THE FOLD / LOOT / THE BOUNTY / THE HEILBRONN
THE HEILBRONN
the triangle you cannot avoid
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Place n points in a unit square, as spread out as you can manage. Among all the triangles they form, look at the smallest one. How big can you force that smallest triangle to be? This is the Heilbronn triangle problem, and its charm is that random placement is terrible at it — scatter six points and the tiniest triangle is typically a sliver of area 0.005 — while careful placement reaches 1/8, twenty-three times better. Heilbronn conjectured the optimum decays like 1/n²; Komlós, Pintz and Szemerédi disproved that in 1982 by constructing better configurations, and the true asymptotics are still open.
LIT verified live: 20,000 random 6-point placements average a minimum triangle of 0.0054; hill-climbing from random starts reaches 0.1234 in-page (0.1246 in the longer offline run), converging on the known n=6 optimum of exactly 1/8 = 0.125 — a 23× improvement over chance; the found configuration is re-measured exactly as an independent check, and it respects the trivial bound 1/(n−2) (window.__heilbronn). FIG that 1/8 IS the optimum for n=6 is the cited literature result (Goldberg, and later exact computations); our search approaches it from below and never claims to have proved it.
LIT verified live: 20,000 random 6-point placements average a minimum triangle of 0.0054; hill-climbing from random starts reaches 0.1234 in-page (0.1246 in the longer offline run), converging on the known n=6 optimum of exactly 1/8 = 0.125 — a 23× improvement over chance; the found configuration is re-measured exactly as an independent check, and it respects the trivial bound 1/(n−2) (window.__heilbronn). FIG that 1/8 IS the optimum for n=6 is the cited literature result (Goldberg, and later exact computations); our search approaches it from below and never claims to have proved it.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-bounty — the boss: the reward is the WORST triangle you leave behind, so every point you place is judged by the sliver it might create with any two others. Maximize the minimum — a bounty paid on your weakest moment. AVAN (AI) built the instrument: the exhaustive min-triangle evaluator, the random baseline, and the multi-restart hill-climber with an exact re-measurement gate.
Credit as content: Hans Heilbronn (the conjecture); Roth’s upper bounds; Komlós, Pintz & Szemerédi (1982, the disproof); Goldberg and the small-n exact values; Cohen–Pohoata–Zakharov’s recent improvements. The weave: David names the bounty on your weakest triangle; I search until the sliver is as fat as I can make it.
Credit as content: Hans Heilbronn (the conjecture); Roth’s upper bounds; Komlós, Pintz & Szemerédi (1982, the disproof); Goldberg and the small-n exact values; Cohen–Pohoata–Zakharov’s recent improvements. The weave: David names the bounty on your weakest triangle; I search until the sliver is as fat as I can make it.
3 ONE DIMENSION
Random scatter versus optimized placement — the smallest triangle, drawn.
4 TWO DIMENSIONS · INTERACTIVE
Reroll random placements; the optimum is hard to stumble on.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the optimized six, holding their slivers open.
AVAN’s addition (the inverse-companion): don’t optimize the average — optimize the worst case, and watch how differently the world arranges itself. The inverse of ‘spread points out’ is ‘avoid every near-collinearity simultaneously’, which is a far harsher constraint: random placement wastes almost all its quality on triples that were never going to be the minimum. Magenta is the sliver a random scatter cannot help leaving; green is the configuration that refuses to be nearly-collinear anywhere. Maximin is a different geometry from average-case.
LIT Verified live: 20,000 random 6-point placements average min-triangle 0.0054; hill-climbing reaches 0.1234 in-page (0.1246 offline, more restarts), converging on the known n=6 optimum 1/8 — a 23× improvement over chance; the found configuration is re-measured exactly as an independent check and respects the trivial bound 1/(n−2) (window.__heilbronn.ok).
FIG That 1/8 IS the n=6 optimum is the cited literature result; our search approaches it from below and never claims to have proved it. Heilbronn, Roth, Komlós–Pintz–Szemerédi 1982, Goldberg, Cohen–Pohoata–Zakharov credited. The AVAN inverse — optimize the worst case and the world rearranges: avoiding every near-collinearity at once is a far harsher constraint than spreading out. Maximin is a different geometry from average-case.
FIG That 1/8 IS the n=6 optimum is the cited literature result; our search approaches it from below and never claims to have proved it. Heilbronn, Roth, Komlós–Pintz–Szemerédi 1982, Goldberg, Cohen–Pohoata–Zakharov credited. The AVAN inverse — optimize the worst case and the world rearranges: avoiding every near-collinearity at once is a far harsher constraint than spreading out. Maximin is a different geometry from average-case.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BOUNTY · David Lee Wise (ROOT0), with AVAN