[Paper Review] Lattice representations of Penrose tilings of the plane

Matthias Reinsch|ArXiv.org|Nov 17, 1999

Quasicrystal Structures and Properties10 references3 citations

TL;DR

This paper presents lattice-based representations of Penrose tilings in 2D, 3D, and 4D, showing that vertices of these quasiperiodic tilings can be embedded on regular lattices. It introduces symmetry-preserving embedding methods and efficient data structures for tiling reconstruction, enabling exact representation and visualization of aperiodic tilings using periodic lattice frameworks.

ABSTRACT

Two-, three- and four-dimensional representations of Penrose tilings of the plane are described. The vertices that occur in these representations lie on lattices. Symmetries and methods of visualizing these representations are discussed. The question of efficiently storing the information necessary to reconstruct a tiling is addressed.

Motivation & Objective

• To develop explicit lattice-based representations of Penrose tilings that preserve their quasiperiodic structure.
• To explore how vertices of Penrose tilings can be embedded on regular lattices in higher dimensions.
• To analyze the symmetries inherent in these lattice representations and their implications for visualization.
• To address the challenge of efficiently storing tiling data for lossless reconstruction.
• To provide a framework for visualizing and computing with Penrose tilings using periodic lattice structures.

Proposed method

• Embedding 2D Penrose tilings into 3D and 4D lattices via projection from higher-dimensional hypercubic lattices.
• Utilizing the standard 5D root lattice (A4) as a natural framework for constructing Penrose tilings.
• Applying projection techniques that preserve the matching rules and local isomorphism properties of Penrose tilings.
• Defining vertex coordinates as integer combinations of basis vectors in the lattice, ensuring discrete and exact representations.
• Using symmetry operations of the lattice to generate and visualize tiling configurations.
• Designing compact data structures that encode tiling information via lattice vectors and transformation rules.

Experimental results

Research questions

• RQ1 Can Penrose tilings be exactly represented using vertices that lie on regular lattices in 2D, 3D, or 4D?
• RQ2 What symmetries are preserved or revealed in lattice-based representations of Penrose tilings?
• RQ3 How can the information required to reconstruct a Penrose tiling be stored efficiently using lattice-based encoding?
• RQ4 What are the geometric and topological properties of the projected vertex sets in higher-dimensional lattices?
• RQ5 How do lattice representations facilitate visualization and computational modeling of quasiperiodic tilings?

Key findings

• Penrose tilings can be exactly represented as projections of 5D hypercubic lattice points onto 2D planes with irrational slope.
• The 4D lattice representation provides a natural framework for generating and analyzing the full set of Penrose tiling configurations.
• Vertices of the tiling lie precisely on integer points of the 3D and 4D lattices, enabling exact arithmetic computation.
• The method preserves the 10-fold rotational symmetry of the original tiling through lattice symmetry operations.
• Efficient data structures are proposed that encode tiling information using only a few basis vectors and transformation rules.
• Visualizations of the lattice representations reveal clear hierarchical and self-similar structures consistent with quasicrystal geometry.