The zeta flow
One Möbius map, T(z) = z/(z + 1), takes the upright Riemann sphere to the zeta picture. It is a shear with one fixed point, at 0, so it sits in a flow: Tt(z) = z/(tz + 1). t = 0 is the sphere, t = 1 is the zeta picture, t = 2 is the full step. Every Tt is a real ball on the plane: tipped by 2·arctan t, radius 1/(1 + t²), resting at t/(1 + t²). The free pole walks the harmonic ladder 1/t; the equator is a straight line only at t = 1.
Zero, half, full
t = 0. The Riemann sphere as usual: radius 1, resting at 0, axis upright. Latitudes cast circles around 0, longitudes cast rays. Poles at 0 and ∞. Sphere phase.
t = 1. The zeta picture: tipped a quarter turn, half size, resting at ½. Poles at 0 and 1; the equator casts the critical line, the only moment in the flow when it is straight, because the point going to the lamp (the old −1/t) is on the equator exactly when t = 1. Plane phase.
t = 2. The full step: tipped 127° (past flat), radius ⅕, resting at ⅖. Poles at 0 and ½; the equator casts the circle through ⅓ and 1. Sphere phase again, on the far side.
As t grows the ball keeps tipping toward upside-down, shrinking like 1/t², and sliding back toward 0; the second pole walks 1, ½, ⅓, … into the first. The flow never closes. Compare the dial, which turns about ½–∞ and does close.
Key
● poles and their shadows · — equator and its shadow · ● lamp · ● the ball's lowest point (real origin, 1/(t+1))
Cool latitudes are the hemisphere around the fixed pole 0, warm the hemisphere around the walking pole. "Rays" draws a few lamp rays through the equator so you can see the projection happen. The ladder marks 1, ½, ⅓, … on the plane.
Plane units: the standard sphere's equator casts the unit circle, so 1 on the plane is the grid line labelled 1.