Constructing Interlocking Assemblies with Crystallographic Symmetries

TL;DR

The paper develops a general method to construct interlocking 3D assemblies from planar wallpaper (wallpaper) groups by extending the group action to $\mathbb{R}^3$ and interpolating between parallel fundamental domains. The core idea, the Escher Trick, produces new fundamental domains $F'$ from an initial domain $F$, enabling the creation of 3D blocks $X_\lambda$ whose boundaries are triangulated and which yield assemblies under $G$. It introduces VersaTiles, a versatile tile framework that supports non-unique tilings, including the RhomBlock whose assemblies correspond to lozenge tilings; it also presents limit cases for $p4$ and $p3$ that generate the Versatile Block and RhomBlock, respectively. The work connects geometric construction with combinatorial tiling theory (Truchet and lozenge tilings) and suggests pathways to space-filling 3D assemblies, modular design, and reconfigurable structures with clear mathematical foundations in crystallographic and three-dimensional symmetry groups.

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.

Figures (77)

• Figure 1:
• Figure 2:
• Figure 3:
• Figure 4:
• Figure 6:

Theorems & Definitions (51)

• Definition 1.1: GoertzenFIB
• Example 2.1
• Remark 2.2
• Definition 2.3
• Remark 2.4
• Example 2.5
• Remark 2.6
• Example 2.7
• Definition 2.8
• Lemma 2.9: PleskenKristallographischeGruppen