The Honeybee Conjecture

Wild honeybees, usually choose to construct their nests in hollowed-out wood, rock crevices, the underside of roofs, and generally anywhere which offers protection from the elements. After finding a place they feel is suitable for their hive, bees will begin the construction of their nest from the top, downwards.

Worker bees prepare the space by covering its walls with a thin layer of propolis. This substance is made from plant resins collected by the beeswax secreted from glands in their abdomens and their saliva. Bees use propolis to cover surfaces on the inside of their hive at various stages of the build as it has antimicrobial properties which help protect the colony from harmful germs that have the potential to be introduced from the outside once the hive is finished. Next, the bees will chew the wax they secrete until it is soft, bonding bits of it together to eventually form individual cells. These cells will be used to store nectar, pollen, water, honey, eggs, and larvae. As bees age, they produce a lower quality of wax — consequently the bees in charge of building comb are usually between two to three weeks old. Once constructed, the hive will normally have just one entrance and will be occupied by the colony for several years.

The walls of a finished honeycomb can support up to 30 times their own weight and will contain honey in their upper sections, pollen in the rows below this, followed by worker brood cells, drone brood cells, and finally queen cells at the bottom of the structure.

The hexagonal shapes created during the hive-building process have been a cause for debate in the scientific world since at least the 4th century AD, when Greek mathematician Pappus of Alexandria stated that bees had “a certain geometrical forethought”. Some individuals believed that bees made cells this shape because it enabled them to store the greatest quantity of food while using as little wax in their construction as possible. Others argued that bees made round cells, but the surface tension at junctions where cells met pulled the circles into hexagons. In addition to this, there were also people who thought that hexagonal shapes were the automatic result of each bee trying to make the cell it was working on as large as possible, with the edges of each pressing up against the next.

In July 2013, a study headed by Engineer Bhushan Karihaloo at the University of Cardiff, UK, seemed to bring this dispute to an end simply by using a smoker. This piece of equipment is used by apiarists who want to collect honey and other useful substances from hives by moving bees away from important sections of the comb. Blowing smoke into and around the hive from this hand-held container will repel the bees, as well as make them more docile. As part of this experiment, Karihaloo’s team deliberately disturbed a colony of bees, which were in the process of making comb, by smoking them away from certain sections of the structure. By doing this, they observed that the most recently built cells were circular, whereas those that had been constructed just a little beforehand were hexagonal.

This research found that the heat generated by the bees while they worked caused the wax of the comb to melt, ultimately leading to the cell walls becoming flattened and hexagonal in shape. This would suggest that surface tension does indeed play a part in this phenomenon, however, it is still not entirely clear whether the bees do this on purpose, or whether the heat they exude naturally during this process happens to produce this result.

Nevertheless, research has proven that bees are able to measure the depth of each cell by crawling into them and can determine the width of comb walls using various other parts of their bodies. They also seem to know when to change the tilt of cells to prevent honey from dripping out. With this much instinctual supposed mathematical skill, there is a possibility that bees do more to influence the shape of their comb’s cells than meets the eye.

Propolis: Propolis is a 100% natural, antimicrobial substance made from plant resins, which is collected by honeybees along with the nectar and pollen they need for food. It is also known as ‘bee glue’ and sometimes as ‘hive dross’.

Honeycomb is not created by blind physics but its an intriguing product of Physics which have been a source of speculations and theories dating back to 40 B.C, when a Roman scholar, Terentius Varro, proposed an answer, which ever since has been called “The Honeybee Conjecture.”
So what can be the basis of honeycombs being hexagonal! Is it that maybe a honeycomb built of hexagons can hold more honey or it might be that it require less building wax?

The regular hexagons of honeycombs might owe more to the laws of physics than to honeybees’ engineering prowess.

The perfect hexagonal array of bees’ honeycombs, admired for millennia as an example of natural pattern formation, owes more to simple physical forces than to the skill of bees according to studies.

Bees simply make cells that are circular in cross section and are packed together like a layer of bubbles. The wax, softened by the heat of the bees’ bodies, then gets pulled into hexagonal cells by surface tension at the junctions where three walls meet.

A regular geometric array of identical cells with simple polygonal cross sections can take only one of three forms: triangular, square or hexagonal. Of these, hexagons divide up the space using the smallest wall area, and thus, for a honeycomb, the least wax. Hexagonal shape would result automatically because from the pressure of each bee trying to make its cell as large as possible.

It might seem like there is not much left for the bees to do once they’ve made the circular cells. But they do seem to be expert builders. They can, for example, use their head as a plumb-line to measure the vertical, tilt the axis of the cells very slightly up from the horizontal to prevent the honey from flowing out, and measure cell wall thicknesses with extreme precision.

The reason might be that for bees to assemble a honeycomb the way bees actually do it, it’s simpler for each cell to be exactly the same. If the sides are all equal — “perfectly” hexagonal — every cell fits tight with every other cell. Everybody can pitch in. That way, a honeycomb is basically an easy jigsaw puzzle. All the parts fit.
But that just clears the question regarding that all honeycombs are equal-sized, but why is there a preference for hexagons? Is there something special about a six-sided shape?
According to the Solid State Chemistry that we studied regarding lattices, there are some inherently bad shapes like a honeycomb built from spheres would have little spaces between each unit creating gaps that would need extra wax for patching .
Pentagons, octagons also produce gaps. So what then?
As we can observe, that there are only three geometrical figures with equal sides that can fit together on a flat surface without leaving gaps: equilateral triangles, squares and hexagons.
So which one to choose? The triangle? The square? Or the hexagon? Which one is best? 
Here’s when mathematics came into play through the honeycomb conjecture — which was not solved till the turn of this millennium, almost 2000 years after the conjecture came into existence which proposed that a structure built from hexagons is probably a bit more compact than a structure built from squares or triangles. A hexagonal honeycomb, would probably have the smallest total perimeter. Well, the conjecture was proved and now we have a mathematical formula for the proof of this conjecture

The equality is attained for the regular hexagonal tile which is the shortest possible value.

Hence, in other words, “Any partition of the plane into regions of equal area has perimeter at least that of the regular hexagonal grid”.

Compact matters since the more compact the structure, the less wax we need to construct the honeycomb.

We don’t really know it for sure, but certain studies show that for a bee to produce an ounce of wax it must consume about eight ounce of honey and that’s a lot!

So, rather than being just a special case of surface tension, it is far beyond that.

As Mr. Charles Darwin himself once wrote, the honeycomb is a masterpiece of engineering, it is “ absolutely perfect in economizing labor and wax”.

And rightly said, in this modern era, honeycomb still is of cardinal importance to the Aeronautical Industry with wings and fuselages incorporating the honeycomb structure to lower density and making aircraft lighter.

Advantages of the hexagonal shape:-

  • The geometry of this shape uses the least amount of material to hold the most weight
  • It takes the bees quite a bit of work to make the honeycomb. The wax comes from glands on the bees’ bellies, or abdomens. Honeybees have to make and eat about two tablespoons of honey to make one ounce of wax. Then they can add this wax to the comb as they build. A bee colony can produce 100 pounds of honey. In some places they can even produce 300 to 500 lbs. The structure is important to hold all this weight and protect the honey, especially during winter.
  • Hexagons and honeycomb shapes are also useful for building things humans use, too, like bridges, airplanes, and cars. It gives materials extra strength.
  • After all, materials made with hexagon shapes can also handle a lot of force, even if they are made out of a lighter material.

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.