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Honey bees consume about 8.4 lb (3.8 kg) of honey to secrete 1 lb (450 g) of wax, [1] and so beekeepers may return the wax to the hive after harvesting the honey to improve honey outputs. The structure of the comb may be left basically intact when honey is extracted from it by uncapping and spinning in a centrifugal honey extractor .
A regular hexagonal grid This honeycomb forms a circle packing, with circles centered on each hexagon. The honeycomb conjecture states that a regular hexagonal grid or honeycomb has the least total perimeter of any subdivision of the plane into regions of equal area. The conjecture was proven in 1999 by mathematician Thomas C. Hales. [1]
Hexagonal tiling is the densest way to arrange circles in two dimensions. The honeycomb conjecture states that hexagonal tiling is the best way to divide a surface into regions of equal area with the least total perimeter.
Unlike the worker bees, drones do not sting. Honey bee larvae hatch from eggs in three to four days. They are then fed by worker bees and develop through several stages in hexagonal cells made of beeswax. Cells are capped by worker bees when the larva pupates. Queens and drones are larger than workers, so require larger cells to develop.
Similarly, in a proper honeycomb, there must be no edges or vertices lying part way along the face of a neighbouring cell. Interpreting each brick face as a hexagon having two interior angles of 180 degrees allows the pattern to be considered as a proper tiling. However, not all geometers accept such hexagons.
The rhombic dodecahedron is a space-filling polyhedron, meaning it can be applied to tessellate three-dimensional space: it can be stacked to fill a space, much like hexagons fill a plane. It is a parallelohedron because it can be space-filling a honeycomb in which all of its copies meet face-to-face. [7]
The cells in the paper nests of social wasps, and the wax cells in honeycomb built by honey bees are well-known examples. Among animals, bony fish, reptiles or the pangolin , or fruits like the salak are protected by overlapping scales or osteoderms , these form more-or-less exactly repeating units, though often the scales in fact vary ...
However, a regular hexagon can be dissected into six equilateral triangles, each of which can be dissected into a regular hexagon and three more equilateral triangles. This is the basis for an infinite tiling of the hexagon with hexagons. The hexagon is therefore an irrep-∞ or irrep-infinity irreptile.