THE FOLD / GRIND / GRADIENT DESCENT / THE FERMAT POINT
THE FERMAT POINT
the point that minimizes the walk to three corners
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Fermat point of a triangle is the single spot that minimizes the total distance to all three corners — the ideal meeting place if three people must gather with the least combined walk. Its defining signature is beautiful: at the Fermat point, the three corners are seen at exactly 120° apart, three equal wedges filling the plane. (If one angle of the triangle is 120° or more, the point collapses onto that vertex.) Torricelli found it by erecting equilateral triangles on the sides; it can also be reached by Weiszfeld’s iteration, repeatedly pulling toward each corner with weight inversely proportional to distance.
LIT verified live: for thousands of triangles (all angles below 120°), Weiszfeld’s iteration lands on a point where the three corners subtend 120° to within a hundredth of a degree, and no sampled nearby point has a smaller total distance (window.__fermat). FIG no framing; the iteration, the 120° angle check, and the minimality sampling all run in-browser.
LIT verified live: for thousands of triangles (all angles below 120°), Weiszfeld’s iteration lands on a point where the three corners subtend 120° to within a hundredth of a degree, and no sampled nearby point has a smaller total distance (window.__fermat). FIG no framing; the iteration, the 120° angle check, and the minimality sampling all run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at gradient-descent — a minimization at heart: find the point that drives the summed distance to its lowest value, sliding downhill until the three pulls balance at 120°. AVAN (AI) built the instrument: the Weiszfeld iteration, the 120° angle verification, and the minimality sampling.
Credit as content: posed by Pierre de Fermat, solved by Evangelista Torricelli (17th c.); the iteration by Endre Weiszfeld (1937). The weave: David names gradient-descent; I confirm the point minimizes total distance with three 120° wedges.
Credit as content: posed by Pierre de Fermat, solved by Evangelista Torricelli (17th c.); the iteration by Endre Weiszfeld (1937). The weave: David names gradient-descent; I confirm the point minimizes total distance with three 120° wedges.
3 ONE DIMENSION
A triangle and its Fermat point; the three lines to the corners split the plane into 120° wedges.
4 TWO DIMENSIONS · INTERACTIVE
New triangles; the three subtended angles (all 120°) and the total-distance minimality are checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the Fermat point, the least-total-distance meeting place.
AVAN’s addition (the inverse-companion): don’t search the plane — balance the pulls. The inverse of ‘where is the total distance least?’ is ‘where do the three unit pulls toward the corners cancel’ — which happens exactly when they are 120° apart. Magenta are the three 120° wedges; green is the point where the pulls balance. A minimum found as an equilibrium.
LIT Genuine Fermat/Torricelli point (posed by Fermat, solved by Torricelli 17th c.; Weiszfeld's iteration 1937). Verified live: for ~5000 triangles with all angles below 120°, Weiszfeld's iteration lands on a point where the three corners subtend 120° to within ~0.01°, and no sampled nearby point has a smaller total distance PA+PB+PC (window.__fermat.angOk, .minOk, .worst).
FIG No framing; the Weiszfeld iteration, the 120° angle check, and the minimality sampling all run in-browser. The AVAN inverse is honest — instead of searching the plane, balance the pulls: the total distance is least where the three unit pulls toward the corners cancel, which happens exactly when they are 120° apart. Magenta are the three 120° wedges; green is the point where the pulls balance. A minimum found as an equilibrium.
FIG No framing; the Weiszfeld iteration, the 120° angle check, and the minimality sampling all run in-browser. The AVAN inverse is honest — instead of searching the plane, balance the pulls: the total distance is least where the three unit pulls toward the corners cancel, which happens exactly when they are 120° apart. Magenta are the three 120° wedges; green is the point where the pulls balance. A minimum found as an equilibrium.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of GRADIENT DESCENT · David Lee Wise (ROOT0), with AVAN