One sphere, four turns
+ζ painted on the Riemann sphere, and nothing else: the sphere stands with 0 at the bottom, ∞ at the top, and the four real points 0, 1, ∞, −1 a quarter turn apart on the real circle — 1 is half way to ∞. The three struts are the three axes you can turn it about. A quarter turn about the ±i strut tips it on its side: 0 and ∞ land on the unit circle together, and the shadow is zeta's picture. Another quarter turn and ∞ is at the bottom: Li's chart. Turning about the 0–∞ strut slides the touch point along the unit circle, through 1, i, −1, −i. Turning about the ±1 strut is the dial whose half turn is s ↦ 1 − s. The ball is the thing; every chart is its shadow from the top.
+ + + +The ball
+seen from the i side; the lamp is at the top, the touch point at the bottom
+Its shadow: the chart
+stop 0 · w = s / (1 − s): ζ's 0 at the centre, its pole at the lamp
+ +hover the chart for the point's two names
+ +The four stops are four charts
+The quarter turn about the ±i strut is w ↦ (w − 1)/(w + 1): it fixes ±i and cycles the real circle 0 → −1 → ∞ → 1 → 0. Read off the shadow at each stop: 0: w = s/(1 − s), ζ's 0 at the centre, its pole at the lamp, the critical line the unit circle. ¼: w = 2s − 1, the zeta orientation — 0 and ∞ of the sphere on the unit circle at ∓1 (the scale-and-shift puts them on 0 and 1), the critical line standing up through the touch point. ½: w = 1 − 1/s, Li's chart, ζ's pole at the centre and ζ's 0 at the lamp. ¾: w = 1/(1 − 2s). Then home.
+Three struts, three turns
+Tip about ±i (the stops above). Slide about 0–∞: the point touching the plane runs round the unit circle, 1 → i → −1 → −i. Dial about ±1: its half turn is s ↦ 1 − s, the functional equation. The three half turns are s ↦ (1 − s)/(1 − 2s), s ↦ s/(2s − 1) and s ↦ 1 − s; each is the product of the other two. On the critical line they read t ↦ 1/4t, t ↦ −1/4t, t ↦ −t.
+1 is half way to ∞
+On the sphere the arc from 0 to 1 equals the arc from 1 to ∞: a quarter turn each. The beads on the rim show it. In ζ's names the four real points are 0, ½, 1 and ∞ — the two poles of its picture and its two units — so the point of the s-plane called ∞ is the sphere's −1, a quarter turn from either pole, no farther than ½ is.
+Where the zeros are
+In the sphere's own frame every nontrivial zero sits on the unit circle, and all of them crowd the point −1: the first within 4° of it, the fortieth within half a degree. The trivial zeros sit on the real circle and crowd −1 from the other side, at −2/3, −4/5, −6/7, … So the one place on the ball where ζ is busy is the bead opposite 1. The lamp page is a magnifying glass held over that bead.
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