{n}\}\) is an assembly of blocks that satisfies  
\[\bigcup_{i=1}^{n}B_{i}=\mathcal{L}.\]  
Another interpretation of this modified Escher-Like approach is that for \(1\leq i\leq n\), the block \(B_{i}\) is obtained by interpolating between the tiles \(f(0,\tau_{i})\) and \(f(1,\tau_{i})\). Moreover, this interpolation is specified by the function \(f\) and is carried out in such a way that for every \(t\in[0,1]\), the set \(\{B_{1}\cap\mathcal{S}_{t},\ldots,B_{n}\cap\mathcal{S}_{t}\}\) forms a tiling of the surface \(\mathcal{S}_{t}\). For simplicity, the interpretation just described, omitting the function \(f\), is used to present the following examples of TIA. Note that an assembly of blocks resulting from this construction is not necessarily a TIA.