THE FOLD / RESPAWN / THE CONTINUE / THE FAGNANO
THE FAGNANO
the min-perimeter inscribed triangle is the orthic
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Fagnano’s problem asks: of all triangles inscribed in a given acute triangle — one vertex on each side — which has the smallest perimeter? The answer is the orthic triangle, whose vertices are the feet of the three altitudes. It is also the path a light ray traces bouncing inside the triangle: at each side the incoming and outgoing segments make equal angles, so the orthic triangle is the unique closed billiard orbit. Its perimeter has a clean closed form: a·cos A + b·cos B + c·cos C.
LIT verified live two ways: the orthic triangle’s perimeter (from the altitude feet) equals a·cos A + b·cos B + c·cos C to ~1e-15, and across thousands of acute triangles no randomly-sampled inscribed triangle ever has a smaller perimeter than the orthic (window.__fagnano). FIG no framing; the altitude feet, the closed-form perimeter, and the minimality sampling all run in-browser.
LIT verified live two ways: the orthic triangle’s perimeter (from the altitude feet) equals a·cos A + b·cos B + c·cos C to ~1e-15, and across thousands of acute triangles no randomly-sampled inscribed triangle ever has a smaller perimeter than the orthic (window.__fagnano). FIG no framing; the altitude feet, the closed-form perimeter, and the minimality sampling all run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-continue — the light ray that reflects off each side and returns to where it began: the orthic triangle is the closed orbit that continues forever. AVAN (AI) built the instrument: the altitude feet, the closed-form perimeter, and the minimality check.
Credit as content: Giovanni Fagnano (1775); the reflection view via Hermann Schwarz and Lipót Fejér. The weave: David names the returning orbit; I confirm the orthic triangle is the minimum-perimeter inscribed triangle.
Credit as content: Giovanni Fagnano (1775); the reflection view via Hermann Schwarz and Lipót Fejér. The weave: David names the returning orbit; I confirm the orthic triangle is the minimum-perimeter inscribed triangle.
3 ONE DIMENSION
An acute triangle, its altitudes, and the orthic triangle (feet of the altitudes) — the closed billiard path.
4 TWO DIMENSIONS · INTERACTIVE
New acute triangles; orthic perimeter vs the a·cosA+b·cosB+c·cosC form, and vs sampled inscribed triangles.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the orthic triangle — the minimum-perimeter inscribed path.
AVAN’s addition (the inverse-companion): don’t search all inscribed triangles — drop the altitudes. The inverse of ‘the minimum-perimeter inscribed triangle’ is ‘the feet of the three altitudes’, which is also the closed light path that reflects off every side. Magenta are the three altitudes; green is the orthic triangle they land on. Minimality read as reflection.
LIT Genuine Fagnano's problem (Giovanni Fagnano, 1775; reflection view via Schwarz and Fejér). Verified live two ways: the orthic triangle's perimeter (from altitude feet) equals a·cosA+b·cosB+c·cosC to ~3.6e-15, and across ~1500 acute triangles no sampled inscribed triangle beats the orthic perimeter (window.__fagnano.pf, .mn, .worst, .tested).
FIG No framing; the altitude feet, the closed-form perimeter, and the minimality sampling all run in-browser. The AVAN inverse is honest — instead of searching all inscribed triangles, drop the altitudes: the inverse of 'the minimum-perimeter inscribed triangle' is 'the feet of the three altitudes', which is also the closed light path reflecting off every side. Magenta are the three altitudes; green is the orthic triangle they land on. Minimality read as reflection.
FIG No framing; the altitude feet, the closed-form perimeter, and the minimality sampling all run in-browser. The AVAN inverse is honest — instead of searching all inscribed triangles, drop the altitudes: the inverse of 'the minimum-perimeter inscribed triangle' is 'the feet of the three altitudes', which is also the closed light path reflecting off every side. Magenta are the three altitudes; green is the orthic triangle they land on. Minimality read as reflection.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE CONTINUE · David Lee Wise (ROOT0), with AVAN