How many pairs of points can be exactly one unit apart? Build arrangements, drag vertices, and compare grids with alternative lattices. Explore the deceptively simple problem behind a landmark 2026 AI-generated mathematical discovery.
For distinct points P and Q, a unit-distance pair satisfies (x₁ − x₂)² + (y₁ − y₂)² = 1. The green lines join pairs that satisfy this relation within the small visualization tolerance. A rescaled grid with spacing 1/√5 connects points displaced by (1, 2) grid steps—the familiar chess knight's move. The surprising result: one can beat the ordinary nearest-neighbor grid without any sophisticated mathematics.
Erdős conjectured that the maximum number of unit-distance pairs grows like n1+o(1). A 2026 AI-generated construction, subsequently checked by mathematicians, instead gave at least n1+δ pairs for infinitely many n, with a fixed positive δ. Its proof uses deep algebraic number theory and is not reproduced by this finite-size sandbox. Here we explore intuition and geometry, not the published counterexample.