Constructing Interlocking Assemblies with Crystallographic Symmetries

TL;DR

This paper introduces a novel method for creating interlocking 3D assemblies based on planar crystallographic symmetries, extending wallpaper groups into three dimensions to generate tiles that can tessellate space in complex ways.

Contribution

It presents a new construction technique that extends wallpaper groups into 3D, enabling the design of interlocking blocks with diverse tessellation patterns, including smooth and triangulated surfaces.

Findings

Blocks can be triangulated and approximated with smooth surfaces.

A family of blocks called RhomBlock can tessellate space with lozenge patterns.

The method enables multiple tiling configurations using generalized Truchet tiles.

Abstract

This work presents a construction method for interlocking assemblies based on planar crystallographic symmetries. Planar crystallographic groups, also known as wallpaper groups, correspond to tessellations of the plane with a tile, called a fundamental domain, such that the action of the group can be used to tessellate the plane with the given tile. The main idea of this method is to extend the action of a wallpaper group so that it acts on three-dimensional space and places two fundamental domains into parallel planes. Next, we interpolate between these domains to obtain a block that serves as a candidate for interlocking assemblies. We show that the resulting blocks can be triangulated, and we can also approximate blocks with smooth surfaces using this approach. Finally, we show that there exists a family of blocks derived from this construction that can be tiled in multiple ways, characterised by generalised Truchet tiles. The assemblies of one block in this family, which we call RhomBlock, correspond to tessellations with lozenges.