Erdős Unit Distance Explorer
Published:
Overview
Interactive explorer for unit-distance graphs on two rings: Erdős’s classic 1946 lattice construction, and the construction from OpenAI’s 2025 result that disproved the Erdős unit distance conjecture.
In 1946, Paul Erdős asked how many pairs of points on a plane can be at distance exactly 1 from each other. Every edge drawn in the app connects two points exactly one unit apart — the Grid (Gauss) mode shows Erdős’s original lower bound on the integer lattice; the CM mode shows the cyclotomic-ring construction that proves his conjecture false, with each point participating in far more unit distances than any lattice allows.
Key Features
- Two interactive constructions: Grid (Gauss) and CM (cyclotomic)
- Adjustable scale, glow, and rotation
- Live point/edge counters
- PNG export (phone, desktop, custom sizes)
Impact
Makes an abstract 2025 discrete-geometry result — the disproof of a nearly 80-year-old conjecture — visually intuitive: the density difference between the two constructions is the counterexample, seen rather than just read.