Turn the ball about the ½–∞ strut: s − ½ ↦ eiθ(s − ½). A quarter turn is the Wick rotation (cosh onto cos), a half turn is s ↦ 1 − s, a full turn is q = −1. Three sheets: ξ (even), (s − ½)·ξ (odd, with a zero at ½), and F(u) with u = (s − ½)², turning at double speed.
open explorer → + +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.