Ring Lab
A circular surface with two ends: an inner rim and an outer rim, and waves that go out and come back in. Measure distance across the ring in log-radius — the ring's own stretching scale — and keep only the round waves, the ones that breathe in and out with no ripples around the circle. Then the ring is exactly the log-time string: the inner rim at radius 1 is the corner, the outer rim is the far wall, and each zero's wave is one breathing pattern. Two panels: the breathing ring for a chosen zero, and where the outer rim would have to sit to allow it.
Zero as a breathing pattern
Left. The breathing pattern for pitch E is sin(E · ln r) / √r: at rest on the inner rim, rippling outward in rings whose spacing grows with radius — equal steps in log-radius — and fading like 1/√r. Press Breathe and the rings pulse out and in; that's a standing wave, so the crests don't travel, they swell and subside in place. Snap E to a zero and this is that zero's wave wrapped around a circle. Right. A hard outer rim allows a pattern only where the pattern is at rest, so for pitch E the allowed rim radii are ekπ/E, marked on a log-radius axis for the first six zeros. Read down any vertical line: no single rim radius is allowed for two different zeros at once, apart from the inner rim itself. The outer rim would have to sit at a different place for every zero — the "moving far wall," in ring form. Why only round waves. Tick the ripple box: an angular ripple is a second family of waves with its own pitches, and once they are counted the ring has a surface's number of pitches, far too many. Keeping only the in-and-out waves is what makes the ring one-dimensional and the count come out like the zeros'. That is a legitimate move in physics — you restrict to a symmetry sector — but it means the ring picture is the string picture, not a new one, and the corner rule is still the open part.