A strip across the lattice

Drag the gold ring across the lattice: the strip passes through the point (0, 0) and cuts towards it. The lattice points inside the strip make a staircase; dropped onto the line, a step to the right becomes a long piece L (gold) and a step up a short piece S (ice blue). Aim at (q, p) and the slope is p/q. Shift this stretch of staircase along the coral segment and it sits exactly as before against the strip, so the next stretch follows the same pattern: with p/q in lowest terms, the row repeats every p + q pieces, q long and p short, and those are the pieces lit. Aim at an unreduced point such as (6, 4) and the coral segment stops at (3, 2). Whichever point you aim at, the row repeats; the larger p + q in lowest terms, the longer the stretch that repeats. (The slope goes up to 1, where long and short pieces are equally long and the colours still alternate.) Switch to slope 1/φ: that direction meets no lattice point, and the corner reads NEVER. The small chart on the right counts longer and longer stretches of the row from the left: each count of long against short pieces is a fraction, yet they tend to φ ≈ 1.618, which is none; aimed at a lattice point, they settle on the fraction q/p instead.