THE FOLD / SPAWN / NULL ISLAND / THE POMPEIU
THE POMPEIU
three distances that always form a triangle
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Pompeiu’s theorem is a small gem of Euclidean geometry. Take an equilateral triangle ABC and any point P in the plane. Then the three distances PA, PB, PC can always be arranged into a triangle — they satisfy the triangle inequality. Moreover, that ‘distance triangle’ is degenerate (flat — the longest distance exactly equals the sum of the other two) precisely when P lies on the circumcircle of ABC. Off the circumcircle you get a genuine triangle; on it, the three distances collapse to a straight line.
LIT verified live: for thousands of random points P, the three distances to an equilateral triangle’s vertices satisfy the triangle inequality; when P sits exactly on the circumcircle, the longest distance equals the sum of the other two to ~1e-16 (degenerate); and off the circle the inequality is strict (window.__pompeiu). FIG no framing; the distances, the triangle-inequality test, and the circumcircle degeneracy all run in-browser.
LIT verified live: for thousands of random points P, the three distances to an equilateral triangle’s vertices satisfy the triangle inequality; when P sits exactly on the circumcircle, the longest distance equals the sum of the other two to ~1e-16 (degenerate); and off the circle the inequality is strict (window.__pompeiu). FIG no framing; the distances, the triangle-inequality test, and the circumcircle degeneracy all run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at null-island — the spawn point: drop any P anywhere and a brand-new triangle spawns from its three distances to the equilateral’s corners. AVAN (AI) built the instrument: the three distances, the triangle-inequality check, and the circumcircle degeneracy test.
Credit as content: Dimitrie Pompeiu (1936). The weave: David names the spawn; I confirm PA, PB, PC form a triangle, flat exactly on the circumcircle.
Credit as content: Dimitrie Pompeiu (1936). The weave: David names the spawn; I confirm PA, PB, PC form a triangle, flat exactly on the circumcircle.
3 ONE DIMENSION
An equilateral triangle, a point P, and the triangle built from the three distances PA, PB, PC.
4 TWO DIMENSIONS · INTERACTIVE
Move P; PA,PB,PC always satisfy the triangle inequality — flat exactly when P is on the circumcircle.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the triangle formed by the three distances.
AVAN’s addition (the inverse-companion): don’t just measure the distances — assemble them. The inverse of ‘the three distances from P’ is ‘a triangle with those side lengths’, which flattens exactly when P reaches the circumcircle. Magenta is the equilateral triangle and its circumcircle; green is the distance-triangle it spawns. Three lengths, always a triangle.
LIT Genuine Pompeiu's theorem (Dimitrie Pompeiu, 1936). Verified live: for ~8000 random points P, the distances PA,PB,PC to an equilateral triangle satisfy the triangle inequality; P on the circumcircle gives a degenerate triangle (longest = sum of other two, worst ~1e-16), and off the circle it is strict (window.__pompeiu.ti, .deg, .st, .worst).
FIG No framing; the distances, the triangle-inequality test, and the circumcircle degeneracy all run in-browser. The AVAN inverse is honest — instead of just measuring the distances, assemble them: the inverse of 'the three distances from P' is 'a triangle with those side lengths', which flattens exactly when P reaches the circumcircle. Magenta is the equilateral triangle and its circumcircle; green is the distance-triangle it spawns. Three lengths, always a triangle.
FIG No framing; the distances, the triangle-inequality test, and the circumcircle degeneracy all run in-browser. The AVAN inverse is honest — instead of just measuring the distances, assemble them: the inverse of 'the three distances from P' is 'a triangle with those side lengths', which flattens exactly when P reaches the circumcircle. Magenta is the equilateral triangle and its circumcircle; green is the distance-triangle it spawns. Three lengths, always a triangle.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of NULL ISLAND · David Lee Wise (ROOT0), with AVAN