Honeycomb Conjecture
Regular hexagons give the least-perimeter way to divide the plane into equal-area cells.
The honeycomb conjecture asks why hexagonal cells are optimal for equal-area partitions of the plane. Hales proved that the regular hexagonal tiling minimizes total perimeter.
Formula
For any partition of the plane into equal-area cells, the perimeter-to-area ratio is minimized by regular hexagons.
Thomas Hales proved that the honeycomb (regular hexagonal lattice) achieves the least-perimeter partition into equal areas.
The regular hexagon with side s achieves the exact lower bound — nature exploits this in honeycombs and basalt columns.
Summary
The honeycomb conjecture asks why hexagonal cells are optimal for equal-area partitions of the plane. Hales proved that the regular hexagonal tiling minimizes total perimeter.
