The economics of a honeycomb

"Bees. . . by virtue of a certain geometrical forethought . . . know that the hexagon is greater than the square and the triangle and will hold more honey for the same expenditure of material in constructing each."                                                     ~Pappus of Alexandria (290-350 AD)

I came across this quote recently, and to be honest, it kind of shocked me for two reasons.  

First...why had I never heard this!!  I've seen honeycombs plenty of times before, I buy honey frequently, but I don't remember anyone every relating to me this quasi-mystical observation about bees' almost human-level of reasoning.  

Second, I found Pappus' observation frustratingly difficult to get my head around!  It's not intuitively obvious what he means.  Minutes after reading that quote, I found myself angrily scribbling shapes and figures on scrap paper, trying desperately to understand the concept.

Took me a beat, but...I finally got it.  And it kinda flabbergasted me.  If you want to understand it too, do read on...

Bees are excellent factory managers

The honeybee's penchant for hexagonal patterns is actually a very clever solution of a common problem in manufacturing economics:  how to produce the highest quantity of goods, while spending as little as possible?

Turns out bees can solve that exact problem:

By structuring honeycombs in sheets and sheets of tiled hexagons, the bees are actually maximizing their honey producing real estate, while minimizing the amount of wax material needed to build!

The production of honey is subject to a constraint - wax isn't free.  The bees need time, energy, and resources to create the wax hives are made of.  Its a limited resource.

But given that constraint, the bees have worked out the mathematically optimal method of producing as much honey as possible!

Area and perimeter

Imagine a square that is 2 cm by 2 cm.  

It's area, which represents the total amount of material the square can contain, is 4 cm squared.  

But the perimeter - the distance around the outside of the shape - is 8 cm.  

Meaning that if the bees shaped their honeycombs in cells in squares, they would need 8 cm of wax to cordon off 4 square cms of honey.

Why don't bees built honeycomb cells like this?  Row after row of squares, like a big checker board?

Because its an economically inefficient use of material!  By avoiding squares the bees can store the same amount of honey, but use less wax!

A bit of geometry...

By adding more sides to any polygon, you can increase its area, while decreasing the perimeter.  In essence, you can fence in more space with less fence.

Don't believe me?  Try it out!  

Remember, a square measuring 8 around closed off an area of 4.

But with an eight-sided octagons, that same 4 cm squared of area can be roped off with just 7.3 cms of material.  Same amount of honey, but nearly a full unit less wax needed to contain it!

Keep adding sides, and you'll see the same pattern.  You can hold the area constant, while shrinking the perimeter.  

Take it right up the a circle - the polygon with infinite sides!  A circle with an area of 4 has an circumference of 7.08! 

So adding sides to the honeycomb cell clearly has an advantage here.  The more sides added, the more efficient the hive's construction - its using less resources and enabling the same amount of honey-making space.  

And yet the cells of a honeycomb are just six-sided hexagons.  Why stop at six?  Considering the advantages of more sides, why not a nine-, twelve- or twenty-sided cell?  Why not circles?

At this point, you may be losing some faith in this insect's geometric capacity.  But not so fast!  The bee's design choices are actually highly sensible, once you consider...tessellation!

Tessellation

Of course, the many cells of a honeycomb are not free-standing, individual little containers.  They're jammed into a larger grid.  Every cell is surrounded by other cells.  The cells are a bunch of containers that have to fit inside of another container - the hive.

Tessellation is a fancy mathy word for tiling.  Its the repetition of a shape, or group of shapes, over and over in a 2D pattern.

Tessellation examples

Notice how the tessellation in the top right leaves little gaps between the cells? 

The other patterns fit together perfectly, however.  The tessellation leaves no empty spaces.

And did you see the pattern in the top left?  The red, blue and yellow hexagons?  Yeah, hexagons...the bee's preferred shape?  No gaps between the cells, right?

This is the key to the bee's design genius.  Its the thing that impressed Pappus so.

It turns out that a hexagon is the most-sided polygon that can be laid in perfect tessellation.

As soon as you go past six sides, you start creating gaps, or irregular courses in your tessellations.  You can see that for yourself by playing with this Tessellation Creator.

If the bees built cells with more than six sides, they would be wasting space in their hives.  Yes, each honeycomb cell would hold the same amount of honey, while using less material...but it would also waste space inside the hive by leaving gaps between the cells.

The hexagonal system is the best possible solution, given the fact that the bees real-estate is limited to the inside of the hive!

Optimization given constraint

The hexagon is the perfect shape, given the circumstances.  Any fewer sides than 5 (so a square, rectangle or triangle), and the bee wastes wax.  Any more than 5, and they waste hive space.

The honeybee's have solved two tricky math problem at once!  In economics jargon, we'd say they are "maximizing a production function given a constraint."  Mathematicians would say the have solved a "discrete optimization" problem.

But think about this...an animal with a TINY brain, no, tools, no ability to plan or reason has found the perfect balance between production cost and production benefit.  This is something many human businessmen and economists struggle to do!!

And its not just a subjectively awesome solution.  Its the best solution conceivable.  Its a mathematical, intrinsic, objectively correct solution!  The bees got the right answer...there's no way to improve on their technique!

To us humans, wild animals often seem be leading a haphazard existence.  Turns out that's not always true!  

Who'da thunk it...inside that honeybee hive is a team of production administrators with a perfect understanding of the a priori nature of geometry.  If only human economists were always that good...