Why are beehives hexagonal shapes?

[Stephanus]

Do anybody know why beehive is hexagonal?
The angle of polygon are determine by this formula: (I use degree instead of radian)
$Angle = (Sides -2) * 180$ where sides are integer.
the inner angle is
$A_{inner} = \frac{Angle}{Sides}$,
So I compile a list of polygon that if put side by side, will fill a circle.
Polygon, inner angle, number of polygon
Triangle, 60^0, 6.
Rectangle, 90⁰, 4
Pentagon, 108⁰, 3.333
Hexagon, 120⁰, 3
Seven-gon, 128.57⁰, 2.8
Octagon, 135⁰, 2.666
The more, the closer the result to 2.
So, only triangle, rectangle, and hexagon can fit full cirle.
So, hexagon is the most stable polygon that can fit full circle. The most sides that can fit a circle. Bees do not know about math, how can they choose hexagon?
How do they create the hive?
Or, it's only random evolution, the species with hexagon survive.

 

[Astronuc]

They seem to do it naturally. It provides a strong structure and good packing fraction.

 

[Stephanus]

They seem to do it naturally. It provides a strong structure and good packing fraction.

Yes, but how?
I once read about the pyramid construction. How can we build a flat, floor balanced, structure so large (at that time) without altimeter? It turned out that ancient egypt built trench along the area, fill it with water. Then, they have balance floor.
So, no technical explanation?

 

[Astronuc]

It seems to fit their size. Somehow Nature has endowed bees with a sense of building the honeycomb that way. It could be just that the bee ancestors were successful, so they survived to pass on whatever gene favors the building of honey combs.

I'm satisfied to simply marvel at how bees do it, and I appreciate the honey.

I've noticed that some wasps also make hexagonal structures.

 

[256bits]

According to this, the bees construct circular cells, and not hexagons.

Engineer Bhushan Karihaloo at the University of Cardiff, UK, and his co-workers say that bees simply make cells that are circular in cross section and are packed together like a layer of bubbles. According to their research, which appears in the Journal of the Royal Society Interface1, 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.

http://www.nature.com/news/how-honeycombs-can-build-themselves-1.13398

So the marvel in bee engineering has a physical explanation, not a biological or evolutionary.

 

[Helios]

There's an economy principle to consider. The bees are going to want to enclose the most volume with the least amount of wax. Hexagons are superior to squares or triangles. Furthermore, there is one unique height for a hexagon prism that would also maximize volume and minimize wax.

 

[zoobyshoe]

According to this, the bees construct circular cells, and not hexagons.http://www.nature.com/news/how-honeycombs-can-build-themselves-1.13398

So the marvel in bee engineering has a physical explanation, not a biological or evolutionary.

Yes, the bees don't make hexagons, they make circles and the eventual hexagonal configuration is the default behavior of the material under the circumstances.

The team interrupted honeybees making a comb by smoking them out of the hive, and found that the most recently built cells have a circular shape, whereas those just a little older have developed into hexagons. The authors say that the worker bees that make the comb knead and heat the wax with their bodies until it reaches about 45 oC — warm enough to flow like a viscous liquid.

They mention later that you can see a form of this dynamic by squeezing a bunch of drinking straws together: they'll naturally shift from circles to hexagons. There's no marvelous geometric calculating on the bee's part.

 

[collinsmark]

It's also a matter of tiling.

The post by @256bits in post #5 doesn't explain the full story, why there is typically one circle surrounded by 6 circles, as opposed to being surrounded by 5 or 8 circles. Although it does explain part of it (see below).

Only 3-sided (triangles), 4-sided (squares) and 6-sided (hexagons) are the two-dimensional geometric cell shapes capable of tiling. So that rules out 5-sided (pentagons), 7-sided (heptagons) and 8-sided (octagons), and all higher sided shapes right there; they simply don't tile.

Of those shapes that do tile, the hexagonal shape is the most efficient here. And that's where the circles in @256bits in post #5 do come into play. Take a handful of pennies (or other coins, all of equal denomination) and surround one coin by other coins (all touching the central coin) with as many coins as you can fit without overlapping. You'll see that a hexagonal pattern naturally forms: the central coin is surrounded by exactly 6 other coins.

 

[Almeisan]

Whatever the engineering rationale or the physical mechanism, it is a product of evoution. If bees make circular structures that then become hexagonal by surface tension, evolution doesn't care. The bee doesn't know it is making a circular structure. Bee behavior and the chemicals in the wax, both are selected upon by evolution.

It is probably way harder to evolve bee behavior that constructs hexagonal cells.