# On the Penrose and Taylor-Socolar Hexagonal Tilings

## TL;DR

This paper explores the deep connection between Penrose and Taylor-Socolar hexagonal tilings, providing a unified framework that clarifies their relationship and simplifies proofs of their fundamental properties.

## Contribution

It introduces a unified approach combining geometric and algebraic methods to relate and analyze both tiling types.

## Key findings

- Unified framework for Penrose and Taylor-Socolar tilings
- Simplified proofs of basic properties
- Clarified relationship between the two tilings

## Abstract

We study the intimate relationship between the Penrose and the Taylor-Socolar tilings, within both the context of double hexagon tiles and the algebraic context of hierarchical inverse sequences of triangular lattices. This unified approach produces both types of tilings together, clarifies their relationship, and offers straightforward proofs of their basic properties.