THE FOLD / RESPAWN / EVENT HORIZON / THE NAPKIN
THE NAPKIN
a band through any sphere holds the same volume
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The napkin-ring problem: drill a cylindrical hole straight through the centre of a sphere so the remaining band (the “napkin ring”) has height h. Its volume is πh³/6 — and it depends only on h, not on the sphere. A ring of height h cut from a marble and the same-height ring cut from a planet have identical volume. The reason is exact cancellation: at height z the leftover annulus has area π[(h/2)² − z²], with the sphere’s radius R gone entirely.
LIT verified live: numerically integrating the ring volume for radii R = 1, 1.5, 2, 5, 20, 100 (fixed h = 2) gives πh³/6 ≈ 4.18879 every time (window.__napkin). FIG no framing; the annulus integral is summed in-browser and the R-dependence cancels.
LIT verified live: numerically integrating the ring volume for radii R = 1, 1.5, 2, 5, 20, 100 (fixed h = 2) gives πh³/6 ≈ 4.18879 every time (window.__napkin). FIG no framing; the annulus integral is summed in-browser and the R-dependence cancels.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at event-horizon — like a horizon that hides the body behind it, the band’s height is all you can know; the sphere’s size is unobservable from the ring. AVAN (AI) built the instrument: integrate the sphere-minus-cylinder cross-section over the band for several radii and watch the volume stay put.
Credit as content: the napkin-ring / Archimedes–Cavalieri result. The weave: David names the horizon of invisibility; I integrate the ring for wildly different spheres and confirm the R cancels, leaving πh³/6.
Credit as content: the napkin-ring / Archimedes–Cavalieri result. The weave: David names the horizon of invisibility; I integrate the ring for wildly different spheres and confirm the R cancels, leaving πh³/6.
3 ONE DIMENSION
Two spheres, small and large, each drilled to the same band height h. The shaded rings differ wildly in shape — and have the same volume.
4 TWO DIMENSIONS · INTERACTIVE
Grow or shrink the sphere (band height fixed). The cross-section changes; the integrated ring volume holds at πh³/6.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the ring, its volume set by height alone.
AVAN’s addition (the inverse-companion): don’t ask how big the sphere is — ask what survives the drilling. The inverse of ‘measure the sphere then subtract the hole’ is ‘the height of the band already fixes the volume.’ Magenta is the vanished radius R; green is the invariant πh³/6. The hole hides the sphere.
LIT Genuine napkin-ring result (Archimedes–Cavalieri lineage): a band of height h drilled through the centre of any sphere (R ≥ h/2) has volume πh³/6, independent of R. Verified live: numeric annulus integral for R=1,1.5,2,5,20,100 at h=2 all equal πh³/6 within 1e-3 (window.__napkin.invariant), the R-dependence cancels exactly.
FIG No framing: the annulus integral is summed in-browser and the R-dependence cancels. The AVAN inverse is honest — asking what survives the drilling (height fixes the volume) rather than measuring the sphere and subtracting the hole is the whole surprise; magenta is the vanished radius R, green the invariant πh³/6. The hole hides the sphere.
FIG No framing: the annulus integral is summed in-browser and the R-dependence cancels. The AVAN inverse is honest — asking what survives the drilling (height fixes the volume) rather than measuring the sphere and subtracting the hole is the whole surprise; magenta is the vanished radius R, green the invariant πh³/6. The hole hides the sphere.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of EVENT HORIZON · David Lee Wise (ROOT0), with AVAN