From Tilings of Orientable Surfaces to Topological Interlocking Assemblies
Abstract
:1. Introduction
2. Symmetries and the Escher Trick
3. Constructing Planar Topological Interlocking Assemblies
4. Constructing Spherical Topological Interlocking Assemblies
4.1. The Modified Escher-like Approach
- for two real numbers the surfaces and are disjoint, i.e., the intersection is empty,
- the set defined as is a connected set in .
4.2. Examples of TIA That Arise from the Modified Escher-like Approach
4.2.1. TIA Based on an Archimedean Solid
- For , the tile is constructed by deforming the edges of the hexagonal face such that the projection of the tile onto a plane containing the face is the polygon that is indicated by the blue dotted lines in Figure 9a.
- Furthermore, for , the tile is obtained by deforming the edges of the pentagonal face such that the projection of the modified tile onto a plane containing the face is the polygon that is indicated by the blue dotted lines in Figure 9b.
4.2.2. TIA Based on the Unit Sphere
- For , the intersection between the hexagonal face and the spherical hexagon satisfies That means that this intersection contains the vertices of the hexagonal face
- For , the intersection between the pentagonal face and the spherical pentagon satisfy the equality , which means that this intersection contains the vertices of the pentagonal face
4.2.3. TIA Based on a Catalan Solid
5. Conclusions and Outlook
- The successful construction of planar and non-planar TIA using the Escher-like method;
- The exploration of a semi-regular tiling yielding an interlocking structure;
- The extension of the method to non-planar surfaces, such as the unit sphere and various convex polyhedra, demonstrates the feasibility of constructing spherical and polyhedral TIA.
- Practical insights gained from 3D-printed models provides a profound understanding of the assembly and interlocking processes.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Akpanya, R.; Goertzen, T.; Niemeyer, A.C. From Tilings of Orientable Surfaces to Topological Interlocking Assemblies. Appl. Sci. 2024, 14, 7276. https://doi.org/10.3390/app14167276
Akpanya R, Goertzen T, Niemeyer AC. From Tilings of Orientable Surfaces to Topological Interlocking Assemblies. Applied Sciences. 2024; 14(16):7276. https://doi.org/10.3390/app14167276
Chicago/Turabian StyleAkpanya, Reymond, Tom Goertzen, and Alice C. Niemeyer. 2024. "From Tilings of Orientable Surfaces to Topological Interlocking Assemblies" Applied Sciences 14, no. 16: 7276. https://doi.org/10.3390/app14167276