Saddle Lab
This is the surface I keep pointing at: the modular surface, a saddle-shaped sheet with one edge that runs off to infinity, built by gluing the sides of a single tile of the hyperbolic plane. It has closed loops that behave like primes, standing waves whose pitches behave like zeros, and a formula connecting the two that reads line for line like Riemann's. For its own zeta function the Riemann hypothesis is a theorem. Three tabs: a path bouncing on it, its primes, and its pitches next to the real zeros and a sphere's.
Geodesic flow on the modular surface exact
The picture is the upper half-plane with the hyperbolic ruler: distances shrink as you go up, so the strip between the two vertical lines above the arc is one tile, and it is infinitely tall in ordinary inches but finite in hyperbolic length. The surface is that tile with its left edge glued to its right edge and the two halves of the arc glued to each other. A straight line on this surface is a circle arc that meets the bottom at right angles; when it leaves the tile it re-enters from the glued edge — that is the "bounce." Unlike the circle billiard, there is no angle that repeats: the path is chaotic, and the button direction + 0.01° shows what that means — two paths that start almost identically part ways after a handful of bounces. Chaotic motion of this kind is what produces vibration pitches with the zeros' statistics, and what a sphere — where every path is regular — never produces. The spike at the top, where the tile narrows to nothing, is the cusp: the surface's one edge, running off to infinity.
The closed loops of the surface, and the primes they play exact
Some paths on the surface come back to their start and close up. Each closed loop has a length L, and the number N = eL — the "norm" — plays the role a prime plays for the integers, with L playing log p. The count of loops with length at most L grows like eL/L, which is the prime number theorem with eL in place of x. In Selberg's formula for this surface, the standing-wave pitches sit on one side and a sum over these loops, weighted by their lengths, sits on the other — exactly where the primes and their powers sit in Riemann's explicit formula. The loops are labelled by an integer, the trace: trace 3 is the shortest, and the loop for it is drawn on the left as the arc it traces before closing. Here the "primes" are things you can walk.
Pitches of three instruments tabulated values
Left: the standing-wave pitches of an ordinary sphere. They come in a rigid ladder, and each rung is shared by many waves at once (3, 5, 7, 9 …); the ladder is regular and the sharing is total. Middle: the heights of the zeta zeros — no two alike, irregular spacing, and a measured tendency to repel each other (small gaps are rarer than chance would give), which is the fingerprint of chaotic vibration. Right: the standing-wave pitches of the modular surface, computed numerically over the last decades. Like the zeros they are irregular and never shared — but look closely and they do not repel: close pairs occur (19.42 and 19.48, for instance). That is a famous surprise. The modular surface's motion is chaotic, yet its extra arithmetic symmetries make its pitch spacings look random-and-independent rather than repelling; a generic saddle surface without those symmetries shows the zeros' repulsion. So the sphere's ladder is the wrong kind of thing; a generic saddle has the zeros' statistics but no primes; the modular surface has the primes but the wrong statistics. No known instrument has both, and "find the one that does" is a fair one-sentence statement of the open problem.