Two poles of one sphere
The Śrī Yantra turned inside out by circle inversion, then wrapped onto a ball by stereographic projection. On the sphere the inversion is just a flip, and the bindu and infinity are the two poles.
Explore it
Spin the yantra and its inversion on the Riemann sphere, with the plane underneath. Slide the lift, compare stereographic with great circles, switch the sphere off to see just the plane, and add or remove the outer yantra (lotus rings and bhupura).
open explorer →Overlays
The yantra laid over its own inversions: one in the outer rim, one in a smaller circle tuned so the trefoil's lobe meetings land on the outer triangle's midpoints. Toggle any of the three.
open overlays →Symmetry costs a line
Force up–down mirror symmetry on the yantra while keeping all 29 triple points exact, and two of its bases are pulled into one line, carrying one point on the axis and a mirror pair: 1 + 2.
From Riemann to ζ
The plane is a ball with a lamp on top. Feed ζ a grid and it draws latitude and longitude of a globe lying on its side: the usual stereographic picture tipped a quarter turn, halved, and slid by ½, so the poles land on 0 and 1. The equator casts the critical line; its two ends are ½ and ∞; the functional equation is the half-turn that swaps the poles and keeps the equator. The Riemann hypothesis, on the ball: every zero lives on the equator, climbing toward the lamp.
Sphere, plane, sphere
The pattern underneath. Unit quaternions projected from −1; the pure imaginaries are a sphere. Multiply on both sides (locked) and it only spins. Multiply on one side, q = eiθ, and it drifts through the projection point: sphere, plane, inside-out sphere, plane, sphere — the four Wick turns. The plane is a chart missing one point; the sphere is projective; putting the point back is analytic continuation. In the plane phase the grid about j is zeta's grid, poles on 0 and 1. Overlay every turn and the spheres are a pencil threaded on the jk ring, centres sliding along i; the full circle sweeps out the hypersphere. Every prime gives ζ a ball wrapping at rate log p (the harmonics); the lamp, ∞, has a ball with no period, and its factor is the gamma function. Fold it in and ξ lives on the sphere, the far side of the equator the alias of the near side, folded at the Nyquist line ½.
Lamp at ½
The same ball, projected from the other pole. Move the lamp to the touch point ½ and hold the screen above, tangent at ∞: the frame does not move — the poles land directly above where they were — only ∞ and ½ trade places, and near trades with far. The zeros, which climbed away forever in the old chart, fall into the centre: the trivial ones along the real axis from one side, the critical ones along the strut from both. The functional equation is a point reflection, the dial a plain turn of the page. And ξ draws its own light cone: a bell curve shrinking along the critical line, cosh growing along the real one, |ξ| = 1 a cone between them — the quarter turn swaps inside for outside.
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.
Zeta ball · four turns
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. Then the Wick turn: multiply on one side only, p ↦ eiθp, and the ball swells through the lamp into the plane (the lamp’s shadow), turns inside out, flattens into the plane projected from ½, and comes home — the four corners of the quaternion explorable, with q₁ = eiθq and q₂ = q̄ read out. 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 →The zeta strut
ζ, ξ and Davenport–Heilbronn painted in the s-plane, in w = 1/(s − ½), and on the Riemann sphere with ½ in front; Hardy's Z(t) along the strut, with the off-line zero at t ≈ 85.7 marked.
open explorer →The zeta flow
Sphere → zeta picture → the full step, as one Möbius flow z ↦ z/(tz + 1). At every t the ball is placed in space (tipped 2·arctan t, radius 1/(1 + t²), resting at t/(1 + t²)) and its shadow follows. The equator is straight only at t = 1; the free pole walks 1, ½, ⅓, …
open explorer →The zeta dial
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 →Lamp at ½
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 →four upward triangles · five downward · the bindu
Triangle coordinates follow a numerical solution of the yantra's concurrency conditions (after Kulaichev and Huet), drawn as 27 lines with 29 triple points.