The Heilbronn Problem
math.tejstead.com- Category
- Education
- Rank
- No. 265Apps board
Previous survey · No. 247 ·
- Type
- APP
- Date
About
Visual reference and browser verifier for the Heilbronn problem: arrange points so that the smallest triangle formed by any three is as large as possible. The archive covers square, triangle and convex containers, with coordinates, attribution and proof references where available. Candidate verification is separate from discovering or proving a global optimum.
At a glance
The Heilbronn Problem is a visual reference and calculator for a deceptively simple geometry puzzle: place points so that even the smallest triangle made from any three is as large as possible. It is worth exploring if you enjoy mathematical optimization or need inspectable candidate arrangements. Pictures come with coordinates and a verifier, so you can check a configuration instead of judging it by eye.
Good fit
Students exploring point placement and researchers comparing candidate configurations in a square, triangle or convex region. The archive separates recorded arrangements from their supporting references. Its browser verifier enumerates the triangles in a supplied configuration and reports the smallest area as a fraction, along with whether the points fit the selected container. That makes it useful for understanding why a plausible picture may still contain a tiny triangle.
Consider something else when
You want the browser to discover a new optimal arrangement or prove that no better one exists. Checking a candidate establishes its value and feasibility, not global optimality. Search tools and formal proofs are separate parts of the project. A match with the site’s recorded best value is not a substitute for reading the relevant proof or checking whether that entry is merely best known.
Start here
Open Verifier with the square container selected. Enter 0 0, 1 0 and 0 1, one pair per line, then choose Verify. In the browser trial, this returned a feasible configuration with exact area 1/2. Replacing 1 0 with 2 0 produced an out-of-bounds rejection. Use those small cases to understand the output before pasting a larger candidate.
Cost and requirements
The public tables and verifier opened without an account, payment or API key. The selected verifier source computes the triangle areas in the browser and fetches the site’s stored values for comparison. It accepts decimal coordinate literals and caps inputs at eighty points. The project’s code is MIT licensed; contributed coordinate data carries separate attribution. No local installation is needed to use the public verifier.
Limits to know
Compare the same container and normalization: the site reports values relative to a unit-area region. Decimal inputs are treated as exact, so rounded coordinates define a different candidate. Record matching uses tolerances, and the equilateral frame uses a numerical approximation. The browser trial covered two simple square cases. It did not validate every archived configuration, run the search toolkit or check the formal proofs.
Sources checked 2026-10-31. Source-based assessment.
Why it made the leaderboard
Coordinates and a triangle verifier make candidate arrangements inspectable instead of relying on a persuasive diagram. Useful for exploring the geometry and checking feasibility; matching a recorded value does not prove that no better arrangement exists.
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