The Honeycomb Conjecture

Abstract

This article gives a proof of the classical honeycomb conjecture: any partition of the plane into regions of equal area has perimeter at least that of the regular hexagonal honeycomb tiling.

Influences on the formation and evolution of Physarum polycephalum inspired emergent transport networks

“…When both RA and SA are equal to 45°, the spontaneous branching behaviour does not occur and the closure of lacunae continues until a minimal surface is reached. With periodic boundary conditions (diffusion continues across the boundary of the lattice to the opposite edge) this surface connects all sides of the twodimensional (toroidal) environment which, when tiled, approximates a hexagonal pattern, known to be the minimal connectivity in two-dimensions (Hales 2001) (Fig. 4).…”

Image Segmentation Applied to the Study of Micrographs of Cellular Solids

“…The hexagonal structure is giving a partition of a surface with equal-sized cells, which are minimizing the total perimeter of the cells. This is known as the "honeycomb conjecture" [21][22][23]. Thus, the bees need to use in the hexagonal structure the least material to create the cells within a given volume.…”

A comparison of MAE Based Image Search for Hexagonally and Regularly Structured Pixel Data

“…For example, the honeycomb, known as the best 2D example of hexagonal lattice in nature, has led to the famous honeycomb conjecture which states that a hexagonal grid is the best way to divide a surface into regions of equal area with the least total perimeters [1]. The honeycomb conjecture was proposed by Pappus of Alexandria [2] and proved by mathematician Thomas C. Hales elegantly in 1999 [3].…”