Mathematical Foundations of Interlocking Assemblies

TL;DR

This paper develops a rigorous mathematical framework for interlocking assemblies, especially those with crystallographic symmetries, and proves the interlocking property for specific block configurations like the RhomBlock.

Contribution

It introduces a comprehensive mathematical theory for interlocking assemblies, including definitions, methods, and proofs for properties of assemblies with symmetries.

Findings

Verified interlocking properties for assemblies with crystallographic symmetries.

Proved the interlocking property for the RhomBlock using combinatorial theory.

Established a framework for future research on interlocking assemblies.

Abstract

The study of interlocking assemblies is an emerging field with applications in various disciplines. However, to this day, the mathematical treatment of these assemblies has been sparse. In this work, we develop a comprehensive mathematical theory for interlocking assemblies, providing a precise definition and a method for proving the interlocking property based on infinitesimal motions. We consider assemblies with crystallographic symmetries and verify interlocking properties for such assemblies. Our analysis includes the development of an infinite polytope with crystallographic symmetries to ensure that the interlocking property holds. For a certain block, called the RhomBlock, that can be assembled in numerous ways, characterised by the combinatorial theory of lozenges, we rigorously prove the interlocking property. By conclusively showing that any assembly of the RhomBlock is interlocking, we provide a robust framework for further exploration and application of interlocking assemblies.