Abstract This paper identifies a hidden topological property within the class of even-vertex aperiodic tilings, such as Penrose P3 and Ammann-Beenker A5 systems. We prove that these tilings possess a global parity state that is non-locally derivable from the tiling’s local geometry. We present two primary results: first, a transformation Φ that generates a distinct, non-MLD “companion” tiling by mapping vertex parity to face color; and second, a constructive algorithm for deriving the corresponding companion tileset. By analyzing the “Ordering and Resolution” of matching rules, we demonstrate why certain systems (P3) generate these companions while others (P2) remain degenerate. Finally, a computational search within a generalized search space suggests that this single parity bit represents the exhaustive non-MLD companion state for this class of systems.
Andrew Bayly (Wed,) studied this question.