Mathematical Foundations and Construction of Interlocking Assemblies

Location

SMRI Seminar Room (A12 Room 301)

Abstract

Interlocking assemblies - systems of rigid bodies constrained by their geometric arrangements - are important in fields such as architecture and materials science, where they are also known as topological interlocking assemblies. Despite their practical relevance, the mathematical principles governing these assemblies are not fully developed. In this talk, we present a detailed mathematical framework for understanding interlocking assemblies in three-dimensional Euclidean space.

We define an interlocking assembly as a collection of bodies together with a particular subset called the "frame". When the frame is fixed, the remaining blocks cannot move without causing face intersections, thus ensuring the rigidity of the assembly. Focusing on three dimensions, we examine various examples of interlocking assemblies and develop methods for their construction and verification.

Central to our approach is the theory of planar crystallographic groups, also known as wallpaper groups. By extending these groups to three-dimensional space groups and continuously deforming their fundamental domains, we construct interlocking blocks with specific properties. This approach not only allows the creation of complex interlocking structures, but also provides a systematic method to prove their interlocking properties using criteria based on infinitesimal motions.