erdős — unit distance

Interactive explorer for unit-distance graphs on two rings.

Gauss — classic Erdős construction (1946). Points on the Gaussian integer lattice; edges connect pairs at unit distance.

CM — construction from OpenAI's 2025 result that disproves the Erdős unit distance conjecture. Paper · Article

Adjust scale, glow, and rotation. Export as PNG (phone · desktop · custom).

What you're seeing

In 1946, Paul Erdős asked a deceptively simple question: place n points on a plane — how many pairs of them can be at distance exactly 1 from each other? Every line you see in this app is one of those pairs: an edge drawn between two points exactly one unit apart.

Grid (Gauss) shows Erdős's original lower bound. Points sit on the integer lattice, and each point connects to its four immediate neighbors. As you raise n, watch the counter in the corner: points grow as (2n+1)², while edges stay close to twice the point count — each interior point touches only four unit neighbors, and every edge is shared by two points. Erdős conjectured that no arrangement could do fundamentally better than a lattice like this.

CM shows why that conjecture is now false. In 2025, an OpenAI model found a construction based on a cyclotomic ring (complex numbers built from roots of unity) where each point participates in far more unit distances than any lattice allows. Switch to it and compare the stats: roughly the same number of points, but each one is far more connected — the visual density you see is the counterexample.

Things to try: rotate the Grid 45° to see the lattice diagonally; crank the scale on CM to see its layered, almost-crystalline internal structure; lower the glow to count edges at a single node by eye.

Run

npm install
npm run dev

Then open http://localhost:5173.

Run the unit tests with npm test.