2026
CLM-LR-DRAFT-COVERING-1= 2π/√27EXACT VALUERegular circle coveringsSource: Thinnest covering of the plane by congruent circles ↗
PROVEN
Limits RegistryPUBLIC RECORDLimits RegistryPUBLIC RECORDLimits RegistryPUBLIC RECORDLimits RegistryResearch Console →Covering the plane with congruent circles and leaving no gaps forces the circles to overlap. The thinnest such covering — the arrangement with the least overlap, achieved by centering circles on a hexagonal lattice — has density 2π/√27 ≈ 1.209, meaning the circles' total area exceeds the plane's area by about 21%.
The question, scope, and sources behind this Registry record.
What is the minimum-density arrangement of congruent circles that covers the entire Euclidean plane with no gaps?
Current frontiers derived from accepted Claims.
The accepted equality closes this optimization result.
Assertions tied to evidence, attribution, and review.
The frontier as it changed over time.
Only accepted Claims matching the current specification contribute to the displayed bounds. Strict inequalities remain open; contradictory Claims require editorial review.