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<title>Two Poles of One Sphere</title>
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  <header>
    <div class="eyebrow">maths · projective geometry</div>
    <h1>Two poles <i>of one sphere</i></h1>
  </header>
  <p class="lede">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.</p>
  <video id="v" controls playsinline preload="none" src="https://teahub.xyz/maths/videos/raw/commit/ec0bb56210abe087f4731777ffba8d1fa6bab3df/yantra-two-poles.mp4"></video>
  <div class="cards">
    <a class="card" href="sphere/">
      <h2>Explore it</h2>
      <p>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).</p>
      <span class="go">open explorer →</span>
    </a>
    <a class="card" href="overlay/">
      <h2>Overlays</h2>
      <p>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.</p>
      <span class="go">open overlays →</span>
    </a>
  </div>
  <h2 style="font-family:var(--display);font-weight:500;font-size:28px;margin:8px 0 0">Symmetry costs a line</h2>
  <p class="lede">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.</p>
  <video controls playsinline preload="none" src="https://teahub.xyz/maths/videos/raw/commit/a7601b040a95725e40e58f6ac214a2fe096a3a3e/yantra-symmetry.mp4"></video>
  <h2 style="font-family:var(--display);font-weight:500;font-size:28px;margin:8px 0 0">From Riemann to ζ</h2>
  <p class="lede">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.</p>
  <video controls playsinline preload="none" src="https://teahub.xyz/maths/videos/raw/commit/4c7ac00ce465abf403b9caf270e14bc1b8688a53/zeta-ball.mp4"></video>
  <h2 style="font-family:var(--display);font-weight:500;font-size:28px;margin:8px 0 0">Sphere, plane, sphere</h2>
  <p class="lede">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 = e<sup>iθ</sup>, 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.</p>
  <video controls playsinline preload="none" src="https://teahub.xyz/maths/videos/raw/commit/a8ddd96ef040299dd47e42a7167dca6e881ea1da/wick.mp4"></video>
  <p class="lede">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.</p>
  <div class="cards">
    <a class="card" href="strut/">
      <h2>The zeta strut</h2>
      <p>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.</p>
      <span class="go">open explorer →</span>
    </a>
    <a class="card" href="flow/">
      <h2>The zeta flow</h2>
      <p>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, …</p>
      <span class="go">open explorer →</span>
    </a>
    <a class="card" href="dial/">
      <h2>The zeta dial</h2>
      <p>Turn the ball about the ½–∞ strut: s − ½ ↦ e<sup>iθ</sup>(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.</p>
      <span class="go">open explorer →</span>
    </a>
    <a class="card" href="lamp/">
      <h2>Lamp at ½</h2>
      <p>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.</p>
      <span class="go">open explorer →</span>
    </a>
    <a class="card" href="zeta/">
      <h2>Zeta ball</h2>
      <p>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.</p>
      <span class="go">open explorer →</span>
    </a>
  </div>
  <section class="notes">
    <p><span class="dot" style="background:var(--up)"></span>four upward triangles · <span class="dot" style="background:var(--down)"></span>five downward · <span class="dot" style="background:var(--bindu)"></span>the bindu</p>
    <p>Triangle coordinates follow a numerical solution of the yantra's concurrency conditions (after Kulaichev and Huet), drawn as 27 lines with 29 triple points.</p>
  </section>
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