Penrose tilings obey matching rules — constraints on how tiles can fit together. They also have Ammann bars — lines that run across the tiling at specific angles, continuous if and only if the tiling is correct. And they have height functions — scalar fields that assign a number to each vertex, increasing or decreasing according to edge crossings. These three frameworks were developed independently, each capturing some aspect of what makes the tiling work.
Pardo-Guerra, Washburn, and Allahyarov prove they are the same thing.
The unification goes through cochains. Assign a signed bar-crossing count to each directed edge of the tiling. This produces an antisymmetric 1-cochain. The matching rules hold if and only if the cochain is closed — meaning its value around any cycle is zero. This is a cocycle condition, the discrete analog of a conservative vector field. And when the cocycle condition holds, the discrete Poincaré lemma guarantees a scalar potential, which is exactly the height function.
Four descriptions: matching rules, Ammann bar continuity, cocycle closure, height function existence. Each is equivalent to the others. The result extends beyond Penrose to canonical projection tilings from Z^N, where the lattice-coordinate cochains reconstruct vertex positions and, for generic windows, form a Z-basis.
The conjecture at the end is sharper: this four-way equivalence may characterize precisely the Pisot substitution tilings — the tilings whose expansion factor is a Pisot number (an algebraic integer greater than 1 whose conjugates all lie inside the unit circle). If true, the equivalence isn't just a property of known examples but a structural theorem about which tilings can exist.
The matching rules were always algebraic topology. It took 50 years to see it.