Constructing Topologically Interlocking Blocks from Tetrahedra and Octahedra in the Tetroctahedrille

Core Concepts

This paper presents a modular approach to construct non-convex interlocking blocks by combining tetrahedra and octahedra within the tetrahedral-octahedral honeycomb, also known as the tetroctahedrille. The resulting interlocking blocks are versatile and allow for various topological interlocking assemblies.

Abstract

The paper introduces the tetroctahedrille, a three-dimensional lattice formed by alternating octahedra and tetrahedra, and demonstrates how it can be used to construct interlocking blocks. Several examples of such blocks are presented, including the kitten, cushion, and shuriken, along with their corresponding topological interlocking assemblies.
The key highlights are:

The tetroctahedrille provides a framework for constructing interlocking blocks by combining tetrahedra and octahedra.
The kitten, cushion, and shuriken are examples of interlocking blocks that can be created within the tetroctahedrille.
Topological interlocking assemblies are presented for each of the constructed blocks, demonstrating their versatility.
Modifications such as truncation and continuous deformation are explored to generate new interlocking blocks from the existing ones.
The paper discusses the potential for approximating geometric objects using the tetroctahedrille and the resulting interlocking assemblies.

Stats

The tetroctahedrille can be described by the set of integer points ⟨e1 + e2, e1 + e3, e2 + e3⟩Z, where {e1, e2, e3} is the standard basis of the Euclidean 3-space.
The vertices of the tetrahedra in the tetroctahedrille can be described by the sets T1, T2, T3, T4, and the octahedra can be described by the set O.

Quotes

"Exploiting a modular approach to construct new polyhedra by combining finitely many polyhedra, such as the Platonic solids, can yield geometric shapes that allow various applications."
"Since regular tetrahedra and octahedra are convex, they can be described by their sets of incident vertices."
"Proving that an assembly of blocks is a TI assembly turns out to be a high-dimensional problem as motions of any group of blocks have to be considered simultaneously."

Key Insights Distilled From

Topologically Interlocking Blocks inside the Tetroctahedrille

by Reymond Akpa... at arxiv.org 05-06-2024

https://arxiv.org/pdf/2405.01944.pdf

Deeper Inquiries

How can the group-theoretic properties of the tetroctahedrille be further exploited to generate new families of interlocking blocks

The group-theoretic properties of the tetroctahedrille can be further exploited to generate new families of interlocking blocks by considering the symmetries and transformations that preserve the interlocking property. One approach is to analyze the group actions on the tetroctahedrille and identify subgroups that correspond to specific transformations of the blocks. By understanding the group structure, we can systematically generate new families of interlocking blocks by applying group operations to the existing blocks. This can lead to the creation of more complex and versatile interlocking assemblies with unique geometric properties.

What are the limitations of the tetroctahedrille-based approach, and how can it be extended to handle more complex geometric shapes

The limitations of the tetroctahedrille-based approach lie in its restriction to simple polyhedral shapes like tetrahedra and octahedra. To handle more complex geometric shapes, the approach can be extended by incorporating a wider variety of polyhedra or by introducing deformations and modifications to the existing blocks. For instance, by allowing for non-convex polyhedra or curved surfaces within the interlocking assemblies, the approach can accommodate a broader range of geometric structures. Additionally, integrating computational tools for geometric modeling and optimization can enhance the design flexibility and efficiency of creating interlocking blocks for complex shapes.

Can the insights from this work be applied to the design and construction of real-world structures, such as buildings or bridges, that utilize topological interlocking assemblies

The insights from this work can be applied to the design and construction of real-world structures that utilize topological interlocking assemblies, such as buildings or bridges, in several ways. Firstly, the concept of topological interlocking can be employed in architectural and structural engineering to create innovative and sustainable designs that rely on geometric interlocking for stability and load-bearing capacity. By leveraging the principles of interlocking blocks, architects and engineers can develop structures that are structurally efficient, aesthetically pleasing, and environmentally friendly.
Furthermore, the computational methods and geometric approximations presented in the research can be utilized in the digital design and fabrication processes of complex structures. By using advanced modeling software and digital fabrication techniques, architects and engineers can translate the theoretical concepts of topological interlocking into practical construction solutions. This can lead to the realization of novel architectural forms, efficient structural systems, and customized building components that optimize material usage and construction processes. Overall, the application of topological interlocking assemblies in real-world projects has the potential to revolutionize the way we design and build structures, offering new possibilities for sustainable and resilient architecture.