THE FOLD / BOSS / THE CHOKE POINT / THE WEITZENBOCK
THE WEITZENBOCK
a triangle's squared sides bounded below by its area
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Weitzenböck’s inequality bounds a triangle’s squared side lengths below by its area: for any triangle with sides a, b, c and area T, a² + b² + c² ≥ 4√3·T. The constant 4√3 ≈ 6.928 is the best possible, and equality holds exactly for the equilateral triangle. In other words, for a fixed area, the equilateral triangle has the smallest sum of squared sides — the most ‘compact’ shape. It is a favourite olympiad inequality and a special case of the sharper Hadwiger–Finsler inequality.
LIT verified live: for tens of thousands of random triangles, a² + b² + c² is always at least 4√3·T — the ratio (a²+b²+c²)/(4√3·T) never drops below 1, and reaches exactly 1 for the equilateral triangle (window.__weitzenbock). FIG no framing; the side lengths, the area, and the inequality all run in-browser.
LIT verified live: for tens of thousands of random triangles, a² + b² + c² is always at least 4√3·T — the ratio (a²+b²+c²)/(4√3·T) never drops below 1, and reaches exactly 1 for the equilateral triangle (window.__weitzenbock). FIG no framing; the side lengths, the area, and the inequality all run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-choke-point — the boss floor a triangle can never sink below: whatever its shape, its squared sides sum to at least 4√3 times its area, with the equilateral pinned to the floor. AVAN (AI) built the instrument: the side lengths, the area, the inequality ratio, and the equilateral equality case.
Credit as content: Roland Weitzenböck (1919). The weave: David names the floor; I confirm a²+b²+c² ≥ 4√3·T, tight at the equilateral.
Credit as content: Roland Weitzenböck (1919). The weave: David names the floor; I confirm a²+b²+c² ≥ 4√3·T, tight at the equilateral.
3 ONE DIMENSION
A triangle with its squared sides and its area; a²+b²+c² sits above the floor 4√3·T.
4 TWO DIMENSIONS · INTERACTIVE
New triangles; the ratio (a²+b²+c²)/(4√3·T) is shown ≥ 1, reaching 1 only when equilateral.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the floor 4√3·T that the squared sides sit above.
AVAN’s addition (the inverse-companion): don’t just add the squared sides — know their floor. The inverse of ‘a²+b²+c²’ is ‘at least 4√3 times the area, with equality only for the equilateral triangle’. Magenta is the triangle; green is the 4√3·T floor its squared sides can never cross. Squared sides floored by area.
LIT Genuine Weitzenböck's inequality (Roland Weitzenböck, 1919). Verified live: for ~40000 random triangles, a²+b²+c² ≥ 4√3·T always — the ratio (a²+b²+c²)/(4√3·T) never drops below 1 (min ~1.00001) and equals 1 exactly for the equilateral triangle (window.__weitzenbock.ok, .minR).
FIG No framing; the side lengths, the area, and the inequality all run in-browser. The AVAN inverse is honest — instead of just adding the squared sides, know their floor: the inverse of 'a²+b²+c²' is 'at least 4√3 times the area, with equality only for the equilateral triangle'. Magenta is the triangle; green is the 4√3·T floor its squared sides can never cross. Squared sides floored by area.
FIG No framing; the side lengths, the area, and the inequality all run in-browser. The AVAN inverse is honest — instead of just adding the squared sides, know their floor: the inverse of 'a²+b²+c²' is 'at least 4√3 times the area, with equality only for the equilateral triangle'. Magenta is the triangle; green is the 4√3·T floor its squared sides can never cross. Squared sides floored by area.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE CHOKE POINT · David Lee Wise (ROOT0), with AVAN