The Heilbronn Problem
A new tracker website catalogs record-breaking configurations and exact coordinates for the classic Heilbronn geometry problem.
summary is enough The site is a neat database, but the real value is in the comment thread's overview of the problem's asymptotic bounds.
summary
The Heilbronn problem asks how to position n points in a unit-area region to maximize the area of the smallest triangle formed by any three points. A new site catalogs optimal configurations, recent record-breaking coordinates, and in-browser verifiers for squares, triangles, and convex regions. Recent records are driven by amateur mathematicians and AI assistance. The hacker news thread provides valuable historical context on the gap between upper and lower bounds.
key points
- The site tracks record configurations with exact coordinates and an in-browser rational arithmetic verifier for squares, triangles, and convex regions.
- Recent records feature contributions from amateur mathematicians and AI models like Claude Opus.
- Commenters detail the asymptotic history of the problem, noting recent upper bound improvements by MIT researchers after a 40-year stall.
who should read it
Mathematicians and optimization enthusiasts interested in extremal geometry and recreational mathematics.
discussion warm
consensus
Commenters enthusiastically discuss the mathematics of the Heilbronn problem, with a detailed breakdown of asymptotic bounds and historical breakthroughs.
It's cool to see a piece of mathematics that I am personally very interested in show up on HN!
From a pure mathematics standpoint, I am most interested in asymptotics (what happens as n becomes very large).
Some tidbits which might be interesting:
* Here's an argument saying there must always be a triangle of area less than about 1/n. Divide the unit square into n/3 vertical strips. By the pigeonhole principle one of the strips must contain three points. The strip has area about 1/n, so the triangle also has area at most 1/n.
* In contrast, the best known lower bound is something like …
It provides a clear mathematical primer on the problem's asymptotics and highlights recent progress after a 40-year stall.
Generated by an LLM from the article and thread. Check the sources before relying on it.