# 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
#### 4.2.2 TIA Based on the Unit Sphere  
As a next example, the unit sphere is examined. In particular, the unit sphere is used to construct a spherical example of a TIA that consists of two different block types. For this goal, the truncated icosahedron is very useful. The existence of the truncated icosahedron establishes that the unit sphere can be tiled with 20 spherical hexagons \(h^{\prime}_{1},\ldots,h^{\prime}_{20}\) and 12 spherical pentagons \(p^{\prime}_{1},\ldots,p^{\prime}_{12}\); see Figure 12.  
Hence, this tiling can be exploited to construct the desired TIA. We recall that the hexagonal faces of the truncated icosahedron \(\mathcal{I}\) are given by \(h_{1},\ldots,h_{20}\) and that the pentagonal faces of this polyhedron are given by \(p_{1},\ldots,p_{12}\). Moreover, recall that the modified  
Figure 11: Two exploded views of the TIA that realises the surface of the truncated icosahedron.  
Figure 10: (**a**) Different views of the block that result from interpolating between a pentagonal face of the truncated icosahedron and a modified tile that is contained on the surface of a scaled truncated icosahedron (**b**) Different views of the block that result from interpolating between a hexagonal face of the truncated icosahedron and a modified tile that is contained on the surface of a scaled truncated icosahedron.  
Figure 9: Applying the Escher-Like approach to hexagons (**a**) and pentagons (**b**) of the truncated icosahedron (**c**).  
tiles that are obtained from these hexagonal and pentagonal faces are denoted by \(t_{1},\ldots,t_{20}\) and \(t^{\prime}_{1},\ldots,t^{\prime}_{12}\). In order to achieve the goal of constructing a TIA that is based on the unit sphere, the following two assumptions are made:  
* For \(1\leq i\leq 20\), the intersection between the hexagonal face \(h_{i}\) and the spherical hexagon \(h^{\prime}_{i}\) satisfies \(|h_{i}\cap H^{\prime}_{i}|=6\). That means that this intersection contains the vertices of the hexagonal face \(h_{i}\).
* For \(1\leq j\leq 12\), the intersection between the pentagonal face \(p_{j}\) and the spherical pentagon \(p^{\prime}_{j}\) satisfy the equality \(|p_{j}\cap p^{\prime}_{j}|=5\), which means that this intersection contains the vertices of the pentagonal face \(p_{j}\).  
Thus, the desired interlocking blocks are constructed by interpolating between the tiles \(h_{i},2\cdot t_{i},3\cdot h^{\prime}_{i}\) for \(1\leq i\leq 20\) and between the tiles \(p_{j},2\cdot t^{\prime}_{j},3\cdot p^{\prime}_{j}\) for \(1\leq j\leq 12\). Figure 13 shows different views of the block that result from interpolating between the tiles \(p_{1},t^{\prime}_{1},3\cdot p^{\prime}_{1}\).  
The TIA that arises from the above construction is illustrated in Figure 14. The TIA arising from the truncated icosahedron and the above construction have been 3D-printed, see Figure 15.  
Figure 14: Two exploded views of the TIA that is based on the unit sphere.  
Figure 12: Tiling of (approximated) unit sphere consisting of 20 spherical hexagons and 12 spherical pentagons.  
Figure 13: Three different views of the interlocking block that results from interpolating between a hexagonal face of the truncated icosahedron, a modified tile of a scaled truncated icosahedron and a spherical hexagon that forms a tile of the unit sphere.  
In order to establish the interlocking property of the given assembly, it is sufficient to choose a frame that consists of any two blocks of the assembly.