Honeycomb
From Academic Kids

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A honeycomb is a mass of hexagonal wax cells built by honeybees in their nests to contain their larvae and stores of honey and pollen. The term is also used for manmade materials that resemble it in appearance or structure.
Honeycomb is essentially the furniture and pantry in the bees' home. Beekeepers may remove the entire honeycomb to harvest honey. The honey can be extracted from the comb by uncapping and spinning in a centrifugal machine  the honey extractor. Fresh, new comb is sometimes sold and used intact as comb honey, especially if the honey is being spread on bread rather than used in cooking or to sweeten tea. Some believe that this benefits one's physical and mental health.
Broodcomb becomes dark over time, because of the cocoons embedded in the cells and the tracking of many feet. Honeycomb in the "supers" that is not allowed to be used for brood stays light colored.
Honeycomb geometry
The axes of honeycomb cells are always quasihorizontal, and the nonangled rows of honeycomb cells are always horizontally (not vertically) aligned. Thus, each cell has two vertical walls, with "floors" and "ceilings" composed of two angled walls. The cells slope slightly upwards towards the open ends.
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There are two possible explanations for the reason that honeycomb is composed of hexagons, rather than any other shape. One is that the hexagon tiles the plane with minimal perimeter per piece area. Thus a hexagonal structure uses the least material to create a lattice of cells with a given volume. Another, given 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. A third possibility is that honeybees have some kind of understanding of the fundamental geometry which underlies the physical universe.
It is likely that the honeybee constructs the honeycomb based on instinct, and the prevailing theory of biology is that the appearance of such efficient shapes in nature is a result of natural selection.
The closed ends of the honeycomb cells are also an example of geometric efficiency, albeit threedimensional and littlenoticed. The ends are trihedral (i.e., composed of three planes) pyramidal in shape, 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 is approximately 109° 28' 16" (= 180°  arccos(1/3)).)
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A computergenerated model of a honeycomb cell, showing a hexagonal tube terminating in three equal rhombuses that meet at a point on the axis of the cell
The threedimensional geometry of a honeycomb cell.
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.
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A computergenerated model of two opposing honeycomb layers, showing three cells on one layer fitting together with three cells on the opposing layer
Opposing layers of honeycomb cells fit together.
Individual cells do not, of course, show this geometrical perfection: in a regular comb, there are deviations of a few percent from the "perfect" hexagonal shape. When the bees encounter obstacles the shapes are often distorted.
In 1965, L. Fejes Tóth discovered that the trihedral pyramidal shape (which is composed of three rhombuses) used by the honeybee is not the theoretically optimal threedimensional geometry. A cell end composed of two hexagons and two smaller rhombuses would actually be .035% (or approximately 1 part per 2850) more efficient. This difference would be too small to measure on an actual honeycomb.
References
 Thompson, D'Arcy Wentworth (1942). On Growth and Form. Dover Publications. ISBN 0486671356.
 "The Mathematics of the Honeycomb" (June 1985). Science Digest, pp. 7477.de:Bienenwabe