The intimate relationship between the Penrose and the Taylor-Socolar tilings is studied, 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.
Keywords: Penrose tiling; Taylor–Socolar tiling; double hexagon tiling; inverse sequences of hierarchical lattices; nested triangularizations.