The same ball in two explorers: put ½ at the centre and the critical line becomes the strut through 0 and ∞, the functional equation a sign flip through the centre. Swap ζ for a function with the same symmetry and no Euler product (Davenport–Heilbronn) and the zeros step off the strut: symmetry alone does not pin them, the arithmetic is the extra incidence.
The base case. ζ painted on the Riemann sphere, 0 at the bottom, ∞ at the top, and 0, 1, ∞, −1 a quarter turn apart — 1 is half way to ∞. Three sliders turn it about its three struts; the tip about ±i has four stops whose shadows are the charts s/(1 − s), 2s − 1 (zeta’s orientation), 1 − 1/s (Li) and 1/(1 − 2s). The ball beside its shadow, with the six frame points and the zeros as beads.
+ +One sphere, dragged by hand or turned by its three struts. Stand it up for the base case — 0 at the floor, ∞ at the lamp, and 0, 1, ∞, −1 a quarter turn apart, so 1 is half way to ∞ — then tip it through four stops whose shadows are the charts s/(1 − s), 2s − 1 (zeta’s orientation), 1 − 1/s (Li) and 1/(1 − 2s). The slide runs the touching point round the critical circle, and dragging the ball moves the sliders. Live ζ or ξ on the ball and on the floor, the six frame points under both names, the zeros as beads, and 3b1b’s grid as the ball’s own latitude–longitude, bent by η.
open explorer → @@ -89,11 +89,6 @@ a.card p{margin:0;color:var(--muted);font-size:12.5px}The ball projected from the other pole: lamp at the touch point ½, screen at ∞, u = ¼/(s − ½). Same frame, the functional equation a point reflection, the dial a plain rotation. Every zero falls into the centre — trivial along the mirror from one side, critical along the strut from both — and the critical strip becomes the outside of two kissing circles. Paint ζ, ξ, or χ = ζ(s)/ζ(1 − s), the gamma factor on its own.
open explorer → - -The earlier picture: ζ on a ball of diameter ½ resting at ½, turned by quaternion quarter-turns; 3b1b's grid as the ball's own latitude–longitude, bent by η.
- open explorer → -four upward triangles · five downward · the bindu