Honeycomb

The axes of honeycomb cells are always nearly horizontal, with the open end higher than the back end. The open end of a cell is typically referred to as the top of the cell, while the opposite end is called the bottom. The cells slope slightly upwards, between 9 and 14°, towards the open ends.

Two possible explanations exist as to why honeycomb is composed of hexagons rather than any other shape. First, the hexagonal tiling creates a partition with equal-sized cells, while minimizing the total perimeter of the cells. Known in geometry as the honeycomb conjecture, this was given by Jan Brożek and mathematically proven much later by Thomas Hales. Thus, a hexagonal structure uses the least material to create a lattice of cells within a given volume. A second reason, given by D'Arcy Wentworth Thompson, is that the shape simply results from the process of individual bees putting cells together: somewhat analogous to the boundary shapes created in a field of soap bubbles. In support of this, he notes that queen cells, which are constructed singly, are irregular and lumpy with no apparent attempt at efficiency.[3]

The closed ends of the honeycomb cells are also an example of geometric efficiency, though three-dimensional.[4] The ends are trihedral (i.e., composed of three planes) sections of rhombic dodecahedra, with the dihedral angles of all adjacent surfaces measuring 120°, the angle that minimizes surface area for a given volume. (The angle formed by the edges at the pyramidal apex, known as the tetrahedral angle, is approximately 109° 28' 16" (= arccos(−1/3))

The shape of the cells is such that two opposing honeycomb layers nest into each other, with each facet of the closed ends being shared by opposing cells.[4]

Individual cells do not show this geometric perfection: in a regular comb, deviations of a few percent from the "perfect" hexagonal shape occur.[4] In transition zones between the larger cells of drone comb and the smaller cells of worker comb, or when the bees encounter obstacles, the shapes are often distorted. Cells are also angled up about 13° from horizontal to prevent honey from dripping out.[5]

In 1965, László Fejes Tóth discovered that the trihedral pyramidal shape (which is composed of three rhombi) used by the honeybee is not the theoretically optimal three-dimensional geometry. A cell end composed of two hexagons and two smaller rhombi would actually be .035% (or about one part per 2850) more efficient. This difference is too minute to measure on an actual honeycomb, and irrelevant to the hive economy in terms of efficient use of wax, considering wild comb varies considerably from any mathematical notion of "ideal" geometry.[6][7]

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  A Western honey bee on a honeycomb
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  The bees begin to build the comb from the top of each section. When a cell is filled with honey, the bees seal it with wax.
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  Natural honeycombs on a building
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  Honeycomb with eggs and larvae
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  Closeup of an abandoned Apis florea nest in Thailand. The hexagonal grid of wax cells on either side of the nest are slightly offset from each other. This increases the strength of the comb and reduces the amount of wax required to produce a robust structure.
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  Honeycomb
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  The lower part of the natural comb of Apis dorsata has a number of unoccupied cells
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  "Artificial honeycomb" foundation plate in which bees have already completed some cells
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  Honeycomb section containing transition from worker to drone (larger) cells – here bees make irregular and five-cornered cells (marked with red dots)
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  Western honeybees and honeycomb
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  Honeycombs made by machine with beeswax and with the whole structure of the hexagonal cell already built
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  Coat of arms of the town of Viļāni, Latvia

• Honeycomb structure
• Wax foundation
• Hive frame
• Jan Dzierzon