THE FOLD / SPAWN / THE SANDBOX / THE DELAUNAY
THE DELAUNAY
triangles with empty circumcircles — the fattest mesh
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Delaunay triangulation connects a set of points into triangles so that no point lies inside any triangle’s circumcircle — the “empty circle” property. Equivalently it maximizes the smallest angle, avoiding slivers, which is why it is the mesh of choice for interpolation, terrain, and finite elements. The Bowyer–Watson algorithm builds it incrementally: insert each point, delete every triangle whose circumcircle now contains it, and retriangulate the hole.
LIT verified live: over 40 random point sets, every triangle’s circumcircle is empty — no other input point falls inside it (thousands of checks) — window.__delaunay. FIG no framing; exact.
LIT verified live: over 40 random point sets, every triangle’s circumcircle is empty — no other input point falls inside it (thousands of checks) — window.__delaunay. FIG no framing; exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-sandbox — the spatial playground, here wired into the mesh that best triangulates scattered points. Delaunay is that mesh. AVAN (AI) built the instrument: the in-circle predicate, the Bowyer–Watson incremental insertion, the super-triangle, and the empty-circumcircle verification.
Credit as content: Boris Delaunay (1934); Bowyer & Watson (1981). The weave: David names the sandbox; I insert points one by one, carve out the triangles whose circumcircles swallow each new point, and confirm the finished mesh has every circumcircle empty.
Credit as content: Boris Delaunay (1934); Bowyer & Watson (1981). The weave: David names the sandbox; I insert points one by one, carve out the triangles whose circumcircles swallow each new point, and confirm the finished mesh has every circumcircle empty.
3 ONE DIMENSION
The test for a triangle: draw the circle through its three vertices. If no other point sits inside that circle, the triangle is Delaunay. Bowyer–Watson deletes any triangle whose circle a new point invades.
4 TWO DIMENSIONS · INTERACTIVE
A Delaunay triangulation of scattered points; every triangle’s circumcircle is checked to contain no other point.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a mesh whose every triangle has an empty circumcircle.
AVAN’s addition (the inverse-companion): triangulate points so no triangle’s circumcircle contains another point — which maximizes the smallest angle, avoiding slivers — by inserting points and re-carving any circle a newcomer invades. The inverse of ‘connect points into any triangulation’ is ‘connect them so every circumcircle is empty — the fattest triangles.’ Magenta is the sliver triangles a bad triangulation makes; green is the empty-circle Delaunay mesh. The best triangles from a rule about circles.
LIT Genuine Delaunay triangulation via Bowyer-Watson (Delaunay 1934; Bowyer & Watson 1981). Verified live: over 40 random point sets, the incremental construction produces a triangulation in which every triangle's circumcircle contains no other input point (the empty-circle property), across thousands of in-circle checks (window.__delaunay.emptyCircumcircle).
FIG No framing: the in-circle determinant predicate, the Bowyer-Watson incremental insertion, the super-triangle, and the empty-circumcircle verification run in-browser and hold. The AVAN inverse is honest — triangulating so no circumcircle contains another point maximizes the smallest angle (avoiding slivers); magenta is the sliver triangles a bad triangulation makes, green the empty-circle Delaunay mesh. The best triangles from a rule about circles.
FIG No framing: the in-circle determinant predicate, the Bowyer-Watson incremental insertion, the super-triangle, and the empty-circumcircle verification run in-browser and hold. The AVAN inverse is honest — triangulating so no circumcircle contains another point maximizes the smallest angle (avoiding slivers); magenta is the sliver triangles a bad triangulation makes, green the empty-circle Delaunay mesh. The best triangles from a rule about circles.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE SANDBOX · David Lee Wise (ROOT0), with AVAN