THE FOLD / CO-OP / THE SYNC / THE VIVIANI
THE VIVIANI
three distances, one constant sum — the height
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Viviani’s theorem. Stand anywhere inside an equilateral triangle. Drop a perpendicular to each of the three sides and measure the three distances. They always add up to the same total — exactly the triangle’s height — no matter where you stand.
Wander the point around and the three distances trade off: step toward one side and that distance shrinks while the other two grow to compensate, their sum frozen. It is a conserved quantity hiding in plain sight — three freely-changing numbers locked to one constant. It’s also the geometric heart of barycentric coordinates: divide the three distances by the height and you get three weights that always sum to 1 and pin the point’s exact location.
LIT verified live: for hundreds of random interior points the three perpendicular distances sum to the height exactly, and the normalized distances reconstruct the original point (window.__viviani). FIG no framing; the constant-sum invariant and the barycentric reconstruction are exact.
Wander the point around and the three distances trade off: step toward one side and that distance shrinks while the other two grow to compensate, their sum frozen. It is a conserved quantity hiding in plain sight — three freely-changing numbers locked to one constant. It’s also the geometric heart of barycentric coordinates: divide the three distances by the height and you get three weights that always sum to 1 and pin the point’s exact location.
LIT verified live: for hundreds of random interior points the three perpendicular distances sum to the height exactly, and the normalized distances reconstruct the original point (window.__viviani). FIG no framing; the constant-sum invariant and the barycentric reconstruction are exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this in THE SYNC — the co-op domain of quantities kept in lockstep. Viviani is three distances held in perfect sync: push one down and the others rise so the total never drifts. AVAN (AI) built the instrument: the moving point, the three perpendiculars, the barycentric recovery.
The weave: David names the seat (kept in sync); I make the three distances trade off while their sum stays pinned to the height — the stacked bar in 1D, the draggable point in 2D, the barycentric inverse in 3D. The sphere is the seam. Credit: Vincenzo Viviani (1622–1703), a pupil of Galileo.
The weave: David names the seat (kept in sync); I make the three distances trade off while their sum stays pinned to the height — the stacked bar in 1D, the draggable point in 2D, the barycentric inverse in 3D. The sphere is the seam. Credit: Vincenzo Viviani (1622–1703), a pupil of Galileo.
3 ONE DIMENSION
The three distances stacked into one bar. As the point moves, the coloured segments swap size — but the bar’s total length holds fixed at the triangle’s height. Conservation, drawn as a bar that never changes length.
4 TWO DIMENSIONS · INTERACTIVE
An equilateral triangle with an interior point and its three perpendiculars. Move the point and watch d1, d2, d3 trade off while their sum stays exactly equal to the height — the invariant made visible.
5 THREE DIMENSIONS + AVAN’S INVERSE
The point and its three distances turning in 3D — green, the forward map: a location inside the triangle produces three perpendicular lengths that sum to the height.
AVAN’s addition (the inverse-companion): the magenta point is the location rebuilt from the three distances alone. Because the sum is always the height, dividing the distances by it gives three weights that add to 1 — the point’s barycentric coordinates — and those weights place it right back. That is the exact inverse: forward, point → three distances; backward, three distances → point, and both are clean, total maps. Viviani’s constant is precisely the normalization that makes the inverse well-defined — without the fixed sum there would be no way to turn three lengths into one unambiguous position. The magenta reconstruction lands exactly on the green original. Green is where you are; magenta is you, recovered from nothing but your three distances to the walls.
LIT Genuine Viviani's theorem (Vincenzo Viviani, 1622-1703, pupil of Galileo). Verified live: for hundreds of random interior points the three perpendicular distances sum to the height exactly, and the normalized distances (barycentric weights) reconstruct the original point to full precision (window.__viviani.sumEqualsHeight && barycentricRecovers). The constant-sum invariant and the barycentric reconstruction are both exact and cross-checked in-browser.
FIG No framing: the constant sum-of-distances equal to the height, and the invertibility to barycentric coordinates, are real and verified over many interior points. The connection to barycentric coordinates is the genuine mathematical content (the fixed sum is exactly the normalization that makes the inverse map well-defined), demonstrated by reconstruction, not asserted.
FIG No framing: the constant sum-of-distances equal to the height, and the invertibility to barycentric coordinates, are real and verified over many interior points. The connection to barycentric coordinates is the genuine mathematical content (the fixed sum is exactly the normalization that makes the inverse map well-defined), demonstrated by reconstruction, not asserted.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE SYNC · David Lee Wise (ROOT0), with AVAN