On the Penrose and Taylor-Socolar hexagonal tilings

Area of Science:

• Mathematics
• Geometry
• Tiling Theory

Background:

• Penrose tilings are aperiodic structures with unique properties.
• Taylor-Socolar tilings are related to hierarchical lattices.
• Understanding the relationship between different tiling systems is crucial in geometry.

Purpose of the Study:

• To investigate the intimate relationship between Penrose and Taylor-Socolar tilings.
• To develop a unified approach for producing both tiling types.
• To clarify the connection and prove basic properties of these tilings.

Main Methods:

• Utilizing double hexagon tiles for a geometric context.
• Employing hierarchical inverse sequences of triangular lattices for an algebraic context.
• Developing a unified framework to generate and analyze both tiling systems.

Main Results:

• The unified approach successfully produces both Penrose and Taylor-Socolar tilings concurrently.
• The study clarifies the intricate relationship between these two types of tilings.
• Straightforward proofs for the fundamental properties of both tilings are provided.

Conclusions:

• A unified method effectively generates and explains Penrose and Taylor-Socolar tilings.
• The research deepens the understanding of aperiodic tilings and their mathematical underpinnings.
• The findings offer a simplified framework for studying these complex geometric structures.