From af75c05c7f40201b35b83de697836d4647bfb000 Mon Sep 17 00:00:00 2001 From: teahubxyz Date: Wed, 28 Oct 2026 05:13:01 +0000 Subject: [PATCH] =?UTF-8?q?index:=20"Lamp=20at=20=C2=BD"=20video=20(3:07)?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Co-Authored-By: Claude Fable 5.1 Claude-Session: https://claude.ai/code/session_01FAqjv3n4yABxvJPJHFRhVA --- index.html | 3 +++ 1 file changed, 3 insertions(+) diff --git a/index.html b/index.html index 25a6170..44c03d8 100644 --- a/index.html +++ b/index.html @@ -59,6 +59,9 @@ a.card p{margin:0;color:var(--muted);font-size:12.5px}

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.

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Lamp at ½

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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.

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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.