May 21, 2013 12:42 PM Subscribe

People have long been interested in the architectural endeavors of animals. The internal structure of bee hives, the hexagonal combs of wax, have been amongst these ponderings, going back to Marcus Terentius Varro's *Rerum Rusticarum Libri Tres*, a volume on
Roman farm management. He wrote, "The geometricians prove that this hexagon inscribed in a circular figure encloses the greatest amount of space," and over the years, mathematicians have studied the hexagonal structures made by bees, and in 1998, Thomas Hales produced a mathematical proof for the classical hexagonal honeycomb conjecture, which "asserts that the most efficient partition of the plane into equal areas is the regular hexagonal tiling."

Pappus of Alexandria also wrote upon the sagacity of bees, in their selection of the hexagon for their honeycombs.

In 1913, Virginia Farmer produced*Roman Farm Management: the Treaties of Cato and Varro, done into English, with notes of modern instances*, and the section "Of bees" contains notes of later studies, namely Maurice Maeterlinck's book, *La vie des abeilles*, or *The Life of the Bee*, which was published in 1901. That volume notes that the French scientist René Antoine Ferchault de Réaumur posed to the mathematician Johann Samuel König the following question (Google books preview):

The Colorado State Beekeepers Association has a PDF with more mathematicians and their studies of hexagonal honeycombs.

Hales' honeycomb conjecture was last revised in 2002, and the paper is available on Arxiv in a more legible format than that found on Hales' website, where you can find more background on the Hexagonal Honeycomb conjecture.

The recent NPR story on this topic also references the article titled "Symmetrical Universe" by Alan Lightman in Orion Magazine. An article excerpt is available on Longreads, or you can read the whole article in the gorgeous Orion Magazine Digital.
posted by filthy light thief (25 comments total)
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Pappus of Alexandria also wrote upon the sagacity of bees, in their selection of the hexagon for their honeycombs.

In 1913, Virginia Farmer produced

Of all possible hexagonal cells with pyramidal base composed of three equal and similar rhombs, to find the one whose construction would need the least material.König's findings were close to the earlier calculations of the French-Italian astronomer and mathematician, Giacomo Filippo Maraldi (or Jacques Philippe Maraldi), which matched the design of honeycombs.

The Colorado State Beekeepers Association has a PDF with more mathematicians and their studies of hexagonal honeycombs.

Hales' honeycomb conjecture was last revised in 2002, and the paper is available on Arxiv in a more legible format than that found on Hales' website, where you can find more background on the Hexagonal Honeycomb conjecture.

The recent NPR story on this topic also references the article titled "Symmetrical Universe" by Alan Lightman in Orion Magazine. An article excerpt is available on Longreads, or you can read the whole article in the gorgeous Orion Magazine Digital.

Probably the best advertisement ever for natural selection.

Out of all the crazy insect nest-builders in the past 150 million years, Mother Nature favors those who do the least work for the most babies.

Their geometry is beautiful because we don't see all the "mistakes".

posted by weapons-grade pandemonium at 1:02 PM on May 21, 2013 [1 favorite]

Out of all the crazy insect nest-builders in the past 150 million years, Mother Nature favors those who do the least work for the most babies.

Their geometry is beautiful because we don't see all the "mistakes".

posted by weapons-grade pandemonium at 1:02 PM on May 21, 2013 [1 favorite]

Sadly, the divine-seeming regularity of most Langstroth-type hives comes not from the natural instinct of the bees, but rather the stupid foundation that we impose on them to trick them into doing what we want them to do instead of what is natural to them. To each their own, of course, but the reason I went to top bar beekeeping and Warre hives was that when they draw their own comb their own way, they solve some of the "intractable" problems of beekeeping on their own, and from a strictly artistic point, when they go off-book, they make some amazingly beautiful structures.

posted by sonascope at 1:04 PM on May 21, 2013 [1 favorite]

posted by sonascope at 1:04 PM on May 21, 2013 [1 favorite]

weapons-grade pandemonium, are you suggesting that we may discover fossilized remains of bee hives that have cells that are triangular, quadrilateral and so on?

posted by No Robots at 1:13 PM on May 21, 2013

posted by No Robots at 1:13 PM on May 21, 2013

No, I think wgp means that we don't see the mistakes because the mistakes don't lead to successful reproduction? maybe?

posted by elizardbits at 1:15 PM on May 21, 2013

posted by elizardbits at 1:15 PM on May 21, 2013

So... we assume the existence of triangular-cell beehives *because* we have no evidence for them?

posted by No Robots at 1:18 PM on May 21, 2013 [1 favorite]

posted by No Robots at 1:18 PM on May 21, 2013 [1 favorite]

filthy light thief: "*asserts that the most efficient partition of the plane into equal areas is the regular hexagonal tiling*"

What does this mean, exactly? How is it more efficient than just squares?

posted by Joakim Ziegler at 1:38 PM on May 21, 2013 [1 favorite]

What does this mean, exactly? How is it more efficient than just squares?

posted by Joakim Ziegler at 1:38 PM on May 21, 2013 [1 favorite]

Thomas Hales' Honeycomb Conjecture (PDF) is a short paper that starts with a brief overview of the history of the conjecture, and is in clear enough language, before getting into the math of the theorms themselves.

posted by filthy light thief at 1:52 PM on May 21, 2013 [2 favorites]

What does this mean, exactly? How is it more efficient than just squares?

Efficiency is measured as follows (according to the paper): You have to fence off a piece of land into plots that each measure 1 square unit. You define

What he proves is that, in the long run, no matter how you shape the plots, the Efficiency is at least as large as^{4}$\sqrt{\mathrm{12}}$. Then he notes that the Efficiency for a hexagonal honeycomb is exactly equal to ^{4}$\sqrt{\mathrm{12}}$, so you can never do than that.

posted by benito.strauss at 1:59 PM on May 21, 2013 [1 favorite]

Efficiency is measured as follows (according to the paper): You have to fence off a piece of land into plots that each measure 1 square unit. You define

Efficiency = the length of fencing used / area enclosed.Plus, you have to keep fencing off larger and larger plots of land until you cover the whole plane.

What he proves is that, in the long run, no matter how you shape the plots, the Efficiency is at least as large as

posted by benito.strauss at 1:59 PM on May 21, 2013 [1 favorite]

You don't exactly have a lot of options when making repeating patterns out of a single equilateral shape: Triangles, squares, and hexagons. Hexagons are the most like circles (ie the shape of most insects when seen from the front), therefore they are the best. QED bitches.

posted by cman at 2:02 PM on May 21, 2013 [3 favorites]

posted by cman at 2:02 PM on May 21, 2013 [3 favorites]

Maybe I mis-scanned the paper, but I don't think the result depends on using regular shapes or a repeating pattern. It's a very strong result.

posted by benito.strauss at 2:08 PM on May 21, 2013 [1 favorite]

posted by benito.strauss at 2:08 PM on May 21, 2013 [1 favorite]

Fascinating post, thanks **filthy light thief**.

As a side note, I was typing a (now redundant) reply to**Joakim** and realised that bubbles should behave the same way. A few minutes of ~~playing~~ rigorous experimentation in my kitchen sink suggests that they do, at least in areas where all the bubbles are pretty much the same size. Awesome.

posted by metaBugs at 2:09 PM on May 21, 2013

As a side note, I was typing a (now redundant) reply to

posted by metaBugs at 2:09 PM on May 21, 2013

Bees do not actually make hexaons on purpose. It just happens because it is the most efficient distribution of materiel when bees are making the comb shape. Much like if you look at bubbles in foam they form regular geometric patterns.

The bees build up the comb from the base up, they make roundish bowls as they deposit materiel to build up the walls, as these bowls are packed very tight, if two bowls share an edge they will become a flattened line, bees also want to use an existing wall to start a new bowl. As they build up these walls they are continually stopping to push their bodies into the hole to make sure it has a large enough volume to eventually hold eggs, pollen or honey.

If you made two bowls out of clay or another soft materiel and pressed them together side to side you'd end up with a straight line instead of two curves. If you continue to wedge more bowls in there or had a bunch of people also building up their own bowls, each person trying to wedge theirs so as to share sides with your existing ones, you would end up with a pretty regular hexagon pattern.

As an aside, I don't think I've ever seen a comb in one of my grandmother's hives that had a perfect hexagon pattern, it's always going a bit screwy from place to place.

Bees are not master mathematicians, but the results are still amazing.

posted by kzin602 at 2:33 PM on May 21, 2013 [5 favorites]

The bees build up the comb from the base up, they make roundish bowls as they deposit materiel to build up the walls, as these bowls are packed very tight, if two bowls share an edge they will become a flattened line, bees also want to use an existing wall to start a new bowl. As they build up these walls they are continually stopping to push their bodies into the hole to make sure it has a large enough volume to eventually hold eggs, pollen or honey.

If you made two bowls out of clay or another soft materiel and pressed them together side to side you'd end up with a straight line instead of two curves. If you continue to wedge more bowls in there or had a bunch of people also building up their own bowls, each person trying to wedge theirs so as to share sides with your existing ones, you would end up with a pretty regular hexagon pattern.

As an aside, I don't think I've ever seen a comb in one of my grandmother's hives that had a perfect hexagon pattern, it's always going a bit screwy from place to place.

Bees are not master mathematicians, but the results are still amazing.

posted by kzin602 at 2:33 PM on May 21, 2013 [5 favorites]

Thank you so much. I was mistakenly imagining a crack team of top bees sweating out the details in one of their many brainstorming sessions.

posted by gilrain at 2:53 PM on May 21, 2013

posted by gilrain at 2:53 PM on May 21, 2013

Of the three ways to tile a plane into a wax-based structure: triangles, squares, and hexagons, shouldn't a hexagonal tessellation naturally be the most efficient, since of the triangle, square and hexagon each enclosed in a unit circle, the hexagon's perimeter most closely approximates the perimeter of an infinitely-sided regular polygon (i.e., the circumference of a circle)?

There must be something I'm missing: Is there a mathematician who can explain why the proof for this conjecture needs to be so complex?

posted by Blazecock Pileon at 3:16 PM on May 21, 2013

Blazecock Pileon: considering all non-uniform combinations is what makes the proof need to be so complex (ie. a mixture of squares and triangles can tile an arbitrary area...).

posted by idiopath at 3:47 PM on May 21, 2013 [1 favorite]

posted by idiopath at 3:47 PM on May 21, 2013 [1 favorite]

I love bees, and I love this post. If I had photoshop skills I'd take a photograph of bees in a hive, change the shape of each comb to, say, an octagon and see how long it took people to notice.

posted by variella at 4:02 PM on May 21, 2013

posted by variella at 4:02 PM on May 21, 2013

a mixture of squares and triangles

The sides don't even have to be straight. You're allowed any (piecewise) smooth curves. (Though quickly scanning more of the paper, it looks like he does straighten all the edges at one stage.)

posted by benito.strauss at 4:43 PM on May 21, 2013 [1 favorite]

The sides don't even have to be straight. You're allowed any (piecewise) smooth curves. (Though quickly scanning more of the paper, it looks like he does straighten all the edges at one stage.)

posted by benito.strauss at 4:43 PM on May 21, 2013 [1 favorite]

Tom Noddy demonstrating with bubbles (from BBC's The Code, ep 2)

posted by blind.wombat at 6:11 PM on May 21, 2013 [3 favorites]

posted by blind.wombat at 6:11 PM on May 21, 2013 [3 favorites]

As these sites show, a wild honeycomb's external shape (as opposed to the internal cells) is not geometric at all, but free-form and organic in shape, often hanging in lobes, rather than the rectilinear hive we're accustomed to in domesticated bees. Getting bees to build in rectangular frames is just convenient for the beekeepers.

(warning: first site's photos load slowly)

posted by bad grammar at 6:59 PM on May 21, 2013

(warning: first site's photos load slowly)

posted by bad grammar at 6:59 PM on May 21, 2013

> Is there a mathematician who can explain why the proof for this conjecture needs to be so complex?

Here's the thing - you have no idea what the shape could be. It doesn't have to be regular, or repeating, it could even be something pathological.

"It stands to reason" that it has to be one of triangles, squares or hexagons - but that isn't a proof.

Consider the Jordan curve theorem which says that a simple closed curve divides the plane into two regions, the "inside" and the "outside" - which share a common boundary, the curve itself.

It seems "obvious" but finalizing a proof took decades and if you study it today in a topology class it will take you at least a week go through,.

posted by lupus_yonderboy at 8:11 PM on May 21, 2013 [1 favorite]

Here's the thing - you have no idea what the shape could be. It doesn't have to be regular, or repeating, it could even be something pathological.

"It stands to reason" that it has to be one of triangles, squares or hexagons - but that isn't a proof.

Consider the Jordan curve theorem which says that a simple closed curve divides the plane into two regions, the "inside" and the "outside" - which share a common boundary, the curve itself.

It seems "obvious" but finalizing a proof took decades and if you study it today in a topology class it will take you at least a week go through,.

posted by lupus_yonderboy at 8:11 PM on May 21, 2013 [1 favorite]

Honey

Only calmness will reassure

the bees to let you rob their hoard.

Any sweat of fear provokes them.

Approach with confidence, and from

the side, not shading their entrance.

And hush smoke gently from the spout

of the pot of rags, for sparks will

anger them. If you go near bees

every day they will know you.

And never jerk or turn so quick

you excite them. If weeds are trimmed

around the hive they have access

and feel free. When they taste your smoke

they fill themselves with honey and

are laden and lazy as you

lift the lid to let in daylight.

No bee full of sweetness wants to

sting. Resist greed. With the top off

you touch the fat gold frames, each cell

a hex perfect as a snowflake,

a sealed relic of sun and time

and roots of many acres fixed

in crystal-tight arrays, in rows

and lattices of sweeter latin

from scattered prose of meadow, woods.

-- Robert Morgan

posted by islander at 10:00 PM on May 21, 2013 [6 favorites]

Only calmness will reassure

the bees to let you rob their hoard.

Any sweat of fear provokes them.

Approach with confidence, and from

the side, not shading their entrance.

And hush smoke gently from the spout

of the pot of rags, for sparks will

anger them. If you go near bees

every day they will know you.

And never jerk or turn so quick

you excite them. If weeds are trimmed

around the hive they have access

and feel free. When they taste your smoke

they fill themselves with honey and

are laden and lazy as you

lift the lid to let in daylight.

No bee full of sweetness wants to

sting. Resist greed. With the top off

you touch the fat gold frames, each cell

a hex perfect as a snowflake,

a sealed relic of sun and time

and roots of many acres fixed

in crystal-tight arrays, in rows

and lattices of sweeter latin

from scattered prose of meadow, woods.

-- Robert Morgan

posted by islander at 10:00 PM on May 21, 2013 [6 favorites]

Why don't you try that and get back to us.

Oh, and fetch the cannon report while you're out, kid.

posted by Herodios at 5:59 AM on May 22, 2013 [1 favorite]

/slight derail

I will say that, for me at least, board games with hexagonal spaces have always been much more appealing than those with squares or irregularly shaped land masses. This one was fun, and of course this one seems to be a favorite amongst my more nerdlier friends.

I do believe that they owe it all to this little guy though.

posted by Blue_Villain at 6:01 AM on May 22, 2013

I will say that, for me at least, board games with hexagonal spaces have always been much more appealing than those with squares or irregularly shaped land masses. This one was fun, and of course this one seems to be a favorite amongst my more nerdlier friends.

I do believe that they owe it all to this little guy though.

posted by Blue_Villain at 6:01 AM on May 22, 2013

B_V, Avalon Hill published hex board games in the early 60s

posted by wilful at 3:53 AM on May 23, 2013

posted by wilful at 3:53 AM on May 23, 2013

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Origin of Species(Archive.org view of the book).posted by filthy light thief at 12:51 PM on May 21, 2013 [2 favorites]