Heilbronn's triangle problem
About this app
n points in a square, the smallest triangle glows: how large can it be at most? Grid (area 0), random, Erdős parabola and random + delete compared, an optimiser lifts the smallest triangle, chart n²·Δ and exponent ladder. New: Δ(n) ≥ c·n^(−2+η) with η ≈ 3·10⁻²³⁶, the “almost n⁻²” conjecture is refuted. openai/math family 191 (formalised in Lean).
What it is about
Place n points in a square of area 1. Every three points form a triangle; how large can the smallest one be at best? The best possible value is called Δ(n). Hans Heilbronn asked this around 1950. The Erdős parabola achieves order n^(−2), and in 1982 Komlós, Pintz and Szemerédi gained a factor of log n, disproving Heilbronn’s original conjecture. What remained open was whether one can do better by at most a factor n^ε. An AI-generated preprint in github.com/openai/math (family 191) claims Δ(n) ≥ c·n^(−2+η) with η ≈ 3·10⁻²³⁶, disproving the “almost n^(−2)” conjecture. The catalogue lists a partial statement as formalised in Lean; the preprint is not peer-reviewed.
What the app shows
The smallest triangle in the square always lights up, and its area is measured exactly. Arrangements for comparison: a grid (many collinear points, area 0), pure random, the Erdős parabola x ↦ x² mod p, and random plus deletion. An optimiser pushes the corners of the smallest triangles apart and finds good, but not provably best, positions. A chart shows n²·Δ against n alongside known best values from the literature, and an exponent ladder places the lower and upper bounds.
Try it yourself
Pick the grid and see that three points on one line make the arrangement worthless. Drag points by hand and try to reach the best known value for your n, then start the optimiser. Compare the Erdős parabola with random plus deletion in the chart. No simulation can show the new theorem: with η ≈ 3·10⁻²³⁶ the effect is invisible for any n you can draw, so it appears only as a statement on the exponent ladder.
Frequently asked questions
- What is the Heilbronn triangle problem?
- The Heilbronn triangle problem asks how to place n points in a square of area 1 so that the smallest triangle formed by three of them is as large as possible. The best achievable value is called Δ(n). The main question is how fast Δ(n) goes to zero as n grows.
- What is the Erdős parabola in the Heilbronn problem?
- For a prime p, the Erdős parabola takes the points (x, x² mod p) for x = 0, …, p−1 and scales them into the unit square. No three of these points are collinear, and every triangle has area at least 1/(2p²). This gives an arrangement with Δ of order n^(−2).
- Has the conjecture Δ(n) ≈ n^(−2) been disproved?
- An AI-generated preprint in the catalogue github.com/openai/math (family 191) claims Δ(n) ≥ c·n^(−2+η) with a fixed, tiny power η ≈ 3·10⁻²³⁶, which would disprove the “almost n^(−2)” form. The catalogue lists the statement for an unbounded sequence of n as formalised in Lean. The preprint has not yet been peer-reviewed.
- What can I try out in this app?
- You choose the number of points n and an arrangement, drag points and see the smallest triangle light up. An optimiser tries to enlarge it, and a chart compares n²·Δ with known best values. An exponent ladder shows what is known about Δ(n).
Subject: Open problems, solved by AI
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