# From Tilings of Orientable Surfaces to Topological Interlocking Assemblies
## 4 Constructing Spherical Topological Interlocking Assemblies
### Examples of TIA That Arise from the Modified Escher-like Approach
#### 4.2.1 TIA Based on an Archimedean Solid  
In literature, the Archimedean solids form a class of convex uniform polyhedra. That means that these polyhedra are vertex-transitive and that the faces of these polyhedra form regular polygons. There are exactly 13 Archimedean solids. The solid that forms the focus of this work is the truncated icosahedron \(\mathcal{I}\). This polyhedron consists of 20 hexagonal and 12 pentagonal faces; see Figure 8.  
Hence, these polygons form a tiling of the surface of the truncated icosahedron. The truncated icosahedron is therefore a suitable candidate for constructing a non-planar TIA that consists of two different block types. It is assumed that the polyhedron \(\mathcal{I}\) is centred at the origin and that all vertices lie on the sphere with radius 1. In order to construct the desired TIA, let \(h_{1},\ldots,h_{20}\) be the 20 hexagonal faces and \(p_{1},\ldots,p_{12}\) be the 12 pentagonal faces of the polyhedron \(\mathcal{I}\). Hence, the faces of the scaled polyhedron \(2\cdot\mathcal{I}\) are given by \(2\cdot h_{1},\ldots,2\cdot h_{20}\) and \(2\cdot p_{1},\ldots,2\cdot p_{12}\). The surface of the polyhedron \(2\cdot\mathcal{I}\) can be tiled by tiles \(t_{1},\ldots,t_{20}\) and \(t_{1}^{\prime},\ldots,t_{12}^{\prime}\) that are obtained as described in the following:  
* For \(1\leq i\leq 20\), the tile \(t_{i}\) is constructed by deforming the edges of the hexagonal face \(2\cdot h_{i}\) such that the projection of the tile \(t_{i}\) onto a plane containing the face \(2\cdot h_{i}\) is the polygon that is indicated by the blue dotted lines in Figure 9a.
* Furthermore, for \(1\leq j\leq 12\), the tile \(t_{i}^{\prime}\) is obtained by deforming the edges of the pentagonal face \(2\cdot p_{j}\) such that the projection of the modified tile \(t_{j}^{\prime}\) onto a plane containing the face \(p_{j}\) is the polygon that is indicated by the blue dotted lines in Figure 9b. Figure 9c illustrates that the set \(\{t_{1},\ldots,t_{20}\}\cup\{t_{1}^{\prime},\ldots,t_{12}^{\prime}\}\) forms a tiling of the surface \(\mathcal{S}_{1}\). By interpolating between the hexagonal face \(h_{i}\) and the modified tile \(t_{i}\) for \(1\leq i\leq 20\), and between the pentagonal face \(p_{j}\) and the modified tile \(t_{j}^{\prime}\) for \(1\leq j\leq 12\), copies of the blocks shown in Figure 10a,b are constructed.  
The process of interpolating between the above tiles can be carried out in such a way that the resulting blocks admit an assembly that realises the surface of the truncated icosahedron; see Figure 11.  
It is not clear how a minimal frame of the TIA illustrated above has to be chosen in order to establish the interlocking property of the given assembly. By employing experiments with 3D-printed assemblies, we observed that the modified Escher-like approach can be iterated in order to construct a TIA with a reduced number of frame blocks. For instance, for \(1\leq i\leq 20\) and for \(1\leq j\leq 12\), it is possible to interpolate between the tiles  
Figure 8: The truncated icosahedron, i.e., the Archimedean solid consisting of 20 hexagonal and 12 pentagonal faces.  
\(h_{i},t_{i},3\cdot h_{i}\) and between the tiles \(p_{j},t^{\prime}_{j},3\cdot p_{j}\) in such a way that a TIA realising the surface of the truncated icosahedron is constructed. A 3D-printed version of the described assembly is illustrated in Figure 15 on the left. The interlocking property of this assembly can be computationally verified by choosing any two blocks as the frame.