Shape Lab
Take any terrain — a row pasted from a sweep's results file, or dials you set here — and look at it as a shape instead of a score. Four views: the terrain with its pitch ladder against the zeros; the mode surface, every standing wave stacked in 3-D (mode number one way, position the other, amplitude as height); the struck surface, the string plucked at a point and followed through time in 3-D; and the swirl, the runners drawn in position-versus-speed, where the round bowl's circle and every other terrain's shape become visible.
The setup — paste a results row (header line + one row), or set the dials
Terrain, and its pitches against the zeros
Export as audio (WAV, 44.1 kHz) — open in Audacity and take the spectrum
Pair correlation — how likely a second pitch is at a given distance from a first (this is where the twin primes leave their mark)
The mode surface — every standing wave, stacked (position across · mode number into the screen · amplitude up) · drag to rotate, scroll to zoom, double-click to reset
The target. The violet button builds the one terrain whose pitches are the zeros: from the zeros' own counting staircase, the semiclassical inversion (Wu and Sprung, 1993) gives the width of the well at every height, and that well — a smooth bowl widening like √V·log V with a fractal fringe of ledges, one ledge per zero — is drawn here and re-solved exactly. Its mode surface, struck surface and swirl are the shapes a correct candidate would have to reproduce. Everything about it comes from the zeros, so it is the answer key, not a discovery; use it to compare shapes, not to score. Row k is the k-th standing wave. Low rows are the slow, broad modes that live in the deepest parts of the terrain; high rows are fast and reach everywhere. Where the terrain is tall, the waves are pushed out and the surface goes flat; where hills are close together, waves get trapped between them and the surface shows islands. This is the system's shape in the literal sense: everything it can ring at, drawn at once.
The struck surface — pluck the string at one point and follow the wave field through time (position across · time into the screen · displacement up) · drag to rotate, scroll to zoom, double-click to reset
The response is the sum over modes of (mode at the pluck point) × (mode at position) × cos(pitch × time) — the exact evolution of a lossless string. On an empty string the pluck splits into two travelling humps that bounce off the ends and recombine; with hills, the humps scatter at every hill and the field fills with interference. The pattern you see repeating along the time axis is the system's ring, laid out as a landscape.
The dome — the energy landscape over position and speed, with the pitches as rings around it (position across · speed into the screen · energy up) · drag to rotate, scroll to zoom, double-click to reset
The surface is energy = speed² + terrain(position): a runner's total energy over the whole position–speed plane. Its contour lines are the swirl's loops, and the violet rings are the contours at the system's own pitches — the only levels a standing wave can occupy. For the target, the rings sit at the zeros' heights (their squares, in this lab's units) and the dome is the log-widening bowl with its ledges; a candidate is a good candidate if its dome has that shape and its rings fall on the same heights. This is the "levels around a spherical surface" picture made literal — a surface in position-and-speed, which is why it can carry the zeros as levels when an ordinary sphere cannot.
The ball — the dome squeezed onto a sphere: rings at the pitches, infinity at the pole, phase around each ring
A map projection of the dome, not a new object. Three squeezes: by count places one ring per level at equal steps — the only squeeze with a meaning, since it is the counting law itself; by the smooth count law places each ring where the T·log T formula says its level should sit, so the rings' unevenness is exactly the primes' ±½ wobble; arctan is a plain drawing choice (the Riemann-sphere map). In each, infinity is the north pole; the ring at each pitch is drawn at its latitude; longitude is the phase — where the runner is in its cycle. The rings crowd toward the pole because the ladder is infinite and a sphere is finite, which is exactly why the real dome is an open bowl. The amber dot runs around the lowest ring; with a twist of 0.5 it comes back to its start only every second lap — the spin-½ behaviour, and the loop lab's twist made visible. The yarn. Tick wind the critical line and the line itself is wrapped onto the ball: height is position along the line (by count, so it reaches ring k exactly at zero k), and longitude is zeta's phase, which turns faster and faster as the line climbs — the Riemann–Siegel theta, ½·log(t/2π) per unit height — and jumps by half a turn at every zero, so the thread hops to the far side of the ball each time it reaches a ring. The white dots are the zeros. This is your spiral, wound on the sphere: the winding is the sphere factor's phase, the hops are the zeros, and the tightening toward the pole is the counting law. It is a picture of the known relation, drawn the way you have always seen it. Ripple bends each ring by its own standing wave — mode k with its k humps on ring k — which is the mode surface wrapped onto the sphere. Full loop adds the mirror half: heights below zero are the reflection of heights above (the functional equation), the sphere is re-read so that the pole is infinity in both directions, and the line becomes a closed great circle through the pole — its two ends meet there with opposite windings, which is the fold once more. Whether a real, lossless system lives on that closed loop is exactly the open question; the loop itself is plain geometry. Prime beads. The primes themselves, placed on the sphere by log p and spun by the flow: as the traveller climbs to height t, bead p turns by t·log p — the projector, on the ball. Their combined pull (weights 1/√p) is the arrow below the sphere; at every zero it swings backward. Twin primes are the amber pairs: two beads at almost the same latitude turning at almost the same rate, so they stay side by side like a double star and drift apart only slowly — one twin pair's beat takes about π·p units of height. They do not move any ring; what they change is the fine correlation between rings, which the pair-correlation panel above draws.
The swirl — runners in position-versus-speed, one loop per pitch
Each closed curve is one runner at one pitch: across is where it is, up and down is how fast it's moving and which way. A round bowl gives circles of the same shape at every energy — the isochronous swirl. A log-widening bowl gives loops that grow slowly and flatten. Hills put dents and pinches in the loops where the runner slows. This is the picture the "circle" you noticed belongs to; it's called phase space.