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
- Dyskin, A.V.; Estrin, Y.; Kanel-Belov, A.J.; Pasternak, E. A new concept in design of materials and structures: Assemblies of interlocked tetrahedron-shaped elements. Scr. Mater. 2001, 44, 2689–2694. [Google Scholar] [CrossRef]
- Weizmann, M.; Amir, O.; Grobman, Y.J. Topological interlocking in buildings: A case for the design and construction of floors. Autom. Constr. 2016, 72, 18–25. [Google Scholar] [CrossRef]
- Miodragovic Vella, I.; Kotnik, T. Stereotomy, an Early Example of a Material System. In Proceedings of the 35th eCAADe Conference, Rome, Italy, 20–22 September 2017; pp. 251–258. [Google Scholar] [CrossRef]
- Tessmann, O.; Rossi, A. Geometry as Interface: Parametric and Combinatorial Topological Interlocking Assemblies. J. Appl. Mech. 2019, 86, 111002. [Google Scholar] [CrossRef]
- Harsono, K.; Shih, S.G.; Wagiri, F.; Alfred, W. Integration of Design and Performance Evaluation for Reusable Osteomorphic-Block Masonry. Nexus Netw. J. 2023, 26, 71–94. [Google Scholar] [CrossRef]
- Hua, H. Porous interlocking assembly: Performance-based dry masonry construction with digital stereotomy. Archit. Intell. 2024, 3, 20. [Google Scholar] [CrossRef]
- Dyskin, A.V.; Estrin, Y.; Pasternak, E.; Khor, H.C.; Kanel-Belov, A.J. The principle of topological interlocking in extraterrestrial construction. Acta Astronaut. 2005, 57, 10–21. [Google Scholar] [CrossRef]
- Carlesso, M.; Giacomelli, R.; Krause, T.; Molotnikov, A.; Koch, D.; Kroll, S.; Tushtev, K.; Estrin, Y.; Rezwan, K. Improvement of sound absorption and flexural compliance of porous alumina-mullite ceramics by engineering the microstructure and segmentation into topologically interlocked blocks. J. Eur. Ceram. Soc. 2013, 33, 2549–2558. [Google Scholar] [CrossRef]
- Dyskin, A.; Estrin, Y.; Kanel-Belov, A.; Pasternak, E. Topological interlocking of platonic solids: A way to new materials and structures. Philos. Mag. Lett. 2003, 83, 197–203. [Google Scholar] [CrossRef]
- Kanel-Belov, A.J.; Dyskin, A.V.; Estrin, Y.; Pasternak, E.; Ivanov-Pogodaev, I.A. Interlocking of Convex Polyhedra: Towards a Geometrical Theory of Fragmented Solids. Mosc. Math. J. 2010, 10, 337–342. [Google Scholar] [CrossRef]
- Glickman, M. The G-block system of vertically interlocking paving. In Proceedings of the Second International Conference on Concrete Block Paving, Delft, The Netherlands, 10–12 April 1984; pp. 10–12. [Google Scholar]
- Gallon, J.G. Machines et Inventions Approuvées par l’Académie Royale des Sciences Depuis son Établissement Jusqu’à Present; Avec Leur Description; l’Académie Royale des Sciences: Paris, France, 1735. [Google Scholar]
- Subramanian, S.G.; Eng, M.; Krishnamurthy, V.R.; Akleman, E. Delaunay Lofts: A biologically inspired approach for modeling space filling modular structures. Comput. Graph. 2019, 82, 73–83. [Google Scholar] [CrossRef]
- Akleman, E.; Krishnamurthy, V.R.; Fu, C.A.; Subramanian, S.G.; Ebert, M.; Eng, M.; Starrett, C.; Panchal, H. Generalized abeille tiles: Topologically interlocked space-filling shapes generated based on fabric symmetries. Comput. Graph. 2020, 89, 156–166. [Google Scholar] [CrossRef]
- Mullins, C.; Ebert, M.; Akleman, E.; Krishnamurthy, V. Voronoi Spaghetti & VoroNoodles: Topologically Interlocked, Space-Filling, Corrugated & Congruent Tiles. In Proceedings of the SIGGRAPH Asia 2022 Technical Communications, SA ’22, Daegu, Republic of Korea, 6–9 December 2022. [Google Scholar] [CrossRef]
- Ebert, M.; Akleman, E.; Krishnamurthy, V.; Kulagin, R.; Estrin, Y. VoroNoodles: Topological Interlocking with Helical Layered 2-Honeycombs. Adv. Eng. Mater. 2023, 26, 2300831. [Google Scholar] [CrossRef]
- Goertzen, T.; Niemeyer, A.; Plesken, W. Topological Interlocking via Symmetry. In Proceedings of the 6th FIB International Congress 2022, Oslo, Norway, 12–16 June 2022; Novus Press: Oslo, Norway, 2022. [Google Scholar]
- Goertzen, T. Constructing Interlocking Assemblies with Crystallographic Symmetries. arXiv 2024, arXiv:2405.15080. [Google Scholar]
- Goertzen, T. Mathematical Foundations of Interlocking Assemblies. arXiv 2024, arXiv:2405.17644. [Google Scholar] [CrossRef]
- Wang, Z.; Song, P.; Isvoranu, F.; Pauly, M. Design and Structural Optimization of Topological Interlocking Assemblies. ACM Trans. Graph. 2019, 38, 1–13. [Google Scholar] [CrossRef]
- Bejarano, A.; Hoffmann, C. A generalized framework for designing topological interlocking configurations. Int. J. Archit. Comput. 2019, 17, 53–73. [Google Scholar] [CrossRef]
- Loing, V.; Baverel, O.; Caron, J.F.; Mesnil, R. Free-form structures from topologically interlocking masonries. Autom. Constr. 2020, 113, 103117. [Google Scholar] [CrossRef]
- Kaplan, C.S.; Salesin, D.H. Escherization. In Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques, New Orleans, LA, USA, 23–28 July 2000; SIGGRAPH ’00. pp. 499–510. [Google Scholar] [CrossRef]
- Smith, D.; Myers, J.S.; Kaplan, C.S.; Goodman-Strauss, C. An aperiodic monotile. arXiv 2023, arXiv:2303.10798. [Google Scholar] [CrossRef]
- Smith, D.; Myers, J.S.; Kaplan, C.S.; Goodman-Strauss, C. A chiral aperiodic monotile. arXiv 2023, arXiv:2305.17743. [Google Scholar] [CrossRef]
- Akpanya, R.; Goertzen, T.; Liu, Y.; Stüttgen, S.; Robertz, D.; Xie, Y.M.; Niemeyer, A.C. Constructing Topological Interlocking Assemblies Based on an Aperiodic Monotile. In Proceedings of the IASS 2024 Symposium: Redefining the Art of Structural Design, Zurich, Switzerland, 26–30 August 2024; Block, P., Boller, G., DeWolf, C., Pauli, J., Kaufmann, W., Eds.; International Association for Shell and Spatial Structures (IASS): Madrid, Spain, 2024. accepted, not yet published. [Google Scholar]
- Akpanya, R.; Goertzen, T.; Wiesenhuetter, S.; Niemeyer, A.C.; Noennig, J. Topological Interlocking, Truchet Tiles and Self-Assemblies: A Construction-Kit for Civil Engineering Design. In Proceedings of the Bridges 2023: Mathematics, Art, Music, Architecture, Culture, Halifax, NS, Canada, 27–31 July 2023; Holdener, J., Torrence, E., Fong, C., Seaton, K., Eds.; Tessellations Publishing: Phoenix, AZ, USA, 2023; pp. 61–68. [Google Scholar]
- Goertzen, T.; Macek, D.; Schnelle, L.; Weiß, M.; Reese, S.; Holthusen, H.; Niemeyer, A.C. Mechanical Comparison of Arrangement Strategies for Topological Interlocking Assemblies. arXiv 2023, arXiv:2312.01958. [Google Scholar] [CrossRef]
- Conway, J.H.; Burgiel, H.; Goodman-Strauss, C. The Symmetries of Things; A K Peters, Ltd.: Wellesley, MA, USA; New York, NY, USA, 2008; pp. xviii+426. [Google Scholar]
- Grünbaum, B.; Shephard, G.C. Tilings and Patterns; A Series of Books in the Mathematical Sciences; An introduction; W. H. Freeman and Company: New York, NY, USA, 1989; pp. xii+446. [Google Scholar]
- Dyskin, A.; Estrin, Y.; Pasternak, E.; Khor, H.; Kanel-Belov, A. Fracture Resistant Structures Based on Topological Interlocking with Non-planar Contacts. Adv. Eng. Mater. 2003, 5, 116–119. [Google Scholar] [CrossRef]
- Stüttgen, S.; Akpanya, R.; Beckmann, B.; Chudoba, R.; Robertz, D.; Niemeyer, A.C. Modular Construction of Topological Interlocking Blocks—An Algebraic Approach for Resource-Efficient Carbon-Reinforced Concrete Structures. Buildings 2023, 13, 2565. [Google Scholar] [CrossRef]
- Akpanya, R.; Goertzen, T.; Niemeyer, A.C. A Group-Theoretic Approach for Constructing Spherical-Interlocking Assemblies. In Proceedings of the IASS Annual Symposium 2023: Integration of Design and Fabrication, Melbourne, Australia, 10–14 July 2023; Xie, Y., Burry, J., Lee, T., Ma, J., Eds.; International Association for Shell and Spatial Structures (IASS): Madrid, Spain, 2023; pp. 470–480. [Google Scholar]
- Viana, V. From Solid to Plane Tessellations, and Back. Nexus Netw. J. 2018, 20, 741–768. [Google Scholar] [CrossRef]
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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
APA StyleAkpanya, R., Goertzen, T., & Niemeyer, A. C. (2024). From Tilings of Orientable Surfaces to Topological Interlocking Assemblies. Applied Sciences, 14(16), 7276. https://doi.org/10.3390/app14167276