Lamp at ½
Three-minute narrated version on the front page: two lamps, the swap, the zeros falling in, and ξ’s light cone.
The zeta ball, projected from the other pole. Put the lamp at the touch point ½ and the screen at the top, tangent at ∞: the chart is u = ¼ / (s − ½). The frame does not move — the poles still sit at ±½, the third strut at ±i/2, the circle |u| = ½ is where both charts agree — only the two ends of the ½–∞ strut have swapped. The old lamp, ∞, is now the centre; the old touch point, ½, is now at infinity. In this chart the functional equation is u ↦ −u and the whole dial is a rotation about the centre. And every zero, trivial and critical, falls into the origin: the trivial ones along the real axis from one side, the critical ones along the strut from both.
u = ¼ / (s − ½), centred on ∞
window ±0.60 · strut gold, mirror white, poles teal, zeros white
hover for u and s
Cross-section through the real axis
The ball of diameter ½ on the s-plane. Old lamp at the top, ∞, casting onto the floor: the poles land on 0 and 1. New lamp at the touch point ½, casting onto the ceiling: the same poles land directly above, at u = ∓½, and ∞ itself lands at the centre. The trivial zeros, bunched near the top of the ball, fall onto the ceiling at −1/(8n+2): −0.1, −0.056, −0.038, … into the centre from the left.
Key
the strut: critical line Re s = ½, now the imaginary axis
the mirror: real axis of s, still the real axis
the ball's poles s = 0, 1 at u = ∓½ (ζ's pole is the right one)
nontrivial zeros at u = ∓i/(4t)
trivial zeros at u = −1/(8n+2)
poles of χ at u = +1/(8n+2), the mirror images of its zeros
the circle |u| = ½: the ball's horizontal equator, the one circle both charts draw in the same place
the strip's edges Re s = 0 and Re s = 1: two circles of diameter ½ kissing at the centre. The critical strip is everything outside them, pinched to nothing at the lamp
Hue is the argument of the value, brightness its size. The dial turns the ball about the ½–∞ strut; the chart's own crosshair stays put.
Same frame, other lamp
Stereographic projection from the top of a ball of diameter d onto the floor, and from the bottom onto the ceiling, differ by z ↦ d²/z̄. With d = ½ that is u = ¼/(s − ½), read from inside the ball so the mirror comes out the right way round. Points on the circle |s − ½| = ½ — through 0, 1 and ½ ± i/2 — are the ball's horizontal equator and land in the same place under both lamps. Everything else trades near for far: the small disc around ½ becomes the whole outside of the lens, and the whole far field of s becomes the small disc around the centre.
So this is the chart centred on the point the s-plane does not have. The real origin ½ has gone to infinity and the imaginary origin ∞ sits at 0.
The dial is a rotation
s ↦ 1 − s is u ↦ −u, a point reflection through the centre. Turning the ball by θ about the ½–∞ strut is u ↦ e−iθu, a rotation of the plane. The quarter turn is the Wick rotation: it lays the strut, with every nontrivial zero on it, along the real axis. No Möbius map, no moving circles. Switch the paint to ξ and the half turn leaves the picture exactly as it was; with ζ it does not, and χ is the difference.
The zeros fall in
Nontrivial zeros ½ ± it sit at u = ∓i/(4t): on the strut, approaching the centre from above and below. Trivial zeros −2n sit at u = −1/(8n + 2): on the mirror, approaching from the left only — the positive real axis carries ζ(2), ζ(3), … → 1 and no zeros at all. One-sided along the real line, two-sided along the imaginary one: the real line reaches the lamp as a ray, the imaginary line as a circle through it. Both sequences converge to a point of this chart. In the s-chart that point is missing, which is the only reason they looked like they ran away.
The strip itself is the outside of two kissing circles, the images of Re s = 0 and Re s = 1. Zoom in and the strip narrows like a horn, its two edges tangent to the strut at the centre: in this chart the strip has no width at the lamp. That is a fact about the chart, not about the zeros — Davenport–Heilbronn's off-line zeros would fall into the same cusp — but it is why, seen from ∞, on the line and near the line are the same thing.
χ, what the half turn leaves over
χ(s) = ζ(s)/ζ(1 − s) = 2sπs−1 sin(πs/2) Γ(1 − s): the gamma factor by itself. Every nontrivial zero cancels out of it. What remains lives on the mirror — zeros at u = −½, −1/(8n+2) on the left, poles at u = +½, +1/(8n+2) on the right, each the point reflection of the other, since χ(s)χ(1 − s) = 1. On the strut |χ| = 1 exactly: pure phase, winding as the Riemann–Siegel θ(t). ζ on the line is that phase times a real amplitude, Hardy's Z. Fold χ in and the picture becomes ξ, symmetric under the half turn.