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.
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 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 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 →Zeta ball
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
Triangle coordinates follow a numerical solution of the yantra's concurrency conditions (after Kulaichev and Huet), drawn as 27 lines with 29 triple points.