Lattice-theoretical insights into the construction of geometries

One of the aims of project A04 is to design piecewise linear blocks satisfying specific boundary conditions and to inspire other projects of the CRC with innovative construction methods. An example of geometries that have been constructed within the project by exploiting crystallographic space groups are topological interlocking blocks. In this context, a topological interlocking assembly is an assembly of blocks together with a fixed frame such that every non-empty set of blocks is kinematically constrained and can therefore not be removed from the assembly without causing intersections between blocks of the assembly. The surfaces of such blocks can be described as triangulated surfaces and then mathematically analysed. Equipping blocks with this additional combinatorial description allows us to apply local modifications to the blocks and simulate their structural behavior.

This seed fund project has explored a lattice-theoretical approach to construct three-dimensional blocks with interlocking properties. A three-dimensional lattice consists of integer linear combina-tions of three linearly independent vectors. Using this new approach, the vertex positions of the constructed blocks can be determined by applying translations only, thus avoiding the need to solve polynomial systems of equations to determine the vertex coordinates. Modular approaches, where larger blocks are constructed by combining smaller blocks, have proven to be fruitful. An example of a structure that results from this modular construction is the face-centred cubic lattice, which is created by systematically assembling regular octahedra and tetrahedra.

The results that have emerged from this seed fund project include:

• Construction of an interlocking block whose copies can be arranged in a spiral form to ensure the interlocking of the entire construction (b)
• Construction of various interlocking blocks that result from exploiting the tetrahedral-octahedral honeycomb (c)
• Modular construction of interlocking blocks by assembling various prisms (d)
• Implementation of a Julia-Package to analyse given blocks and verify the interlocking property of a given assembly, see [2].

Sources:

[1] Vakaliuk, I.; Goertzen, T.; Scheerer, S.; Niemeyer, A. C.; Curbach, M. (2022) Initial numerical development of design procedures for TRC bioinspired shells in: Su-duo Xue, S.-d.; Wu, J.-z.; Sun, G.-j. [eds.] Innovation, Sustainability and Legacy – Proceedings of IASS/APCS 2022, 19.–22.09.2022 in Beijing (China), p. 2597–2608.

[2] Stüttgen, S.; Akpanya, R.; Beckmann, B.; Chudoba, R.; Robertz, D.; Niemeyer, A. C. (2023) Modular Construction of Topological Interlocking Blocks—An Algebraic Approach for Resource-Efficient Carbon-Reinforced Concrete Structures in Buildings 13, issue 10, 2565 – DOI: https://doi.org/10.3390/buildings13102565

[3] Akpanya, R.; Goertzen, T.; Niemeyer, A. C. (2024) Topologically Interlocking Blocks inside the Tetroctahedrille in: arXiv: 2405.01944 [math.CO]

[4] Akpanya, R.; Goertzen, T.; Niemeyer, A. C. (2023) A Group-Theoretic Approach for Constructing Spherical-Interlocking Assemblies in: Xie, Y.; Burry, J.; Lee, T.; Ma, J. [eds.]: Integration of Design and Fabrication – Proc. of the IASS Annual Symp. 2023, 07/2023 in Melbourne (Australia), IASS, 2023, pp. 470–480.

[5] Akpanya, R.; Goertzen, T.; Wiesenhuetter, S.; Niemeyer, A. C.; Noennig, J. (2023) Topological Interlocking, Truchet Tiles and Self-Assemblies: A Construction-Kit for Civil Engineering Design in: Holdener, J.; Torrence, E.; Fong, C.; Seaton, K. [eds.]: Proc. of Bridges 2023: Mathematics, Art, Music, Architecture, Culture, 27.–31.07.2023 in Halifax (Nova Scotia, Canada), Phoenix: Tessellations Publ., p. 61–68.