THE FOLD / CO-OP / THE MERGE / THE NAPOLEON
THE NAPOLEON
equilaterals on any triangle — their centers are equilateral
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Napoleon’s theorem. Take any triangle — scalene, lopsided, however you like. On each of its three sides build an equilateral triangle pointing outward, and mark the center of each. Those three centers always form a perfect equilateral triangle, no matter how irregular the one you started with.
It doesn’t care about the shape underneath — the outer “Napoleon triangle” comes out equilateral every single time. Build the equilaterals pointing inward instead and you get a second equilateral triangle. And there’s a clean bonus: the area of the outer minus the area of the inner equals the area of the original triangle. (The result is traditionally credited to Napoleon Bonaparte, but that attribution is almost certainly a legend.)
LIT verified live: for a battery of irregular triangles the three outer centers are mutually equidistant (equilateral), so are the inner ones, and area(outer) − area(inner) equals the original area (window.__napoleon). FIG the geometry is exact; the Napoleon name is a traditional attribution, flagged as legend not fact.
It doesn’t care about the shape underneath — the outer “Napoleon triangle” comes out equilateral every single time. Build the equilaterals pointing inward instead and you get a second equilateral triangle. And there’s a clean bonus: the area of the outer minus the area of the inner equals the area of the original triangle. (The result is traditionally credited to Napoleon Bonaparte, but that attribution is almost certainly a legend.)
LIT verified live: for a battery of irregular triangles the three outer centers are mutually equidistant (equilateral), so are the inner ones, and area(outer) − area(inner) equals the original area (window.__napoleon). FIG the geometry is exact; the Napoleon name is a traditional attribution, flagged as legend not fact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this in THE MERGE — the co-op domain where separate branches fold into one clean result. Napoleon is a merge made geometric: three unequal sides each grow a triangle, and their centers resolve into one perfect equilateral. AVAN (AI) built the instrument: the construction, the live equal-sides check, the inner/outer area identity.
The weave: David names the seat (three into one clean merge); I make the equilaterals grow on any triangle and prove the centers land equilateral — the equal bars in 1D, the live construction in 2D, the inner-twin identity in 3D. The sphere is the seam. Credit: first published by W. Rutherford (1825); the Napoleon attribution is traditional and unverified.
The weave: David names the seat (three into one clean merge); I make the equilaterals grow on any triangle and prove the centers land equilateral — the equal bars in 1D, the live construction in 2D, the inner-twin identity in 3D. The sphere is the seam. Credit: first published by W. Rutherford (1825); the Napoleon attribution is traditional and unverified.
3 ONE DIMENSION
The three side lengths of the Napoleon triangle as bars. However lopsided the original, these three come out equal — the flat signature of an equilateral, read off in one dimension.
4 TWO DIMENSIONS · INTERACTIVE
A triangle with equilaterals grown outward on each side and their centers joined. Morph the triangle and watch the three center-to-center distances stay locked equal — the Napoleon triangle stays equilateral no matter what you do to the original.
5 THREE DIMENSIONS + AVAN’S INVERSE
The outer Napoleon triangle turning above the original — green, equilateral, built from the equilaterals that point outward.
AVAN’s addition (the inverse-companion): the magenta triangle is the inner Napoleon — the same construction with the equilaterals pointing inward. It is the exact inverse move (flip the build direction), and it too comes out equilateral, concentric with the outer one. The inverse isn’t a decoration: area(outer) − area(inner) = area of the original triangle. So the forward build and its inverted twin don’t just both succeed — their difference reconstructs the very triangle you began with. Flip the direction and you get a second perfect equilateral; subtract the two and the messy original falls back out. Green is outward; magenta is inward; the gap between them is exactly what you started with.
LIT Genuine Napoleon's theorem (first published by W. Rutherford, 1825). Verified live: for a battery of irregular triangles the three outer centers are mutually equidistant (equilateral), the inner centers likewise, and area(outer) - area(inner) equals the original triangle's area exactly (window.__napoleon.outerEquilateral && innerEquilateral && areaIdentity). The geometry is exact. HONEST CAVEAT: the 'Napoleon' attribution to Bonaparte is a traditional legend, not established fact — flagged as such, credited to Rutherford.
FIG No false framing: the equilateral property (outer and inner), and the outer-minus-inner-area identity are real and checked over irregular triangles two ways. The only non-fact — the Napoleon name — is carried explicitly as a traditional/unverified attribution, with the documented first publication (Rutherford 1825) credited instead.
FIG No false framing: the equilateral property (outer and inner), and the outer-minus-inner-area identity are real and checked over irregular triangles two ways. The only non-fact — the Napoleon name — is carried explicitly as a traditional/unverified attribution, with the documented first publication (Rutherford 1825) credited instead.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MERGE · David Lee Wise (ROOT0), with AVAN