Honeycomb theorem

The honeycomb theorem states that the regular hexagonal tiling minimizes total perimeter among partitions of the plane into regions of equal area.

Connect

Honeycomb conjecture: The conjecture that regular hexagons minimize perimeter among equal-area partitions of the plane. The theorem resolves this conjecture by establishing hexagonal optimality.
Thomas C. Hales: An American mathematician whose work includes proofs of the Kepler conjecture and the honeycomb conjecture. Hales proved the honeycomb conjecture in 1999.
Voronoi diagram: A partition of space into regions whose points are closest to designated sites. Voronoi cells show how hexagonal partitions arise from evenly spaced sites.
Honeycomb: A mass of wax cells, usually hexagonal, built by honey bees to store food and rear young. The theorem gives a mathematical account of the perimeter efficiency associated with hexagonal cells.
Kelvin problem: The problem of finding a partition of three-dimensional space into equal-volume cells with minimal average surface area. It is the three-dimensional analogue of the planar honeycomb optimization problem.
Isoperimetric inequality: An inequality stating that a circle encloses the greatest area for a given perimeter. Its area–perimeter bound helps control the shapes of individual cells.
Kepler conjecture: The claim that no arrangement of equal spheres in three-dimensional space has greater density than face-centered cubic packing. Hales's work on this packing problem preceded his proof of the honeycomb conjecture.
Three-dimensional honeycomb conjecture: A family of conjectures asking which partitions of three-dimensional space minimize surface area for equal volumes. It generalizes the planar problem, where the optimal partition is known.
Foam: A material made of gas bubbles separated by thin liquid or solid films. Foam structures form partitions whose boundary area can be studied with honeycomb principles.
Weaire–Phelan structure: A periodic foam structure with two cell shapes that uses less surface area than Kelvin's proposed partition. It demonstrates how the best three-dimensional partition can differ from a simple regular-cell arrangement.