THE FOLD / LOOT / THE JACKPOT / THE HERON
THE HERON
triangle area from its three sides
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Heron’s formula gives a triangle’s area from its three side lengths alone — no height, no angle: with semi-perimeter s = (a+b+c)/2, Area = √(s(s−a)(s−b)(s−c)). It leads to a rare species: Heronian triangles, with integer sides and integer area — 3-4-5 (area 6), 13-14-15 (area 84), 5-5-6 (area 12). Most integer-sided triangles have irrational area (2-3-4 does not qualify); Heronian ones are the exception where both are whole.
LIT verified live: Heron’s formula matches the coordinate (shoelace) area over thousands of triangles, and the Heronian triangles have exactly the integer areas claimed (window.__heron). FIG no framing; exact integer test for Heronian, float agreement for the formula.
LIT verified live: Heron’s formula matches the coordinate (shoelace) area over thousands of triangles, and the Heronian triangles have exactly the integer areas claimed (window.__heron). FIG no framing; exact integer test for Heronian, float agreement for the formula.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-jackpot — the rare payout where a triangle’s sides AND its area all come out whole; Heronian triangles are that jackpot. AVAN (AI) built the instrument: Heron’s formula, the coordinate-area cross-check, and the integer-area (16·Area² a perfect square) test for Heronian triangles.
Credit as content: Heron of Alexandria (c. 60 CE). The weave: David names the-jackpot; I compute area from three sides by Heron’s formula, confirm it equals the coordinate area, and flag the integer-sided triangles whose area is also an integer — the Heronian jackpot.
Credit as content: Heron of Alexandria (c. 60 CE). The weave: David names the-jackpot; I compute area from three sides by Heron’s formula, confirm it equals the coordinate area, and flag the integer-sided triangles whose area is also an integer — the Heronian jackpot.
3 ONE DIMENSION
s = (a+b+c)/2; Area = √(s(s−a)(s−b)(s−c)). 3-4-5 → area 6. 13-14-15 → area 84. 5-5-6 → 12. Heronian = integer sides + integer area.
4 TWO DIMENSIONS · INTERACTIVE
A triangle from its sides; Heron’s area beside the coordinate area, with the Heronian integer-area test.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: area from three sides alone.
AVAN’s addition (the inverse-companion): find a triangle’s area from only its side lengths — no height, no coordinates, no trigonometry — via s and √(s(s−a)(s−b)(s−c)). The inverse of ‘drop a height or place coordinates and use ½bh’ is ‘combine the three sides directly.’ Magenta is the base×height construction; green is Heron’s three-side formula. Area from the sides, nothing more.
LIT Genuine Heron's formula (Heron of Alexandria, c. 60 CE). Verified live: over 5000 random triangles, √(s(s−a)(s−b)(s−c)) from the side lengths equals the coordinate (shoelace) area (window.__heron.formulaMatchesArea, worst ~1e-11), and the Heronian triangles (integer sides + integer area, tested via 16·Area² a perfect square) have exactly the claimed integer areas — 3-4-5→6, 13-14-15→84, etc. (window.__heron.heronianCorrect).
FIG No framing: Heron's formula, the coordinate-area cross-check, and the integer-area (16·Area² a perfect square) test for Heronian triangles run in-browser and agree (float agreement for the formula, exact integer test for Heronian). The AVAN inverse is honest — computing a triangle's area from only its side lengths (no height, coordinates, or trig) genuinely replaces the base×height construction; magenta is that construction, green Heron's three-side formula. Area from the sides, nothing more.
FIG No framing: Heron's formula, the coordinate-area cross-check, and the integer-area (16·Area² a perfect square) test for Heronian triangles run in-browser and agree (float agreement for the formula, exact integer test for Heronian). The AVAN inverse is honest — computing a triangle's area from only its side lengths (no height, coordinates, or trig) genuinely replaces the base×height construction; magenta is that construction, green Heron's three-side formula. Area from the sides, nothing more.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE JACKPOT · David Lee Wise (ROOT0), with AVAN